Math 129 \u0026 Math 155E WB Ed1-

$Page 1: Math 129 \u0026 Math 155E WB Ed1-$
WVU Dept. of Mathematics

Math 129 and Math 155 Joint Workbook

Edition 1.0 August 17, 2016

Gary Ganser, PhD Lori Ogden, PhD

WVU Math 129 & Math 155 Joint Workbook

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Table of Contents Purpose of this Workbook ............................................................................................. 1 How to Use this Workbook ............................................................................................ 1 Calculus 1 (Math 155) Content Overview ..................................................................... 3 1 Algebra ........................................................................................................................ 1-1

1.1 Math 129 Algebra Review ......................................................................................... 1-2 1.2 Math 129 Inequalities ................................................................................................. 1-4 1.3 Math 129 Distance, Midpoint, Lines, and Circles ...................................................... 1-6 1.4 Math 155 Homework .................................................................................................. 1-7 1.5 Hard-Copy Items ........................................................................................................ 1-8

2 Function Basics ......................................................................................................... 2-1 2.1 Math 129 Domain ....................................................................................................... 2-2 2.2 Math 129 Transformations ......................................................................................... 2-3 2.3 Hard-Copy Items ........................................................................................................ 2-6

3 Function Graphs ........................................................................................................ 3-1 3.1 Math 129 Quadratic Functions ................................................................................... 3-2 3.2 Math 129 Polynomial Functions ................................................................................ 3-4 3.3 Math 129 Rational Functions ..................................................................................... 3-7 3.4 Hard-Copy Items ........................................................................................................ 3-10

4 Trigonometry ............................................................................................................. 4-1 4.1 Math 129 Angle Measure ........................................................................................... 4-2 4.2 Math 129 Trigonometry of Right Triangles ............................................................... 4-4 4.3 Math 129 Calculating Trigonometric Values ............................................................. 4-6 4.4 Math 129 Trigonometric Identities ............................................................................. 4-9 4.5 Math 129 Trigonometric Graphs ................................................................................ 4-11 4.6 Math 155 Homework .................................................................................................. 4-13 4.7 Hard-Copy Items ........................................................................................................ 4-15

5 Exponents and Logarithms ...................................................................................... 5-1 5.1 Math 129 Exponential and Logarithmic Functions .................................................... 5-2 5.2 Math 129 Exponential and Logarithmic Equations .................................................... 5-4 5.3 Math 129 Logarithm and Exponential Applications .................................................. 5-5 5.4 Math 129 Graphs of Exponential and Logarithmic Functions ................................... 5-7 5.5 Math 155 Group 02 .................................................................................................... 5-9 5.6 Math 155 Homework .................................................................................................. 5-11 5.7 Hard-Copy Items ........................................................................................................ 5-12

6 Functions and Models .............................................................................................. 6-1 6.1 Math 129 Minimizing Response Time ....................................................................... 6-2 6.2 Math 129 Volume and Area ....................................................................................... 6-4 6.3 Math 129 Building Functions ..................................................................................... 6-5 6.4 Math 155 Group 01 .................................................................................................... 6-7 6.5 Math 155 Homework .................................................................................................. 6-9 6.6 Hard-Copy Items ........................................................................................................ 6-10

7 Limits .......................................................................................................................... 7-1 7.1 Math 129 Indeterminate Forms .................................................................................. 7-2 7.2 Math 155 Group 03 .................................................................................................... 7-4 7.3 Math 155 Group 09 .................................................................................................... 7-6 7.4 Math 155 Homework (to be added) ........................................................................... 7-10


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7.5 Hard-Copy Items ........................................................................................................ 7-11 8 Tangent Lines, Differentiation and Graphs ............................................................ 8-1

8.1 Math 129 Rates of Change & Difference Quotient .................................................... 8-2 8.2 Math 155 Group 03 .................................................................................................... 8-4 8.3 Math 155 Group 04 .................................................................................................... 8-5 8.4 Math 155 Group 05 .................................................................................................... 8-7 8.5 Math 155 Group 06 .................................................................................................... 8-8 8.6 Math 155 Group 08 .................................................................................................... 8-10 8.7 Math 155 Group 20 .................................................................................................... 8-13 8.8 Math 155 Group 18 .................................................................................................... 8-15 8.9 Math 155 Homework (to be added) ........................................................................... 8-17 8.10 Hard-Copy Items ........................................................................................................ 8-18

9 Summation and Integration ..................................................................................... 9-1 9.1 Math 129 Summation Notation .................................................................................. 9-2 9.2 Math 155 Group 30 .................................................................................................... 9-4 9.3 Math 155 Homework (to be added) ........................................................................... 9-6 9.4 Hard-Copy Items ........................................................................................................ 9-7


1

Purpose of this Workbook

The purpose of this workbook is to help students see the connection between a pre-calculus course (Math 129 at WVU) and a calculus course (Math 155 at WVU).

The skills students perfect in pre-calculus provide the foundation necessary for student understanding in calculus. It is our hope that this workbook will clarify how pre-calculus and calculus are related by providing examples of problems that utilize the skills in both courses. For example, students will be introduced to radian measure of angles and trigonometric functions in pre-calculus and will calculate derivatives of trigonometric functions and find angular velocity in calculus.

Students who begin using this workbook in Math 129 at WVU will be expected to continue to use it in Math 155. The pre-calculus portion of the workbook will serve as a reminder and refresher of skills necessary for student success in calculus. Students who do not take Math 129 at WVU will begin using this workbook in Math 155. These students will be responsible for understanding the problems and exercises in the pre-calculus portion of this workbook, as there are problems that students will start in Math 129 and finish in Math 155.

How to Use this Workbook

This workbook should be the physical place where your study of pre-calculus and calculus is contained. The Math 129 portion of the workbook will cover key concepts essential to your understanding of calculus. The Math 155 portion of the workbook will build on the concepts you have learned in Math 129. The workbook will contain group problems, homework problems, and a place to insert “Hard-Copy Items” such as quizzes, tests, and hand-written textbook homework problems.

Although the overarching goal of the workbook is the same for both courses, each course will utilize the workbook in a slightly different way. Both courses will assign group problems. Group problems will be graded, returned and placed back in the workbook. As the semester progresses you will be told which worksheets will be needed for the upcoming class and be given weekly assignments from the textbook to complete and place in your workbook.

In Math 129, you will complete group problems in class at least once a week. It is important to place these problems back in the workbook after they have been graded and returned as they will be referred to in Math 155.

Math 155 students who have taken Math 129 will have already done the pre-calculus portion of the workbook, and will now continue into the calculus portion. It should be an opportunity for you to develop a deeper understanding of the same concepts and skills started in Math 129. A student just starting with Math 155 should recognize most of the pre-calculus material in the workbook from their previous study of pre-calculus. Although Math 155 students who did not take Math 129 are not required to complete the pre-calculus portion you will find that it will be beneficial to review the problems and to try to understand as many as you can.

Each workbook chapter contains a section entitled “Hard-Copy Items” where you should file your quizzes, tests and textbook homework problems that can’t be done in WebAssign. An important part of learning is taking notes. After each test and quiz in your workbook you should


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record the problems that were marked incorrect, what was wrong (and why perhaps) and also the correct steps. These notes should also be filed in the “Hard-Copy Items” section.

The workbook ends with the Integral and the Fundamental Theorem of Calculus. You will do a worksheet that guides you through this theorem. Probably one of the last homework assignments will be to form the basic Table of Integrals. This is the place you will start the next course in calculus - Math 156.

At some point during the semester in Math 155, your teacher or recitation leader will review your workbook and give you a grade. This grade is determined based on workbook homework problems, the detail in the “Hard-Copy Items” section and the overall neatness and care given to your workbook.


