Calculus 1

Math 114/115

Lecture Notes

1 Derivatives

1.1 The Tangent and Velocity Problem (Section 2.1 in text)

Example:

The position of a car is given by the values in the table.

t (seconds)

0

1

2

3

4

5

S (feet)

0

10

32

70

119

178

a) Find the average velocity for the following time periods.

i) 2 5 seconds

ii) 2 4 seconds

iii) 2 3 seconds

b) Sketch a graph of s as a function of t by plotting points and drawing a smooth curve through them.

c) Draw the secant lines whose slopes correspond to the average velocities found in part a).

d)

L

)

x

(

g

lim

a

x

=

Draw a tangent line that touches the curve at the point t = 2, and use its slope to estimate the instantaneous velocity of the car when t = 2.

(

)

x

(

f

lim

a

x

0

0

(

0

b

Example 2 (review of graphs and functional notation): The graph of the function

7

x

2

x

)

x

(

f

3

5

1

-

-

=

is given. Use the graph of estimate answers to a) d) below:

a) For which values of x is f(x) < 0?

b) For which values of x does f(x) have a negative slope?

c) For which value(s) of x does f(x) = 0?

d) For which value(s) of x does f(x) have a slope of 0?

e) Calculate each quantity with a calculator, and use the graph of verify your answer:

i) f(0)

ii) f(5)

)

x

(

f

lim

x

f) Find the equation of the line that passes through the points A(0, f(0)) and B(5,f(5)). Plot the line. Mark the rise, f(5) f(0), and the run, 5 0, on the graph.

Example 3: The graph of the function f(x) = x2 + 1 is given:

a) Calculate f(1) and plot the point P(1, f(1)) on each of the graphs in part b)

b) Calculate f(1 + h) and plot the point Q(1+ h,f(1+h) ) for the following values of h. In each case draw the secant line PQ and calculate its slope:

n

x

b

i. h = 1

f(1 + h) = ___________________

Slope of PQ = ______________

ii.

dx

df

h = .5

f(1 + h) = ___________________

Slope of PQ = ______________

iii.

dx

dy

h = .2

f(1 + h) = ___________________

Slope of PQ = ______________

)

x

(

f

dx

d

iv. h = 0

f(1 + h) = ___________________

Slope of PQ = ________________

c) We would like to know the slope of the tangent line at the point P(1,2). You can see that from part b) that as h get smaller, the secant line gets more and more like the tangent line at P. But if we let h = 0 (as in b iv), what problem is encountered?

1.2 The Limit of a Function (section 2.2 in text)

[

]

)

(

'

)

(

'

)

(

)

(

x

bg

x

af

x

bg

x

af

dx

d

+

=

+

Example 1 : Let

3

x

6

x

x

)

x

(

f

2

-

-

-

=

a) What is the largest natural domain of f(x)?

b) Plot a graph of f(x) (simplify f(x) before graphing)

)

x

cos(

)]

x

[sin(

dx

d

=

c) Explain why

)

x

(

f

lim

3

x

= 5, although f(3) is undefined. (

Example 2: Evaluate

)

x

(

g

lim

a

x

for each number a and for each function g(x) given below. Follow the steps given::

1. Evaluate g(a). If g(a) is equal to a real number, then that number is the value of the limit, and you are done.

2. If g(a) evaluates to

0

0

, then try to simplify g(x) and then evaluate g(a) using the simplified version of g(x). If a real number results, then that number is the value of the limit, and you are done.

a)

6

x

5

x

2

x

x

lim

2

2

2

x

+

+

-

+

-

b)

5

x

2

x

lim

2

3

x

+

-

c)

1

x

1

x

lim

2

1

x

+

-

d)

4

x

10

x

3

x

lim

2

2

2

x

-

-

+

f)

x

0

x

e

1

lim

1.3 Limits Involving Infinity (Section 2.5 in text)

In this section, we consider two types of limits:

1. infinite limits, where f(x) approaches ( or -(

2. limits at infinity, which involve x approaching ( or -(

Note that sometimes f(x) ( ( as x ( ( , combining the two types.

)

x

sin(

)]

x

[cos(

dx

d

-

=

Definition: Let f be a function defined on both sides of a (but not necessarily at a itself). If the values of f(x) can be made as large as we please by taking values of x sufficiently close to a (but not equal to a) then be say:

=

)

x

(

f

lim

a

x

Example 1: Let

2

)

1

x

(

1

)

x

(

f

-

=

. In this example, we will find

)

x

(

f

lim

1

x

.

a) Can you evaluate f(1)?

b) What happens to f(x) as x gets close to 1? Fill in the following table:

X(1-

f(x)

x(1+

f(x)

0

2

.5

1.5

.8

1.2

.9

1.1

.99

1.01

c) Sketch a graph of f(x).

