Chapter 4A Test ~ Part 1 Chapter 4A Test ~ Part 2

Calculus 1 Ms. Konnick Name ________________________________

Spring 2014

The following are the daily homework assignments for Chapter 4A – Graphing (Sections 4.2-4.4)

Section Pages Topics Assignment

4.2

4/9

p.209-216 Extreme values of functions (max and min),

First Derivative Theorem, critical points,

increasing and decreasing

p.216-218: # 1 – 21odd

4.2

4/10

p.209-216 Increasing and decreasing, 1st derivative test,

finding max/min values

p.216-218: # 27 – 33odd, 41 – 43,

49, 50, 55, 56

4.3

4/11

p.218-224 Concavity and the 2nd derivative test, points of

inflection, Intro Special Cases of Second

Derivative Test

p.224-227: # 1-2, 9-11, 13-15

4.3

4/14

p.218-224 Graphing Using First and Second Derivative

Tests

Worksheet

4.3

4/15

p.218-224 Review of First and Second Derivative Tests

Graphing using f’ and f”, Special Cases: Cusps

and Vertical Tangents

Worksheet

p.216 #’s 3 – 5, 7, 29, 31, 35

4.3

4/16

p.218-224 Sketching Graphs when given properties

Sketching Graphs given derivatives

Worksheet

4.2-4.3

4/17

p.209-224 Review and Quiz on Sections 4.2-4.3 NONE

Enjoy your long weekend!

4.4

4/21

p.227-232 Vertical, horizontal, and slant asymptotes,

curve sketching using derivatives

p.233-234: # 1-7 odd, 9, 13, 15,

17, 21, 27, 28

4.4

4/22

p.227-232 Vertical, horizontal, and slant asymptotes,

curve sketching using derivatives

p.233-234: # 1-7 odd, 9, 13, 15,

17, 21, 27, 28

Review

4/23

Review for Chapter 4A Test Complete study guide problems

Test

4/24

Chapter 4A Test ~ Part 1 Continue to Study for Part 2 of

Unit 4A Assessment

Test

4/25

Chapter 4A Test ~ Part 2 Print Notes on Optimization for

class tomorrow!

CALCULUS I -NOTES Name __________________________________

SECTION 4.2-1 Date ______________ Block _______

4.2 Extreme Values of Functions (Day 1)

Let c be a point of the domain D of a function f. The number f(c) is:

Absolute maximum: of f if f(x) f(c) for all x in D; highest point on graph

(no greater value anywhere)

Absolute minimum: of f if f(x) f(c) for all x in D; smallest point on graph

(no smaller value anywhere)

Local maximum: of f if f(x) f(c) for all domain points in some open interval

(no greater value “nearby”; part of the graph that “peaks”)

Local minimum: of f if f(x) f(c) for all domain points in some open interval

(no smaller value “nearby”; part of the graph that “valleys”)

A functions maximum values (maxima) and minimum values (minima) are the function’s extreme values

(extrema). They can occur at interior points and at the endpoints of the domain D.

Does f have a max or min value on [a,b]?

Max-Min Existence Theorem:

If f is continuous on closed interval [a,b] then f attains both an absolute maximum and minimum value.

How do we find Extreme Values (not looking at a graph)?

1) Find critical points

endpoints of the function (closed)

interior points of the domain where f’(x) = 0 (stationary points)

interior points of the domain where f’(x) DNE (singular points)

(sharp corners, jumps, etc.)

2) Plug critical points into f(x) (find the y-values)

3) Largest value = max Smallest value = min

Examples: Find the coordinates of the absolute maximum and absolute minimum on the given interval.

Then graph the function.

1) f(x) = 4 – x2 on [-1, 3]

2) f(x) = 3

2

x3 on [-1,1]

3) f(x) = -x2 + 4x – 1 on [0, 3]

4) f(x) = 51 (2x

3 + 3x

2 – 12x) on [-3, 3]

5) f(x) = sin x on [-, 2]

CALCULUS 1 Name: _______________________________

WORKSHEET 4.2-1 Date: ________________ Block: _______

Find the coordinates of the absolute maximum and absolute minimum on the given interval.

Then graph the function.