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Calculus 1 (Math 155) Content Overview

Functions (Review)

Functions

Limits

Properties of functions and their inverses, specifically for the building block functions: Polynomial, Rational, Trigonometric, Exponential and Logarithmic.

Definition of a function. Mathematical notation used for functions.

Determination of the following additional properties of functions: Continuity, Differentiability and the existence of a Definite Integral. Determination of the relationship between the continuity and differentiability of a function at a point.

Formation of new functions, such as composite functions, from a combination of functions. Domains of composite functions. Use of functions to construct mathematical models.

Concept of a limit: Intuitive definition of a limit, Significance of the limit in expressions such as Formal definitions of Continuity, the Derivative and the Definite Integral.

Ways to Calculate Limits of: Building Block Functions - by using Algebraic and/or Trigonometric manipulation on the functions before applying the Limit Laws, Selected functions - using the Squeeze Theorem, Indeterminate Forms - using L’Hospital’s Rule.

Applications of Limits: Irrational numbers, including expressions such as Use in differentiation and integration.


4

Differentiation

Ways to Calculate the derivatives of: Building Block functions using the Definition of the Derivative, Combinations of building block functions using the Product rule, Quotient rule, and Chain rule, Implicit functions using Implicit Differentiation.

Applications of Differentiation: Maximums, Minimums, Concavity, Graphing, Differentials (local approximations), L’Hospital’s Rule, Related Rates. Finding by first finding and then antidifferentiating .

If follows the natural growth law

then

Concept of the derivative: Slope of Tangent Line, Rate of change of with respect to ,


5

Integration

Concepts of the Integration:

Sum of small changes equals net change,

Ways to Calculate the Definite Integral:

(using standard area formulas),

(or approximate with

large enough),

.

Ways to avoid Calculation of the Definite Integral: Use of the Integral Tables

Applications of Integration: Area, Change in position and Integration of rate of change of gives net

change in : ,

because adding up small changes, in

gives the net change in


1-1

1 1 Algebra

Following are the objectives of this chapter:

a. Solve linear, polynomial, rational, absolute value, radical equations and linear inequalities.

b. Calculate and interpret the slope of a line c. Write equations of and graph: lines, parallel lines, and perpendicular lines.


1-2

1 1.1 Math 129 Algebra Review Name: __________________________

1. In the lab, Yoko has two solutions that contain alcohol and is mixing them with each other. She uses 4 times as much Solution A as Solution B. Solution A is 18% alcohol and Solution B is 14% alcohol. How many milliliters of Solution B does she use, if the resulting mixture has 344 milliliters of pure alcohol?

2. Two trains leave stations 280 miles apart at the same time and travel toward each other. One train travels at 95 miles per hour while the other travels at 105 miles per hour. How long will it take for the two trains to meet?

3. Elsa works as a tutor for $15 an hour and as a waitress for $12 an hour. This month, she worked a combined total of 97 hours at her two jobs. Let t be the number of hours Elsa worked as a tutor this month. Write an expression for the combined total dollar amount she earned this month.

Solve the following for x :

4. 3x − x

4−5= x

6− 2x + 6


1-3

1 1.1 Math 129 Algebra Review (Continued)

5.

2x + 2

− 5x +1

= −5x2 + 3x + 2

6. −3 w

2+ 4 −1= −4

7. Solve for T2 .

P1V1

T1

=P2V2

T2

Solve the following inequalities. Graph your solution set and write your solution set as an inequality and in interval notation.

8. 4z − 9 + 6 > 4

9.

4x +53

− 12≤ 7

6

10. 4 5− 2h − 9 >11


1-4

1 1.2 Math 129 Inequalities Name: __________________________

Mixed Practice: Solve each inequality. Graph your solution set and write your answer in interval notation.

1.

35

x + 12> 3

10 and − 4x >1

2. 3 p + 4 +5< 8

3. 2− 7u + 7 ≤ 4

4. 2 n+ 3 > 7

5. 4z − 9 + 6 > 4


1-5

1 1.2 Math 129 Inequalities (Continued)

Applications: Show work and make sure you answer the question in a complete sentence.

6. As long as the outside temperature is over 45!F and less than 85!F , (45< F < 85) , the city does not issue heating or cooling subsidies for low-income families. What is the

corresponding range of Celsius temperatures C? Recall: F = 9

5C + 32 .

7. According to the official rules for golf, baseball, pool, and bowling, a) golf balls must be within 0.03mm of d=42.7mm, b) baseballs must be within 1.01mm of d=73.78mm, c) billiard balls must be within 0.127mm of d=57.150mm, and d) bowling balls must be within 12.05 mm of d=2171.05mm. Write each statement using an absolute value inequality, then e)

determine which sport gives the least tolerance t, where

t = width of intervalaverage value

⎛⎝⎜

⎞⎠⎟

for the

diameter of the ball.


1-6

1 1.3 Math 129 Distance, Midpoint, Lines, and Circles Name: __________________________

1. Find an equation for the line in Slope-intercept form with the given properties. Sketch the graph.

a) Slope= 2/3; containing the point (2, -1)

b) x − intercept = 3 and y − intercept = − 3

c) Slope undefined; containing the point (1, -5)

d) Perpendicular to the line x + 3y = −8 ; containing the point (0,3)

e) Parallel to y = 3 ; containing the point (-3, 5)

2. Find the intercepts and sketch the graph: 3x −8y = 24

3. Find the center and radius of the following circle and sketch a graph:

2x2 + 2y2 + 4x − 6y + 2 = 0

4. Given the following points: (-2, 5) and (1, -2) a) Find the distance between the two points

b) Find the midpoint


1-7

1 1.4 Math 155 Homework

1. Solve for ′y : sec2 x

y⎛⎝⎜

⎞⎠⎟

y − x ′yy2

⎛⎝⎜

⎞⎠⎟= 1+ ′y

2. Find where

4x(x2 −1)− 2x2 2x(x2 −1)2 is positive, negative or zero.

3. Find where f (x) = (2− x)1/3 − 1

3x(2− x)−2/3 is positive, negative or zero.


1-8

1 1.5 Hard-Copy Items You should file your quizzes, tests and textbook homework problems in this section. After each test and quiz you should record the problems that were marked incorrect, what was wrong (and why perhaps) and also the correct steps. Any notes you have taken should also be filed in this section.


Function Basics 2-1

2

2 Function Basics

Following are the objectives of this chapter:

a. Use and Understand function notation.

b. Manipulate the variables and notation used in the definition of a function to produce new functions.

c. Analyze the graphs of functions specifically finding domain, range, intercepts.

d. Graph the basic building block functions and graph shifts/ translations of new functions based on knowing the graph of the upshifted/untranslated building block function.

e. Derive inverse functions using an algebraic method, and graph a function and its inverse.

f. Combine functions to form new functions such as composite functions, sum, difference, product, and quotient of two functions.


Function Basics 2-2

2

2.1 Math 129 Domain Name: __________________________

1. Find the domains of the following functions.

a) f x( ) = x − 4

x + 3( ) x + 6( )

b) f x( ) = x − 4

x + 3( ) x − 6( )

c) f x( ) = x + 4

x + 3( ) x − 6( )

d) f x( ) = x + 4

x + 4 x + 3( ) x − 6( )


Function Basics 2-3

2

2.2 Math 129 Transformations Name: __________________________

Building Block Functions:

1. Sketch a graph of each of the following functions and state the domain and range of each.

a) f (x) = x2 b) f (x) = x3

domain: domain:range: range:

c) f (x) = x d) f (x) = xdomain: domain:range: range:

e) f (x) = 1x

f) f (x) = xdomain: domain:range: range:

Transformations

2. Do the following:

a) Graph and label the following functions on the same set of axes. Label the axes for scale.

f x( ) = x2 ( ) 32 += xxg ( ) 32 −= xxh

b) Describe the effect that k has on the graph of ( ) kxfy += compared to the graph of

y = f x( ) .