(

)

)

x

cos(

)

x

sin(

dx

d

=

d) Can you find a value of x that makes f(x) > 1,000,000?

Limits at Infinity

In the previous example, we let x approach a particular number, and found that we could make the values of f(x) become arbitrarily large. Now we will let x become large, and see what happens to the values of f(x).

(

)

)

x

cot(

)

x

csc(

)

x

csc(

dx

d

-

=

Definition: Let f be a function defined on some interval ( a , ( ). Then

L

)

x

(

f

lim

x

=

means that the values of f(x) can be made as close to L as we like by taking x sufficiently large.

Example 2: Let

)

1

x

(

1

)

x

(

f

-

=

. In this example, we will find

)

x

(

f

lim

x

.

(

)

)

x

sin(

)

x

cos(

dx

d

-

=

(

)

)

x

tan(

)

x

sec(

)

x

sec(

dx

d

=

a) What happens to f(x) as x becomes large? Fill in the table below and sketch the graph:

b) What is

)

x

(

f

lim

x

?

c) How large must you make x in order to make 0 < f(x) < 10-8? (

(

)

)

x

(

sec

)

x

tan(

dx

d

2

=

A Summary of Methods for Calculating Limits

Exercise: Evaluate each of the following limits:

a)

20

x

x

x

12

x

3

lim

2

2

4

x

-

-

+

-

b)

x

9

x

2

7

x

5

x

3

lim

2

2

x

-

+

-

c)

x

10

x

2

15

x

2

x

lim

2

2

5

x

+

-

+

-

d)

t

2

)

5

t

)(

3

t

(

lim

2

t

-

-

-

-

e)

3

x

2

x

3

x

3

x

8

lim

2

2

2

x

-

-

-

+

Tangents, Velocities and other Rates of Change (Section 2.6 in text)

Example 1: Let f(x) = x2. We will find the equation of the tangent line to f(x) when x = 3:

a) Find the coordinates of the point P and Q. (The x coordinate of P is 3. The coordinates of Q will be in terms of h).

(

)

)

x

(

csc

)

x

cot(

dx

d

2

-

=

b) Find the slope of the secant line PQ (in terms of h)

c) Express the slope of the tangent line at P as a limit of the expression from part

d) Evaluate the limit from part c) to find the slope of the tangent line through point P

e) Find the equation of the tangent line through point P and sketch the line.(

Example 2: We would like to know the slope of the tangent line to the curve f(x) = x2 + 1 at the point P(1,f(1)).

b)

[

]

)

x

(

'

g

))

x

(

g

(

'

f

))

x

(

g

(

f

dx

d

=

If Q is the point Q(1 + h, f(1 + h) ) mark the value 1 + h on the x-axis and f(1 + h) on the y-axis. (we dont know or care exactly what h is equal to)

c) Find the slope of the secant line that passes through the points P(1,f(1)) and Q(1 + h, f(1 + h) ). (Express your answer not as a number, but in terms of f and h)

d) We can see that as h gets smaller, the slope of the secant line gets closer and closer to the the slope of the tangent line at P, since Q slides into P as h gets smaller. What happens to the slope of the secant line if we let h = 0?

e) Since, as we saw in part c), the expression for slope that we came up with in part a) is undefined when h = 0, instead we will take the limit as h approaches zero:

Evaluate

1

)

h

1

(

)

1

(

f

)

h

1

(

f

lim

0

h

-

+

-

+

by first simplifying the expression and then replacing h with 0.

f) What is the slope of the tangent line at P? Find the equation of the tangent line and draw it.

(

)

2

1

x

1

1

)

x

(

tan

dx

d

+

=

-

Example 3: In this exercise will find a general expression for the exact value of the slope of a tangent line. Let P be the point on the graph at which we want to find the slope of the tangent line.

a) Find the coordinates of P and Q (in terms of a, h and f).

b) Find the slope of the secant line PQ (in terms of a, h and f)

c) Express the slope of the tangent line at P as a limit of the expression from part b)

The Derivative (Section 2.7 in text)

(

)

2

1

x

1

1

)

x

(

sin

dx

d

-

=

-

Definition: The derivative of a function f at a number a, denoted f(a) (read as f prime of a), is:

h

)

a

(

f

)

h

a

(

f

lim

)

a

(

'

f

0

h

-

+

=

if this limit exists.

The derivative may be interpreted in several ways: as the slope of the tangent line to the graph when x=a, as the instantaneous velocity at time a, or, more generally, as the instantaneous rate of change of f(x) with respect to x when x = a.