1. 1x6x3xf 2 , [0, 3]

2. 2x4xxf 4 , [−2, 1]

3. osxc2xf ,

2

3,0

4. 2x9xf , [−3, 2]

CALCULUS I -NOTES Name __________________________________

SECTION 4.2-2 Date ______________ Block _______

4.2 Extreme Values of Functions (Day 2)

First Derivative Test for Increasing and Decreasing

If f is continuous on [a,b] and differentiable (no cusps, breaks, asymptotes) on (a,b), then

1) If f’(x) 0 (positive) for all x in (a,b), then f is increasing on [a,b]

2) If f’(x) 0 (negative) for all x in (a,b), then f is decreasing on [a,b]

Where f’(x) = 0 we have local max or min

The First Derivative Test for Local Extreme Values

1) If f’ changes from positive to negative at c, then f has a local max at c

2) If f’changes from negative to positive at c, then f has a local min at c

3) If f’ does not change signs at c, then f has no local extreme value at c

4) At a left endpoint a: 5) At a right endpoint b:

c c

c c

c c

a a b b

f’(c) = 0

f’(c) = 0

f’(c) = 0

f’(c) is und.

f’(c) is und.

f’(c) is und.

Locate all extrema.

Examples: Find the intervals where the function is increasing and decreasing. Find the extreme values.

1) f(x) = 2x3 – 3x

2 – 12x + 7

2) f(x) = -x3 + 12x + 5 on [-3,3]

3) f(x) = x2 – 4x + 2 on [-3, 3]

4) f(x) = (x + 1)3

5) f(x) = 2x

5x 2

CALCULUS I Name __________________________________

WORKSHEET 4.2- 2 Date ______________ Block _______

Find the intervals where the function is increasing and decreasing. Find the extreme values.

1) f(x) = -4x2 – 7x + 5 2) f(x) = x

4 – 8x

2 + 1

3) f(x) = x3 – 4x 4) f(x) = 4x

3 – 6x

2

CALCULUS I -NOTES Name __________________________________

SECTION 4.3-1 Date ______________ Block _______

4.3 How yand y Determine the Shape of a Graph (Day 1)

How do we know if our graph looks like or ?

The difference between the two has to do with concavity (in very simple terms “opening up” or “down”)

Definition:

The graph of a differentiable function y = f(x) is concave up on an interval where y is increasing and

concave down where y is decreasing.

The Second Derivative Test for Concavity:

The graph of y = f(x) is concave down on any interval where f 0 and concave up on any interval

where f 0.

Concave up: Concave down:

CU: f 0 CD: f 0

Consider the curve: We have critical points c1, c2 and c3.

c1 is a max

c3 is a min

the curve is concave down from (-, c2)

the curve is concave up from (c2, )

The place where the concavity changes is at c2.

We call c2 a point of inflection (also a critical point).

Definition:

A point where the graph of a function has a tangent line and where the concavity changes is called a

point of inflection.

Thus a point of inflection on a curve is a point where y is positive on one side and negative on the other.

At such a point, y is either zero or undefined. So y = 0 at a point of inflection.

c c

c1 c2 c3

CALCULUS I -NOTES Name __________________________________

SECTION 4.3-3 Date ______________ Block _______

4.3 How yand y Determine the Shape of a Graph (Day 3)

How to use yand y to Graph a Function

1) Compute the derivative )x(f

Find the first-order critical numbers of f (where )x(f = 0 and )x(f DNE)

Find where the graph is increasing or decreasing

Identify local extrema

2) Compute the second derivate )x(f

Find the second-order critical numbers of f (where )x(f = 0 and )x(f DNE0

Find where the graph is concave up or concave down

Identify the point(s) of inflection

Shapes of Graphs

)x(f 0

graph rises from left to

right (may be wavy)

)x(f 0

graph falls from left to

right (may be wavy)

)x(f 0, )x(f 0

increasing, concave up

)x(f 0, )x(f 0

increasing, concave down

)x(f 0, )x(f 0

decreasing, concave up

)x(f 0, )x(f 0

decreasing, concave down

)x(f = 0, )x(f 0

local maximum

)x(f = 0, )x(f 0

local minimum

)x(f = 0

point of inflection

f(x) = 0

x-intercepts

Find the intervals on which the graph of f is increasing, decreasing, concave upward, and concave downward.

Find the coordinates of any extreme points and point of inflection. Sketch the graph.