Function Basics 2-4

2

2.2 Math 129 Transformations (Continued)


a) Graph and label the following functions on the same set of axes. Label the axes for scale. f x( ) = x3 g x( ) = x + 2( )3

h x( ) = x − 2( )3

b) Describe the effect that h has on the graph of y = f x − h( ) compared to the graph of

y = f x( ) .

4. Do the following: a) Graph and label the following functions on the same set of axes. Label the axes for scale.

f x( ) = x2 g x( ) = 2 x2( ) h x( ) = 0.5 x2( )

b) In detail, describe the effect that a has on the graph of y = af x( ) compared to the graph

of y = f x( ) .


Function Basics 2-5

2

2.2 Math 129 Transformations (Continued)


a) Graph and label the following functions on the same set of axes. Label the axes for scale.

f x( ) = x g x( ) = − x h x( ) = −x

b) In detail, describe how ( )xfy −= differs from the graph of y = f x( ) .

c) In detail, describe how ( )xfy −= differs from the graph of y = f x( ) .

Combined Transformations – Graph/Description from Function

6. Graph and label the following functions on the same set of axes. Label the axes for scale. f x( ) = x g x( ) = −2 x +1 − 4 (Plot and label at least three points)

Combined Transformations – Graph/Function From Description

7. Find the equation for the function, g x( ) , that would match the description below.

Parent Function: f x( ) = x Vertical Shift: Down 2 Horizontal Shift: Right 3 Vertical Stretch/Compression: Compress by a factor of 2 Vertical Reflection: Yes


Function Basics 2-6

2

2.3 Hard-Copy Items You should file your quizzes, tests and textbook homework problems in this section. After each test and quiz you should record the problems that were marked incorrect, what was wrong (and why perhaps) and also the correct steps. Any notes you have taken should also be filed in this section.


Function Graphs 3-1

3

3 Function Graphs

Following are the Math 129 objectives of this chapter:

a. Find and use intercepts, vertex and axis of symmetry to graph quadratic functions,

b. Find and use the degree, the real zeros and their multiplicity, intercepts, and end-behavior to graph polynomial functions.

c. Find and use vertical, horizontal/oblique asymptotes, holes, and intercepts to graph rational functions.


a. Use and understand function notation.

b. Manipulate the variables and notation used in the definition of a function to produce new functions.

c. Graph the basic building block functions and graph shifts/translations of new functions based on knowing the graph of the upshifted/untranslated building block function.


Function Graphs 3-2

3

3.1 Math 129 Quadratic Functions Name: __________________________

Answer all parts of each question completely. Show all work.

1. Depth of a dive: As it leaves its support harness, a mini sub takes a deep dive toward an underwater exploration site. The dive path is modeled by the function d(x) = x2 −12x , where d x( ) represents the depth of the mini sub in hundreds of feet at a distance of x mi from the surface ship.

a) Sketch a graph for d x( ) by calculating (by hand) the vertex and intercepts ( x and y ). Label the x and y axes, all intercepts, and the vertex.

b) What is the practical domain and range of d x( ) ? Express each in interval notation. The practical domain and range deal with numbers that are realistic in a problem situation.

Domain: Range:

c) What is the theoretical domain and range of d x( ) ? Express each in interval notation. The theoretical domain and range does not take the context of the problem into account.

Domain: Range:

d) How far from the mother ship did the mini sub reach its deepest point?

e) How far underwater was the submarine at its deepest point?

f) At x = 4 mi, how deep was the mini sub explorer?


Function Graphs 3-3

3

3.1 Math 129 Quadratic Functions (Continued)

2. Building sheep pens: It’s time to drench the sheep again, so Chance and Chelsea-Lou are fencing off a large rectangular area to build some temporary holding pens. To prep the males, females, and lambs, they are separated into three smaller and equal-sized pens partitioned with in the large rectangle. They have 384 ft of fencing to build these pens. See figure below.

a) Write the function A x( ) that models the area of the larger rectangular area in terms of

the width x of the rectangle in feet.

b) What is the maximum area of the larger rectangular area?

c) What are the dimensions of the outer rectangle?

d) What are the dimensions of the smaller holding pins?


Function Graphs 3-4

3

3.2 Math 129 Polynomial Functions Name: __________________________

Graph/Function From Description


a) Write the equation of a polynomial (in factored form) that has the given characteristics.

Touches at -3 Touches at 4 Crosses at 1 As x →∞, y →−∞ As x →−∞, y →∞

b) Graph the polynomial. Label the axes for scale.

Graph/Function From Description


a) Write the equation of a polynomial (in factored form) that has the given characteristics.

Touches at 3 Crosses at -2 Crosses at 5 As x →∞, y →∞ As x →−∞, y →∞


Function Graphs 3-5

3

3.2 Math 129 Polynomial Functions (Continued)

b) Graph the polynomial. Label the axes for scale.

Graph/Description from Function

3. f x( ) = x3 + x2 − 6x

a) Factor the function completely. Find each of the zeros and the corresponding multiplicities.

b) Determine whether the polynomial will cross or touch at each zero.

c) Describe the end behavior of the polynomial.

d) Using the information from a) – c), graph the function. Label the axes for scale.

Graph/Description from Function

4. f x( ) = x3 + 4x2 + 4x +16

a) Factor the function completely. Find each of the zeros and the corresponding

multiplicities.




Function Graphs 3-6

3

3.2 Math 129 Polynomial Functions (Continued)

d) Using the information from a) – c), graph the function. Label the axes for scale. (Notice that there is little information to draw the graph accurately. Calculus has techniques to give more detail.)

Function/Description from Graph

5. f x( ) = − 1

2x3 − 4x2 −8x

a) Using the graph of the given polynomial, find each of the zeros and the corresponding multiplicities.



d) Using the information from a) – c), find the factored form of the polynomial.

(Hint: It would be unadvisable to try to factor this by hand.)


Function Graphs 3-7

3

3.3 Math 129 Rational Functions Name: __________________________

Answer all questions completely. Show your work.

1. Suppose the cost (in thousands of dollars) of manufacturing x thousand of a given item is modeled by the function C(x) = x2 + 4x + 3 . The average cost of each item would then be

expressed by: A(x) = x2 + 4x + 3

x

a) Graph the function A x( ) (by hand…not with a graphing utility).

i. Identify the domain of A x( ) . Write it in interval notation.

ii. Find all x and y intercepts. Write them as ordered pairs. iii. Find all asymptotes (vertical and horizontal/oblique).

iv. Sketch the graph of A x( ) below. Label the axes (with words), asymptotes, and

intercepts and plot and label at least two points on either side of the vertical asymptote.


Function Graphs 3-8

3

3.3 Math 129 Rational Functions (Continued)

b) Find how many thousand items are manufactured when the average cost is $8.

c) Use the graph to estimate how many thousand items should be manufactured to minimize the average cost.

2. The concentration C of a certain medicine in the bloodstream x hours after being injected

into the shoulder is given by the functions: C(x) = 2x2 + x

x3 + 70

a) Go to Desmos.com and graph C(x) . Click on the “wrench” icon in the top right hand portion of your screen and change your viewing window to −10 ≤ x ≤ 26 , step 2 and −0.4 ≤ y ≤ 0.4 , step 0.1. It should match the picture below.


Function Graphs 3-9

3

3.3 Math 129 Rational Functions (Continued)

b) Sketch your graph. Label each axis (hours & concentration).

c) Consider your graph to answer the following questions. Note: You can retrieve an ordered pair from the graph by clicking on the graph and hovering around that point.

i. Approximately how many hours after injection did the maximum concentration

occur? _____________

ii. What was the maximum concentration? ___________________

iii. What happens to the concentration C as the number of hours becomes infinitely large?

iv. What role does the x-axis play for this function? What does this mean?


Function Graphs 3-10

3



Trigonometry 4-1

4

4 Trigonometry


a. Define the six trigonometric functions in terms of a point on the unit circle and the angle in radian measure off the x –axis and in terms of a right triangle.

b. Sketch and analyze the graphs of y=sin t, y=cos t, and y=tan t and their reciprocal functions using special values, periodicity and symmetry.

c. Find and graph the inverse trig functions, apply the definition and notation of inverse trig functions to simplify compositions of trig functions.

d. Solve trigonometric equations for roots on specific intervals and over the real numbers.

e. Use inverse trig functions to solve trig equations for the principal root.

f. Solve trig equations using identities.

g. Use trigonometry to solve problems.


a. Summarize the properties and definitions of the basic trigonometric functions including the inverse functions.

b. Explain radian measure, why it is used and its relationship to degrees.

c. Evaluate the basic trigonometric functions and their inverses corresponding to the simple

angles 0,

π4

, π3

, π2

….


Trigonometry 4-2

4

4.1 Math 129 Angle Measure Name: __________________________

On topographical maps, each closed figure represents a fixed elevation (a vertical change) according to a given contour interval. The measured distance on the map from point A to point B indicates the horizontal distance or the horizontal change between point A and a location directly beneath point B, according to a given scale of distances.

In the figure shown below, the contour interval is 1:250 (each figure indicates a change of 250m in elevation) and the scale of distances is 1cm = 625m.

1. Find the change of elevation from A to B

2. Use a proportion to find the horizontal distance between points A and B if the measured distance on the map is 1.6 cm

3. Draw the corresponding right triangle and use a special triangle relationship to find the length of the trail up the mountainside that connects A and B. (Round your answer to the nearest hundredth of a meter)


Trigonometry 4-3

4

4.1 Math 129 Angle Measure (Continued)

4. A water sprinkler is set to shoot a stream of water a distance of 12m and rotate through an angle of 40 degrees. a) What is the area of the lawn it waters? (Round your answer to the nearest hundredth.)

b) For r=12 m, what angle (in degrees and radians) is required to water twice as much area?

c) For θ = 40! , what range for the water stream is required to water twice as much area?

(Round your answer to the nearest hundredth.)

5. The planet Jupiter’s largest moon, Ganymede, rotates around the planet at a distance of about 656,000 miles, in an orbit that is perfectly circular. If the moon completes one rotation about Jupiter in 7.15 days, a) Find the angle θ that the moon moves through in 1 day, in both degrees and radians.

(Round to the nearest hundredth.)

b) Find the angular velocity of the moon in radians per hour. (Round to 4 decimal places)

c) Find the moon’s linear velocity in miles per second as it orbits Jupiter. (Round to the nearest hundredth)


Trigonometry 4-4

4

4.2 Math 129 Trigonometry of Right Triangles Name: __________________________

1. Use the diagram given to derive a formula for the height h of the taller building in terms of the height x of the shorter building and the ratios tan(u) and tan(v) (must show all work to receive credit):

a) h = ____________________________(round to the nearest hundredth).

b) Use the formula that you derived in a) to find h given the shorter building is 75m tall with u = 40 degrees and v = 50 degrees.


Trigonometry 4-5

4

4.2 Math 129 Trigonometry of Right Triangles (Continued)

2. From her elevated observation post 300 feet away, a naturalist spots a troop of baboons high up in a tree. Using the small transit attached to her telescope, she finds the angle of depression to the bottom of this tree is 14 degrees, while the angle of elevation to the top of the tree is 25 degrees. The angle of elevation to the troop of baboons is 21 degrees. Use this information to find the following.

(Draw a picture first…no picture, no points! Show all work!)

a) Height of the observation post:____________

b) Height of the baboons’ tree:____________

c) Height of the baboons above the ground (round to the nearest hundredth):_________


Trigonometry 4-6

4

4.3 Math 129 Calculating Trigonometric Values Name: __________________________

Calculate Trigonometric Values

1. By hand, calculate the following values. Show your work and draw the appropriate triangles.

a) sin 7π

4⎛⎝⎜

⎞⎠⎟

b) cos

2π3

⎛⎝⎜

⎞⎠⎟

c) tan 7π

6⎛⎝⎜

⎞⎠⎟

d) csc 5π

6⎛⎝⎜

⎞⎠⎟

e) sec 4π

3⎛⎝⎜

⎞⎠⎟

f) cot

5π4

⎛⎝⎜

⎞⎠⎟

Inverse Trig of Same Trig


a) sin−1 sin 7π

4⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟


Trigonometry 4-7

4

4.3 Math 129 Calculating Trigonometric Values (Continued)

b) cos−1 cos

2π3

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

c) tan−1 tan 7π

6⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

d) csc−1 csc 5π

6⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

e) sec−1 sec 4π

3⎛⎝⎜

⎞⎠⎟

f) cot−1 cot

5π4

⎛⎝⎜

⎞⎠⎟

Inverse Trig of Different Trig


a) sin−1 cos

5π4

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

b) cos−1 csc

π3

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟


Trigonometry 4-8

4

4.3 Math 129 Calculating Trigonometric Values (Continued)

c) tan−1 sin π

2⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

d) csc−1 sin 7π

6⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

e) sec−1 tan 3π

4⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

f) cot−1 cos

3π2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟


Trigonometry 4-9

4

4.4 Math 129 Trigonometric Identities Name: __________________________

1. Find the exact value using the sum and difference formula: cos75!

2. Find the exact value using the half-angle formula: sin 5π

8

3. Find the exact value: tan π

12

4. Find sin2x, cos2x, tan2x, if sin x = 3

13.


Trigonometry 4-10

4

4.4 Math 129 Trigonometric Identities (Continued)

5. Use the information below to find the exact value for sin(α + β ) and cos(α − β ) .

cosα = − 35

, with α in quadrant II

sinβ = − 45

, with β in quadrant III

6. Suppose that sinθ = 3

10 and π

2<θ < π , find the exact values of

sinπ

2 and tanπ

2.


Trigonometry 4-11

4

4.5 Math 129 Trigonometric Graphs Name: __________________________

Show all work. No credit will be given for a page of answers with no work shown.

1. Consider the following graph:

a) Write the equation of the above graph as a transformation of a cosine function:

b) Write the equation of the above graph as a transformation of a sine function:


Trigonometry 4-12

4

4.5 Math 129 Trigonometric Graphs (Continued)

2. Sketch the graph: f (x) = −3sin 3

4x

⎛⎝⎜

⎞⎠⎟+ 2

a) Label the x-axis with at least SIX tic marks and the y-axis with at least SIX tic marks.

b) Label FIVE ordered pairs on your graph (ordered pairs should be in terms of π ).

c) What is the period of the function? _________________

d) f −2π

3⎛⎝⎜

⎞⎠⎟

= ________

e) f (0) = _________

f) If f (x) = −1, what is x? ___________________________


Trigonometry 4-13

4

4.6 Math 155 Homework

1. Simplify tan(sin−1 x)

2. On average, the lung of a human male human holds 6 liters of air at maximum inhalation and 1 liter at maximum exhalation. Assume that the volume of air in his lung varies sinusoidally with time as he breathes in and out. Find an expression that models the volume, V , of air (in liters) in the lung of a typical man at a time t (in seconds) if he is taking fifteen deep breaths each minute. Include units in the variables of the formula. For example, t = 30sec , not t = 30 , must be used in the formula to determine the volume at 30sect = .