1) 𝑓 𝑥 = 𝑥2 − 6𝑥 + 8

𝑓 ′ 𝑥 = 𝑓" 𝑥 =

Critical Points: Critical Points:

Increasing: Decreasing:

Local Max(s): pt(s) of inflection:

Local Min(s):

2) 𝑓 𝑥 = 𝑥3 − 12𝑥 − 5

𝑓 ′ 𝑥 = 𝑓" 𝑥 =

Critical Points: Critical Points:

Increasing: Decreasing:

Local Max(s): pt(s) of inflection:

Local Min(s):

3) 𝑓 𝑥 = 𝑥4 − 4𝑥3 + 5

𝑓 ′ 𝑥 = 𝑓" 𝑥 =

Critical Points: Critical Points:

Increasing: Decreasing:

Local Max(s): pt(s) of inflection:

Local Min(s):

4) 𝑓 𝑥 = 𝑥4 + 8𝑥3 + 18𝑥2 − 8

𝑓 ′ 𝑥 = 𝑓" 𝑥 =

Critical Points: Critical Points:

Increasing: Decreasing:

Local Max(s): pt(s) of inflection:

Local Min(s):

5) 𝑓 𝑥 = 1 − (𝑥 + 3)3

𝑓 ′ 𝑥 = 𝑓" 𝑥 =

Critical Points: Critical Points:

Increasing: Decreasing:

Local Max(s): pt(s) of inflection:

Local Min(s):

6) 𝑓 𝑥 = 𝑥2

3 − 1

𝑓 ′ 𝑥 = 𝑓" 𝑥 =

Critical Points: Critical Points:

Increasing: Decreasing:

Local Max(s): pt(s) of inflection:

Local Min(s):

Calculus 1- Homework Name:

Section 4.3: How y’ and y” Determine the Shape of a Graph Date:

Find the intervals on which the graph of f is increasing, decreasing, concave upward, and concave downward.

Find the coordinates of any extreme points and point of inflection. Sketch the graph.

1) 𝑓 𝑥 = 𝑥2 − 8𝑥 − 9

𝑓 ′ 𝑥 = 𝑓" 𝑥 =

Critical Points: Critical Points:

Increasing: Decreasing:

Local Max(s): pt(s) of inflection:

Local Min(s):

2) 𝑓 𝑥 = 𝑥4 − 4𝑥3 + 2

𝑓 ′ 𝑥 = 𝑓" 𝑥 =

Critical Points: Critical Points:

Increasing: Decreasing:

Local Max(s): pt(s) of inflection:

Local Min(s):

3) 𝑓 𝑥 = −𝑥4 + 6𝑥2 − 4

𝑓 ′ 𝑥 = 𝑓" 𝑥 =

Critical Points: Critical Points:

Increasing: Decreasing:

Local Max(s): pt(s) of inflection:

Local Min(s):

4) 𝑓 𝑥 = 𝑥2

3 + 3

𝑓 ′ 𝑥 = 𝑓" 𝑥 =

Critical Points: Critical Points:

Increasing: Decreasing:

Local Max(s): pt(s) of inflection:

Local Min(s):

CALCULUS I -NOTES Name __________________________________

SECTION 4.3-2 Date ______________ Block _______

4.3 How yand y Determine the Shape of a Graph (Day 2)

Vertical tangents

The graph of a continuous function has a vertical tangent at a point where the tangent is vertical and

the concavity is different on both sides.

Example: Identify the intervals where the function is increasing and decreasing and is concave upward

and concave downward. Find the coordinates of any extreme points and points of inflection.

1) f(x) = 31

x

Cusps

The graph of a continuous function has a cusp at a point where the tangent is vertical and the

concavity is the same on both sides. A cusp can be either a local maximum or a local minimum.

Example: Identify the intervals where the function is increasing and decreasing and is concave upward

and concave downward. Find the coordinates of any extreme points and points of inflection.

2) f(x) = 32

x

c c

f’(c) is und f’(c) is und

c

f’(c) is und. f’(c) is und.

c

CALCULUS 1- NOTES NAME:

SECTION 4.3: HOW Y’ AND Y” DETERMINE THE SHAPE OF A GRAPH DATE:

1.) Use the graph to determine where the function meets the criteria

a) 𝑓 𝑥 = 0 b) 𝑓′ 𝑥 = 0

c) 𝑓′′ 𝑥 = 0 d) 𝑓 𝑥 < 0

e) 𝑓 ′ 𝑥 < 0 f) 𝑓 ′′ (𝑥) < 0

g) 𝑓 ′(𝑥) > 0 h) 𝑓 ′′ (𝑥) > 0

i) 𝑓 ′(𝑥) > 0 & j) 𝑓 ′(𝑥) < 0 &

𝑓 ′′ 𝑥 < 0 𝑓 ′′ 𝑥 > 0

2.) Draw a function with the following properties:

𝑓 −3 = −4

𝑓 −2 = −1

𝑓 −1 = 2

𝑓 ′(𝑥) = 0 𝑤ℎ𝑒𝑛 𝑥 = −1,−3

𝑓 ′ 𝑥 < 0 𝑤ℎ𝑒𝑛 𝑥 < −3 𝑎𝑛𝑑 𝑥 > −1

𝑓 ′ 𝑥 > 0 𝑤ℎ𝑒𝑛 − 3 < 𝑥 < −1

𝑓 ′ ′ 𝑥 = 0 𝑤ℎ𝑒𝑛 𝑥 = −2

𝑓 ′ ′ 𝑥 < 0 𝑤ℎ𝑒𝑛 𝑥 > −2

𝑓 ′ ′ 𝑥 > 0 𝑤ℎ𝑒𝑛 𝑥 < −2

3.) Sketch f(x)

𝑓 −7 = −1

𝑓 −4 = 2

𝑓 0 = 8

𝑓 4 = 2

𝑓 7 = −1

4.) Draw a function with the following properties

𝑓 −5 = 0 𝑓 ′ ′ 𝑥 = 0 𝑤ℎ𝑒𝑛 𝑥 = −5

𝑓 ′ 𝑥 < 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 𝜖 ℛ 𝑓 ′ ′(𝑥) < 0 𝑤ℎ𝑒𝑛𝑥 < −5

𝑓 ′ ′ 𝑥 > 0 𝑤ℎ𝑒𝑛𝑥 > −5

5.) Draw a function with the following properties

𝑓 −2 = 8 𝑓 ′ 𝑥 > 0 𝑓𝑜𝑟 𝑥 < −2 𝑓 ′ ′ 𝑥 < 0 𝑓𝑜𝑟 𝑥 < 0

𝑓 0 = 4 𝑓 ′ 𝑥 > 0 𝑓𝑜𝑟 𝑥 > 2 𝑓 ′ ′ 𝑥 > 0 𝑓𝑜𝑟 𝑥 > 0

𝑓 2 = 0 𝑓 ′ 𝑥 < 0 𝑓𝑜𝑟 − 2 < 𝑥 < 2

𝑓 ′ −2 = 𝑓 ′ 2 = 0

CALCULUS 1- HOMEWORK NAME:

SECTION 4.3: HOW Y’ AND Y” DETERMINE THE SHAPE OF A GRAPH DATE:

1.) Use the graph to determine where the function meets the criteria

a) 𝑓 𝑥 = 0 b) 𝑓′ 𝑥 = 0

c) 𝑓′′ 𝑥 = 0 d) 𝑓 𝑥 < 0

e) 𝑓 ′ 𝑥 < 0 f) 𝑓 ′′ (𝑥) < 0

g) 𝑓 ′(𝑥) > 0 h) 𝑓 ′′ (𝑥) > 0

i) 𝑓 ′(𝑥) > 0 & j) 𝑓 ′(𝑥) < 0 &

𝑓 ′′ 𝑥 < 0 𝑓 ′′ 𝑥 > 0

2.) Draw a function with the following properties:

𝑓 −2 = −4 𝑓 ′(𝑥) = 0 𝑤ℎ𝑒𝑛 𝑥 = −2, 5 𝑓 ′ ′ 𝑥 = 0 𝑤ℎ𝑒𝑛 𝑥 = 1.5

𝑓 5 = 7 𝑓 ′ 𝑥 < 0 𝑤ℎ𝑒𝑛 𝑥 < −2 𝑜𝑟 𝑥 > 5 𝑓 ′ ′ 𝑥 < 0 𝑤ℎ𝑒𝑛 𝑥 > 1.5

𝑓 1.5 = 3 𝑓 ′ 𝑥 > 0 𝑤ℎ𝑒𝑛 − 2 < 𝑥 < 5 𝑓 ′ ′ 𝑥 > 0 𝑤ℎ𝑒𝑛 𝑥 < 1.5

3.) Sketch f(x)

𝑓 3 = −2

𝑓 −3 = −2

𝑓 −6 = 4

𝑓 0 = −6

𝑓 6 = 4

4.) Draw a function with the following properties

𝑓 3 = 4 𝑓 ′ ′ 𝑥 = 0 𝑤ℎ𝑒𝑛 𝑥 = 3

𝑓 ′ 𝑥 > 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 𝜖 ℛ 𝑓 ′ ′ 𝑥 < 0 𝑤ℎ𝑒𝑛𝑥 < 3

𝑓 ′ ′ 𝑥 > 0 𝑤ℎ𝑒𝑛𝑥 > 3

5.) Sketch f(x) if f(0) = 0

6.) Sketch a continuous curve with the following properties. Label coordinates where possible. Give the coordinates

of any maximums or minimums.