3. A movie theater has a screen that is 28 ft tall. When you sit down, the bottom of the screen is 6 ft above your eye level. The angle formed by drawing a line from the bottom of the screen to your eye and then to the top of the screen is called the “Viewing Angle”.

The problem is to find the distance from the wall that maximizes the viewing angle.


Trigonometry 4-14

4

4.6 Math 155 Homework (Continued)

a) Discuss here why the problem really has a maximum based just on understanding the nature of the problem. Consider drawing special cases that would imply there is a maximum.

b) Introduce all the necessary variables into the problem, show them on the diagram and describe them here.

c) Find the mathematical function that you want to maximize.


Trigonometry 4-15

4



Exponents and Logarithms 5-1

5

5 Exponents and Logarithms


a. Identify and use intercepts, horizontal asymptote, and end-behavior and to graph exponential functions.

b. Graph shifts/ translations of basic exponential functions.

c. Solve exponential equations.

d. Solve applications involving compound interest.

e. Identify and use intercepts, vertical asymptotes, and end-behavior and to graph logarithmic functions.

f. Graph shifts/ translations of basic logarithmic functions.

g. Solve equations involving logarithms.

h. Solve applications involving exponential growth and decay.


a. Summarize the properties and definitions of the basic exponential functions including the inverse functions-the logarithms.

b. Apply these functions to growth and decay problems.



5

5.1 Math 129 Exponential and Logarithmic Functions Name: __________________________

Rewrite the following expressions using properties of logarithms. Show all work.

1. log

x3 y(z +1)4

2. ln x2 x −13

(x +1)3

⎡

⎣⎢⎢

⎤

⎦⎥⎥

The expressions and equations that follow are taken from the area of calculus noted in italics. Simplify, solve, or verify as indicated.

3. Identities: Verify that

12

(ex + e− x )+ 12

(ex − e− x )

12

(ex + e− x )− 12

(ex − e− x )= e2x

4. Inverse hyperbolic functions: Solve for t: xe2t + x = e2t −1



5

5.1 Math 129 Exponential and Logarithmic Functions (Continued)

5. Carbon-14 is a radioactive compound that occurs naturally in all living organisms, with the amount in the organism constantly renewed. After death, no new carbon-14 is acquired and the amount in the organism begins to decay exponentially. If the half-life of carbon-14 is 5730 years, how old is a mummy having only 30% of the normal amount of carbon-14? (Do not approximate until your final answer. Round your final answer to the nearest hundredth.)



5

5.2 Math 129 Exponential and Logarithmic Equations Name: __________________________

Solve the following equations using the properties of exponentials and logarithms.

1. 42x+1 = 46x−5

2. 22x+3 = 8x

3. 2x = 5

4. log3(2x − 7) = 4

5. log2 x( ) + log2 x − 7( ) = 3



5

5.3 Math 129 Logarithm and Exponential Applications Name: __________________________

1. Suppose Lamar places $2,000 in an account that pays 6% interest compounded each year. Assume that no withdrawals are made from the account. Do not do any rounding.

a) Find the amount in the account at the end of 1 year.

b) Find the amount in the account at the end of 2 years.

2. An amount of $43,000 is borrowed for years at 4.25% interest, compounded annually. If the loan is paid in full at the end of that period, how much must be paid back? Use a calculator and round your answer to the nearest dollar.

3. Suppose that $2000 is invested at a rate of 4.6%, compounded semiannually. Assuming that no withdrawals are made, find the total amount after 7years. Do not round any intermediate computations, and round your answer to the nearest cent.

4. A laptop computer is purchased for $5,000. Each year, its value is 75% of its value the year before. After how many years will the laptop computer be worth $1,000 or less? Use a calculator if necessary.

5. The mass of a radioactive substance follows a continuous exponential decay model, with a decay rate parameter of 3% per day. A sample of this radioactive substance has an initial mass of 4.52 kg. Find the mass of the sample after four days. Round your answer to two decimal places.

6. Suppose that the number of bacteria in a certain population increases according to a continuous exponential growth model. A sample of 2,700 bacteria selected from this population reached the size of 3,304 bacteria in five hours. Find the hourly growth rate parameter.

7. After the wolf population was decimated due to overhunting, the rabbit population in the Boluhti Game Reserve began to double every 6 months. If there were an estimated 120 rabbits to begin,

a) find the growth rate r

b) find the number of months required for the population to reach 2,500



5

5.3 Math 129 Logarithm and Exponential Applications (Continued)

8. The radioactive element iodine-131 has a half-life of 8 days and is often used to help diagnose patients with thyroid problems. If a certain thyroid procedure requires 0.5 g and is scheduled to take place in 3 days, what is the minimum amount that must be on hand now (to the nearest hundredth of a gram)?



5

5.4 Math 129 Graphs of Exponential and Logarithmic Functions Name: __________________________

Show all work. No credit will be given for a page of answers with no work shown.

1. Consider the following graph:

a) Write the equation of the above graph:

b) State the domain of the function in interval notation:

c) State the range of the function in interval notation:



5

5.4 Math 129 Graphs of Exponential and Logarithmic Functions (Continued)

2. Sketch the graph of f (x) = log2 x . (Label your graph on the Cartesian Plane)

a) State the domain of g(x) :

b) State the range of g(x) :

c) Find g(−3) by looking at your graph:

d) Find g(−3) algebraically (show all work):



5

5.5 Math 155 Group 02 Name: __________________________

This worksheet is about exponents and the exponential functions ax .

You are required to read page 45 and 46 of the text.

There are three questions in all.

1. Work the following problems:

a) State what b1

n means when n is a positive integer and b is a positive real number.

b) Estimate 21/5 (give estimate to ±.05 show work on attached sheet!). You can use the multiplication function of a calculator.

c) Using the exponent function of a calculator, determine 21/5 = _____________

2. Work the following problems:

a) What does mnb mean when n and m are positive integers and b is a positive real

number?

b) Estimate .82 using the same method as in 1.b). What error ±( ) does this imply?



5

5.5 Math 155 Group 02 (Continued)

Conceptual Understanding:

3. After reading page 46 in our book, answer the following questions.

The number 23 means (circle all that are true):

a) 31⋅41 since 2 is almost 1.41.

b) 3 2 is not a number since we can never use a number in decimal form to equal it exactly. We can only approximate it with decimals.

c) We can think of 3 2 as the number that is the limit of a sequence of approximating

numbers in decimal form.

d) Numbers such as 5, 3 2 , 5π , etc. have no real practical use since they are always approximated by decimals.



5


1. Find where g(x) = (x +1)ex + ex positive, negative or zero.

2. Graph f (x) = ln x +1 without using a calculator.

3. Sketch y = ex cos x .



5



Functions and Models 6-1

6

6 Functions and Models

Following is the Math 129 objective of this chapter:

a. Build mathematical models using polynomial and rational functions and use them to solve various types of applications


a. Construct complex mathematical models using the building block functions.

b. Create diagrams with variables defined clearly.

c. Outline clearly how functions were derived and their meaning in the context of a word problem.



6

6.1 Math 129 Minimizing Response Time Name: __________________________

1. Lisa is 80 yards away from a straight shoreline when she gets an emergency call from her home, 500 yards down shore. Knowing she can row at 200 yd/min and run at 300 yd/min, how far down shore should she land the boat to make it home in the shortest time possible?

a) Draw a picture to represent this situation:

b) Write an expression to represent the distance that Lisa rows her boat: ________________

c) Write an expression to represent the distance that Lisa will run: ____________________

d) Write a function to express the total time required for Lisa to reach home:

t x( ) = _____________________

** Using the tools of calculus it can be shown that the distance x down shore that results in the

shortest possible time, is a zero of T (x) = x

200 x2 + 6400− 1

300

To answer the question, “how far down shore should she land the boat to make it home in the shortest time possible?” set T x( ) = 0 and solve for x . (Show all work below and answer the question in a complete sentence.)

e) How many yards down shore should she land the boat to make it home in the shortest time possible?



6

6.1 Math 129 Minimizing Response Time (Continued)

2. Suppose the motion of an object floating in turbulent water is modeled by the function

d(t) = t t2 − 9t + 22( ) , where d t( ) represents the displacement (in meters) at t sec.

***Using calculus, it can be shown that the velocity v of the particle is given by

v(t) = 5

2t

32 − 27

2t

12 +11t−

12 .

Find any time(s) t when the particle is motionless ( v = 0 ).



6

6.2 Math 129 Volume and Area Name: __________________________

Volume of a Box

1. A box has the dimensions given in the diagram.

2 ft

x

ft1

ft3 a) Find a formula for the volume of gas in the tank shown as a function of the depth. Keep

units (use 3 ft , not 3) and find the volume in terms of gallons with 1 ft3 = 7.5gal .

V x( ) =

b) What is the practical domain and range of your function? Express each in interval notation. (The practical domain and range deal with numbers that are realistic in a problem situation.)

c) Use the formula in part a) to find the total volume in gallons.

Area of a Combined Shape

2. The region is composed of two simple shapes: A triangle with a base of 2 ft and a height of

2 ft and attached to the base of the triangle a rectangle 2 ft by 1 ft . Find a formula for the area shown included up to x . Note that x can vary from the bottom of the triangle up to the top of the rectangle. (Hint: There will be two separate formulas depending on what value x has.)

ft2 ft1

A x( ) = ft2 x



6

6.3 Math 129 Building Functions Name: __________________________

1. Wire Area: A wire that is 10 cm is cut to create a square and a circle. a) Draw a figure to represent the situation. Label and identify the variables.

b) Create a function to give the combined area of the square and circle based on the radius of the circle.

c) What is the practical domain and range of your function? Express each in interval notation. The practical domain and range deal with numbers that are realistic in a problem situation. Domain: Range:

d) What is the theoretical domain and range of your function? Express each in interval notation. The theoretical domain and range does not take the context of the problem into account. Domain: Range:



6

6.3 Math 129 Building Functions (Continued)

2. Rectangle Inscribed in a Circle II: A rectangle is inscribed in a circle that has a radius of 3, as shown in the figure. Let P be a point in Quadrant I that is a vertex of the rectangle and is on the circle.

P = x, y( ) x

2 + y2 = 9

a) Find an equation for the area of the rectangle based on x .

b) Find an equation for the perimeter of the rectangle based on x .



6

6.4 Math 155 Group 01 Name: __________________________

A diesel tank has a cross section composed of: a triangle with a base of length 2 ft and a height of 2 ft, and a rectangle of width 2 ft and height of 1 ft. The rectangle’s 2 ft width is adjoined to the triangle at the base to form the cross section of the tank.

In operation the rectangular portion forms the top of the tank. The tank is 3 ft long. During field work the gauge stopped working and a stick inserted into the tank from the top is used to determine the volume of fuel remaining. Let d be the depth in feet.

In steps 1. – 3., below, find a formula V d( ) for the volume of fuel in gallons in the tank as a

function of d 1 ft3 = 7.5gal( ).

1. Draw a figure of the tank and indicate dimensions.

2. Find a formula for the total tank volume in gallons.



6


3. Write a formula V d( ) that keeps the dimensions of the variables, for example,

d = 1 ft not d = 1 . Give the volume in gallons.

4. What is the domain of the function V d( ) ?

5. Graph V d( ) as a function of d .



6


1. A cone-shaped cup is made by attaching the two straight segments shown in the picture cut from a circle of radius R . Find a function of one variable that gives the volume of the cone. In a later section you will determine the best way to cut out the wedge to maximize the volume of the cone.



6



Limits 7-1

7

7 Limits


a. Find and use vertical, horizontal/oblique asymptotes, holes, and intercepts to graph rational functions.


a. Explain and write out the intuitive definition of a limit, continuity and a derivative.

b. Calculate limits, including derivatives by the definition, using the limit laws, algebra and trigonometry and the squeeze theorem.

c. Interpret the derivative in two ways and be able to determine the units of the derivative.

d. Understand the graphical relationship between a function and its derivative.

e. Begin to understand the importance of the derivative.

f. Calculate limits involving indeterminate forms using derivatives and the tangent line approximation with L’Hospital’s Rule.


Limits 7-2

7

7.1 Math 129 Indeterminate Forms When an expression is classified as being written in indeterminate form, it means that we cannot determine what the value equals. Depending on how one approaches 0, the answer can literally be any number, and even ∞ or −∞ . Patterns can be used to demonstrate a few of the possible results.

Zero Divided By Zero

1.

00

is classified as being written in indeterminate form.

a) Find the following values to establish a pattern to find a possible answer for

00

.

44=

33=

22=

11=

00=

b) Find the following values to establish a pattern to find a possible answer for

00

.

04=

03=

02=

01=

00=

c) Find the following values to establish a pattern to find a possible answer for

00

.

40=

30=

20=

10=

00=

d) How do the results from a) – c) compare?


Limits 7-3

7

7.1 Math 129 Indeterminate Forms (Continued)

Zero To The Zero Power

2. 00 is classified as being written in indeterminate form.

a) Find the following values to establish a pattern to find a possible answer for 00 .

04 = 03 = 02 = 01 = 00 =

b) Find the following values to establish a pattern to find a possible answer for 00 .

40 = 30 = 20 = 10 = 00 =

c) How do the results from a) and b) compare?

Infinity Divided By Infinity

3. ∞∞

is classified as being written in indeterminate form. For each expression, factor an x2

from both the numerator and denominator, and then evaluate the expression for larger and larger values for x (taking the limit as x goes to infinity).

a)

x3

x2 + 3

b)

4x2 + xx2 + 3

c)

x2 −1x3 + 3x

d) How do the results from a) – c) compare?


Limits 7-4

7

7.2 Math 155 Group 03 Name: __________________________

Limits like limx→0+

xx are very important.

1. Can you find a negative value of x for which xx is not a real number? Why is it not real?

2. Apply your intuition and understanding of the math involved and guess the value of the limit.

The consensus answer of your group is_________________.

What rationale can you offer to justify this answer?

3. Now, use your calculators to estimate the limit. Put your answers in a table that shows how the limit is approached as x → 0+ .

lim xx

x→0+= _________________

x f x( )


Limits 7-5

7


4. To how many decimal places is your answer accurate in 3., or is it exact?

5. Based on your table values alone is it possible (in theory) that the limit really does not exist? Circle the best answer.

NO Once a trend is seen for several values of x close to zero it will continue and the theoretical limit will exist.

YES

Even though x values used are extremely close to zero, it is possible that when x is even closer the function values start bucking the trend and the limit does not exist.

NO

Once a trend exists for any formula it is not necessary to try x values that are so small ( like x=.00000000001) that they are for practical purposes zero. There are no formulas that can really give much different values for x values such as .0000001 and .00000001


Limits 7-6

7

7.3 Math 155 Group 09 Name: __________________________

The problem is to determine limx→a

f (x)g(x)

when limx→a

f (x) = 0 and limx→a

g(x) = 0 and to understand

how we get the answer!