x y Curve

2x Falling, concave up

2 1 Horizontal tangent

42 x Rising, concave up

4 4 Point of Inflection

64 x Rising, concave down

6 7 Horizontal tangent

6x Falling, concave down

CALCULUS 1 - NOTES Name _______________________________

SECTION 4.4 Date ___________________ Block ____

4.4: Graphs of Rational Functions

rational function: f(x) = h(x)

g(x); h(x) 0 (quotient of 2 polynomials)

f(x) = x

1 * undefined at ______ because _________________

* undefined at ______ because _________________

The graph approaches lines called ______________

vertical asymptote: the line _____ is a VA if

)x(flimax

or

)x(flimax

VA: f(x) = h(x)

g(x) VA occurs where x is undefined or h(x) = 0!

set denominator = 0 and solve for x

horizontal asymptote: the line _____ is a HA if b)x(flimx

order of polynomial: (compare highest powers of x)

same

same HA at y = ratio of coefficients

big

small HA at y = 0

small

big no HA

1. 3x

1y

VA: ________

HA: ________

CALCULUS 1 Name: _____________________________

WORKSHEET 4.4 Date: ______________ Block: ______

Write the equation(s) of each of the asymptote(s). Then, using the first derivative, sketch the graph.

1. f(x) = 2x

2

VA: _____

HA: _____

2. f(x) = 6x2

1x4

VA: _____

HA: _____

3. f(x) = 16x

22

VA: _____

HA: _____

slant (oblique) asymptote: the line _____ is a SA if b)x(flimx

if the degree of the numerator is one

more than the degree of the denominator

small

big no HA, may have an SA if given .etc,

x

x,

...x

x,

x...

x

3

4

2

...3

...2

*A function will have a HA or a SA but never both.

1. 1-x

4xy

2

VA: ________

HA: ________

SA: ________

2. 2x

4xy

2

VA: ________

HA: ________

SA: ________

ASYMPTOTES AND HOLES

One VA; One HA

Two VAs; One HA (just the middle)

One VA; one SA

c.p.

+ + c.p.

+ c.p.

+ c.p.

+ c.p. c.p. c.p.

+ c.p. c.p. c.p.

c.p. c.p.

+ c.p. c.p.

+ + c.p. c.p. c.p.

+ + c.p. c.p. c.p.

+ + c.p.

c.p.

max

min

max max

min min

Calculus 1 Name ___________________________________

Chapter 4A Study Guide

Chapter 4A review problems are located on p.267 in your textbook. You should be able to do the following…..

1. Use the first and second derivative test to find intervals of increasing/decreasing, points of maxs/mins, intervals

of concave up/down, points of inflection, and sketch an accurate graph of a polynomial function.

o # 22, 23, 26

2. Use the first and second derivative test to find intervals of increasing/decreasing, points of maxs/mins, intervals

of concave up/down, points of inflection, and sketch an accurate graph when given the first derivative.

o # 32, 33, 36

3. Use the first and second derivative test to find intervals of increasing/decreasing, points of maxs/mins, intervals

of concave up/down, points of inflection, and sketch an accurate graph of a function with fractional exponents.

o # 27, 28

4. Find asymptotes rational functions and use the first derivative test to find intervals of increasing/decreasing and

points of maxs/mins.

o # 46, 48, 52, plus other problems from section 4.4

5. Use a graph to determine where…

a.) absolute maximum:

b.) 0)( xf

c.) 0)(' xf

d.) 0)('' xf

e.) 0)(' xf

f.) 0)('' xf

g.) 0)(' xf

h.) 0)('' xf

i.) 0)('',0)(" xfxf

j.) 0)('',0)(' xfxf

6. When given criteria, sketch the graph of a function.

o Given the criteria below, sketch the graph of a continuous function, f(x), on the interval [-4,6] only. At

what points are there maximum and minimum values? Why?

510)(510)(

310)(310)(

0)5(0)1(0)3(0)1(

2)6(1)5(3)3(1)1(1)1(2)3(8)4(

xwhenxfxandxwhenxf

xwhenxfxandxwhenxf

ffff

fffffff

o Draw a function with the following properties

52 4)1(

11 0)('' 20 ;20)(3)0(

1 ;1 0)('' 2 x 0;x2- 0)('4)1(

1,1 0)('' 2 0, ,2 0)('5)2(

) f(f

xwhenxfxxwhenxff

xxwhenxfwhenxff

xwhenxfxwhenxff

7. Understand how the 1st and 2

nd derivatives are helpful.

o When is there a point of inflection?

o Can a function have both a local and absolute minimum? If yes, draw a graph. If no, explain why.

o Name the three places where a function can have extreme values? (maximums or mimimums)