Consider a simple form of the problem limx→a

6 x −1( )2 x −1( ) and so f x( ) = 6 x −1( ) and g x( ) = 2 x −1( ) .

1. Sketch f x( ) and g x( ) in the axes provided. Assume a is some positive number like 1.

2. The limit in this case is obviously 3. Based on the geometry of the two graphs, give an explanation of why the limit is 3.

x

y


Limits 7-7

7


3. Now consider limx→0

2x −12x

. Again, both the numerator and the denominator have the limit zero.

Sketch both f x( ) = 2x −1 and g x( ) = 2x in the axes provided. Be especially careful when considering the relationship between the two graphs near x = 0 . Find the equation of the tangent lines of each function at x = 0 and add these lines to the graph. Using the above reasoning and replacing f (x) and g(x) with the corresponding tangent line formulas,

evaluate limx→0

2x −12x

.

Tangent line of f x( ) = 2x

Tangent line of g x( ) = 2x −1

limx→0

2x −12x

=


Limits 7-8

7


4. Now suppose there are two functions f x( ) and g x( ) such that

limx→a

f x( ) = 0 and

limx→a

g x( ) = 0 .

What are the equations of the tangent lines to the curves (linearization of the curves) at x = a?

Using the above, what is limx→a

f x( )g x( ) ?


Limits 7-9

7


5. Use what you have discovered above to find the following limits. Remember, you must first

verify that limx→a

f x( ) = 0 and limx→a

g x( ) = 0.

limx→0

e3x −1x

limx→1

ln xsinπ x


Limits 7-10

7

7.4 Math 155 Homework (to be added)


Limits 7-11

7



Tangent Lines, Differentiation and Graphs 8-1

8

8 Tangent Lines, Differentiation and Graphs


a. Find and interpret the slope of the secant line of a function/average rate of change.


a. Know the derivatives of the building block functions.

b. Calculate derivatives using rules (product rule, quotient rule, chain rule, implicitly, etc).

c. Understand the meaning of the derivative in science such as the rate of change of a function and why it is important and in particular apply it in problems involving related rates.

d. Use the tangent line concept to approximate a function locally and to approximate the change in the function locally also called the differential.

e. Explain how the tangent line approximation shows why L’Hopital’s Rule works.

f. Use the interpretation of the derivative as the slope of the tangent line to graph functions and determine maximums, minimums, increasing/decreasing intervals, inflection points, and concavity.

g. Apply graphing skills to solve word problems that require optimization.

h. Recover a function from its derivative using anti-differentiation.



8

8.1 Math 129 Rates of Change & Difference Quotient Name: __________________________

1. If an arrow is shot vertically from a bow with an initial speed of 187 ft/sec, the height of the arrow can be modeled by the function f (t) = −16t2 +187t , where f t( ) represents the height of the arrow after t seconds (assume the arrow was shot from ground level).

a) What is the arrow’s height at t = 1 second?

b) What is the arrow’s height at t = 2 seconds?

c) Use the formula

f (t2 )− f (t1)t2 − t1

to calculate the average rate of change from t = 1 to t = 2 .

d) What is the rate of change from t = 10 to t = 11? Why is your answer negative?



8

8.1 Math 129 Rates of Change & Difference Quotient (Continued)

2. Calculating the average rate of change using

f (t2 )− f (t1)t2 − t1

(in c) and d) above), you found

the average velocity of the arrow for a specified interval of time. In order to consider the

velocity of the arrow at a precise instant, apply the difference quotient f (t + h)− f (t)

h to

f (t) = −16t2 +187t and investigate what happens as h becomes very small ( h→ 0 ).

a) Apply the difference quotient f (t + h)− f (t)

h to f (t) = −16t2 +187t and simplify your

result. Then approximate the velocity of the arrow at the moment t = 2 , by investigating what happens as h→ 0 .

3. Extra Practice: Apply the difference quotient to the functions below and completely simplify your answers.

a) f (x) = 3

x +1

b) f (x) = x2 − 3x



8

8.2 Math 155 Group 03 Name: __________________________

1. Suppose two functions, f x( ) and g x( ), both defined for all real numbers, agree everywhere

everywhere except at five values of x. Further, f x( ) has a limit at every a as x approaches a( ).What can you say about lim

x→1g x( ). Does it exist or not? Give reasons for your answer.

2. True or False: If false, give a counter example.

_______ If limx→a

f (x) = L , then f (a) = L

_______ If limx→a

f (x)g(x)

= 0 , then limx→a

f (x) = 0

_______ If limx→a

f (x) = 0 and limx→a

g(x) = M ≠ 0 , then limx→a

f (x)g(x) = 0

3. Conceptual Understanding

Put a checkmark by all correct statements.

We write limx→a

f x( ) = L only when each value of x closer to a makes f (x) closer to L.

In other words, if x1 is closer to a than x2 , then f x1( ) must be closer to L than f x2( ).

We write limx→a

f x( ) = L. This means that we can make f (x) arbitrarily close to L

by keeping x sufficiently close to a than x2 , with the possible exception of

what happens when x = a.

We cannot write limx→a

f x( ) = L when f x( ) = L for x close to a because f x( ) is not close to L, it equals L at that point.



8

8.3 Math 155 Group 04 Name: __________________________

1. The position of a particle in meters on the s axis is s t( ) = v0t +

12

at2 , where a and v0

are constants and t is time in seconds. Calculate dsdt

using the defintion.

What does v0 represent and what units does it have?

2. If the pressure P in lbf/ft2 in a pipe full of fluid for 0 ≤ L ≤ L is p = 100lbf/ft2 + x

8µQπR2

where 8µQπR2 is a constant and x is in feet. Find the derivative of P using the defintion.

What units does the quantity 8µQπR2 have?

What units does dPdx

have?



8


3. Conceptual Understanding:

a) Give the definition for a function f x( ) to be continuous at x = a.

b) Give the definition for a function f x( ) to have a derivative at x = a.

c) Check all correct statements

If a function is continuous at x = a then the function has a derivative at x = a.

If a function is differentiable at x = a then the function is continuous at x = a.

There is no logical connection between a function being continuous or having a derivative at x = a.

4. Conceptual Understanding:

Explain why a photograph of a car (probably) does not provide enough information to determine the instantaneous velocity of a car. What kind of visual information would be needed?



8

8.4 Math 155 Group 05 Name: __________________________

1. Calculate

d 35

dx35 xsin x( )



8

8.5 Math 155 Group 06 Name: __________________________

The diagram shows the major components of a stationary double-acting steam engine. Steam pressure drives the piston (1) back and forth inside the cylinder. The piston is attached to the piston rod (2) that drives the crosshead end of the connecting rod (4) back and forth horizontally. The crank (5) end of the connecting rod (4) in turn pushes the crank around the axis of the main bearing (6). The flywheel (7) is attached to one end of the crank. The flywheel rotates with the crank and provides the rotational inertia necessary to keep the crank turning when the piston is instantaneously stationary at the end of each stroke.

The dotted triangle shows the spatial relationship between three of the major components of the engine: the crosshead, connecting rod and crank.

The connecting rod is 18 cm long and the throw of the crank crankshaft is 10 cm. The engine turns at a steady 600 rpm.

1. If θ is the angle in the dotted triangle at the main bearing, at what rate is θ changing and how do we write this in calculus notation using seconds?

1 Piston

2 Piston rod

3 Crosshead

4 Connecting rod

5 Crank

6 Main bearing

7 Flywheel 8 Sliding valve

9 Centrifugal governor



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The forces on the crosshead must be determined to ensure a replacement gudgeon pin has sufficient strength. These forces are largely determined by changes in the velocity of the piston.

2. What is the velocity of the piston when θ = π 2 and θ = π ?

3. At what rate is θ changing and how do we write this in calculus notation using seconds?

4. What is the velocity of the piston when θ = π 2 and θ = π ?



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8.6 Math 155 Group 08 Name: __________________________

The problem is to model the temperature of a cup of coffee from the time it is dispensed from the coffee machine

160!F( )

to a time when the temperature of the coffee stops changing. Let

Ts = 70!F be the temperature of the surrounding room.

1. Let T t( ) be the temperature of the coffee as a function of time t .

a) What is the mathematical expression for the initial condition?

b) What is the domain of T t( ) ?

2. Draw two graphs showing how the temperature of the coffee might change over time; one linear and one decaying exponentially. Mark them as Graph A and Graph B. Put values on the t − axis that you consider reasonable for this problem.

3. What are the units of dT / dt ?

PAUSE: The Recitation Leader will now explain how to approach the next question.



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4. How fast the temperature changes depends on the difference between the coffee temperature

T t( ) and the room temperature Ts . To gain insight we will extract information contained in

the graphs A and B above. Draw graphs of dT / dt as a function of Ts −T (t) based on your graphs.

5. Pick the graph in question 4. that you think corresponds to the real behavior and explain why.

PAUSE: The Recitation Leader will now explain how to approach the next question.

T-TsdT/dt



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6. Write the mathematical equation that corresponds to a linear (straight line) relationship

between dTdt

and T −Ts that goes through the origin.

7. Circle the statements that seem correct

a) The rate of change of temperature of the coffee with respect to time is proportional to the difference in temperature between the liquid and room temperature.

b) The rate of change of temperature of the coffee is constant.

c) In the mathematical model, the coffee never reaches the exact temperature of the room.

d) In reality, the coffee reaches the temperature of the room eventually.

8. Which of statements a) or b) (above) corresponds to limt→∞

T t( ) = Ts ? Circle the best answer.

i) a) only,

ii) b) only,

iii) Both a) and b),

iv) Neither a) nor b).



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8.7 Math 155 Group 20 Name: __________________________

A movie theater has a screen that is 28 ft tall. When you sit down, the bottom of the screen is 6 ft above your eye level. The angle formed by drawing a line from the bottom of the screen to your eye and then to the top of the screen is called the “Viewing Angle.” The problem is to find the distance from the wall that maximizes the viewing angle.

1. Introduce all the necessary variables into the problem, show them on the diagram and describe them here.



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2. Find the mathematical function that you want to maximize.

3. Maximize the function showing all relevant calculations neatly.

4. What is the maximum viewing angle (in degrees)?



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8.8 Math 155 Group 18 Name: __________________________

A point, P , moves around the circumference of a circle of radius r centered at the origin with a

constant angular velocity,

dxdt

.

1. Draw a diagram of the problem where x is the horizontal displacement of P from the origin.

2. What does

dxdt

correspond to?

3. What units (length, time, length/time, etc.) are embedded in each of these quantities?

r

θ

cos θ( )

x

dxdt

dθdt

drdt



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4. What is the relationship among r, x and θ ? What equation relates all 3?

5. Find an expression for

dxdt

. Note that r is constant.

6. If θ increases at the rate of 10 deg/sec , how fast is x changing when θ = π 4 with r = 5 cm?

7. If the particle changes speed and begins traveling around the circle at 20 rpm, what would its angular velocity be in radians/second?

8. Draw a diagram of a possible trajectory of the particle if r is not considered constant, but instead, r = at, a > 0 ( dθ dt is still constant) .



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Summation and Integration 9-1

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9 Summation and Integration


a. Write out the terms of a sequence given the general or nth term.

b. Find the partial sum of a series.

c. Use summation notation to write and evaluate series.


a. Understand the geometric interpretation of the integral in terms of net area and be able to determine the units of the integral.

b. Calculate the integral using net area interpretation, Riemann sums, or the Fundamental Theorem of Calculus.

c. Apply the integral to solve problems involving area, distance travelled, and net changes.

d. Understand the meaning of the Fundamental Theorem of Calculus and be able to explain its derivation intuitively.

e. Know the basic integral Tables.



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9.1 Math 129 Summation Notation Name: __________________________

Series from Equation

1. A series is the sum of the terms of a sequence. The sigma Σ( ) is used to denote the

summation. The lower a( ) and upper b( ) indices indicate the beginning and ending values

for n : n=a

b

∑ . Write out the following sums, and then calculate the values. For infinite sums,

only write out the first six terms.

a)

2n2( )n=1

5

∑

b)

3n

4− 2

⎛⎝⎜

⎞⎠⎟n=0

5

∑

c)

5 f n( )− 2( )n=2

6

∑

d)

6− 3 f n( )( )n=1

∞

∑

e)

23

⎛⎝⎜

⎞⎠⎟

n

n=0

∞

∑



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9.1 Math 129 Summation Notation (Continued)

Equation From Series

2. Find the sigma notation for the given series with the indicated starting value. a) Start at n = 1 , −1+ 2+ 7 +14+ 23

b) Start at n = 0 ,

34+ 9

4+ 27

4+ 81

4+ 243

4+ 729

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c) Start at n = 3 , 9− f 3( ) +12− f 4( ) +15− f 5( ) +18− f 6( )

d) Start at n = 1 , f 0( ) + f 1( )

4+

f 2( )9

+f 3( )16

+f 4( )25

+f 5( )36

+…

e) Start at n = 0 , −1+1+ 3+5+ 7 + 9+…



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9.2 Math 155 Group 30 Name: __________________________

Let s t( ) be the postion of a particle on the s-axis and v t( ) = ds dt the velocity, with t the time

such that a ≤ t ≤ b. Let a = t0 < t1 < t2 < ⋅⋅⋅< tn−1 < tn = b be a partition of a,b⎡⎣ ⎤⎦.

1. What does s tn( )− s t0( ) = s b( )− s a( ) correspond to?

2. What does s ti( )− s ti−1( )correspond to?

3. If ti − ti−1 is small enough, why is s ti( )− s ti−1( ) ≅ v ti( ) ti − ti−1( ) = d

dts ti( )Δti ?

4. What does s tn( )− s tn−1( )+s tn−1( )− s tn−2( )correspond to?

5. What is s tn( )− s tn−1( )( )+ s tn−1( )− s tn−2( )( )+ ⋅⋅⋅+ s t2( )− s t1( )( )+ s t1( )− s t0( )( )= s ti( )− s ti−1( )( )

i=1

i=n

∑ ?



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6. Look at question 5. Replace s ti( )− s ti−1( )( ) in the summation with

ddt

s ti( )Δti .

These are equal as can be seen in question 3. What is the resulting form of the summation?

IMPORTANT

Now consider what happens as all the ti → 0. The summation in question 5. will become an

integral.

7. What integral will the summation in question 5. become as all the ti → 0?

8. Use question 5. to show what that integral is equal to.

9. Can you write out what this means in common sense terms?

10. Using what you have done so far, write your informed opinion of what

dfdxa

b

∫ dx is equal to.

dfdxa

b

∫ dx =

11. Use the above results to evaluate the following:

a)

ddx0

π /2

∫ sin x( )dx =

b)

cos x dx =0

π /2

∫

c)

1t

dt =1

3

∫ Hint: What function F t( ) gives dFdt

= 1t?

d)

2ey + 3y( )1

2

∫ dy



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Documents

Math 129 \u0026 Math 155E WB Ed1-