Example 1: Let f(x) = x - x2

a) Use the graph to arrange the following numbers in increasing order: f(0), f(-3), f(1), f(), f(4), f(-4);

b) Find and simplify an expression for f(a)

b) Use the result from part b) to evaluate f(0), f(-3), f(1), f(), f(4), f(-4); (

Example 2: Sketch the graph of a function for which g(0)=0, g(0) = 3, g(1) = 0 and

g(2) = 1

Example 3

Let g(x) = 1 x3 (graph shown below). Find g(0) and use it to find the equation of the tangent line to the curve at the point (.5, g(.5) ). Draw the tangent line on the graph below.

The Derivative as a Function (Section 2.8 in text)

Recall that f(a) is the derivative of the function f(x) at the point x = a, and is equal to the slope of the tangent line to the graph of f(x) at the point

( a, f(a) ) . We can replace the constant a with the variable x to obtain a new function:

h

)

x

(

f

)

h

x

(

f

lim

)

x

(

'

f

0

h

-

+

=

This new function, f(x), is known as the derivative of f(x).

Example 1:

The graph of f(x) (solid line) and its derivative f(x) (dotted line) are given below.

a) Mark the points where f(x) = 0. What do you notice about the graph of f(x) at those points?

b) Mark the interval on which f(x) is less than zero. What do you notice about the slope of f(x) on that interval? What about the intervals on which f(x) is greater than zero?

The second derivative:

If f(x) is the derivative of the function f(x), then the derivative of f(x), denoted f(x), is known as the second derivative of f(x). The second derivative is the rate of change of the first derivative, that is, it is the rate at which the slope of f(x) is changing.

Example 2:

The graphs of f(x), f(x), and f(x) (the dotted straight line) are given below.

a) Mark the interval on which f(x) is positive, and that on which it is negative. What does the sign (positive or negative) of f(x) tell us about f(x)?

b) Put an x on the spot where f(x) = 0. What happens to f(x) at that point? What happens to f(x)?

Example 3:

The graph of f(x) = - x2 x + 6 is given below.

a) Find f(x) using the definition of derivative and plot f(x) on the graph above.

b) Find f(x) using the definition of derivative. Plot f(x) on the graph above.

Math 114

Section 2.10

What Does f Say About f?

What does f say about f?

If f(x) > 0 on an interval, then f is increasing on that interval

If f(x) < 0 on an interval, then f is decreasing on that interval

f(x) has a local maximum at a point if f(x) = 0 at that point, and if f(x) is increasing to the left of the point and decreasing to the right of the point

f(x) has a local minimum at a point if f(x) = 0 at that point, and if f(x) is decreasing to the left of the point and increasing to the right of the point.

What does f say about f?

If f(x) > 0 on an interval, then f is concave upwards on that interval

If f(x) < 0 on an interval, then f is concave downwards on that interval(

Example:

a) Sketch a graph of a function for which f < 0 and f < 0 on all intervals

b) Sketch a graph of a function for which f > 0 and f > 0 on all intervals

c) Sketch a graph of a function for which f > 0 and f < 0 on all intervals

d) Sketch a graph of a function for which f < 0 and f > 0 on all intervals

Example:

The graphs of f(x) (solid line) and f(x) (dotted line) are shown below. Answer the following questions abut the function f(x):

a) On what intervals if f increasing?

b) On what intervals if f decreasing?

c) On what intervals is the graph of f concave up?

d) At what value(s) of x does f have a local minimum?

e) At what value(s) of x does f have a local maximum?

f) Sketch a possible graph of f

Example:

The graphs of f, f, f are given. Answer the following questions to help identify which graph is which.

a) At approximately which values of x does function 1 have a tangent with slope of zero (flat spots) ?

b) We know that f is always equal to the slope of the tangent to f, at any point x. Which function is equal to zero (has x-intercepts) at the same places as function 1 has flat spots?

c) At approximately which values of x does function 2 have a tangent with slope of zero?

d) Which function is equal to zero (has x-intercepts) at the same places as function 2 has flat spots?

e) Identify each graph as f, f or f: 1_______ 2_________ 3_________

f) On what interval is f( 0?________ On what interval is f decreasing?________

g) Verify that f = 0 at the places where f changes concavity.

Math 114

Section 3.1 Derivatives of Polynomials and Exponential Functions

The Power Rule

Exercise 1 In each case, use the definition of derivative to find f(x):

a) f(x) = x-1

b) f(x) = x1

c) f(x) = x2

The results of the last exercise are generalized in the power rule for differentiation, which if given here without proof:

Exercise 2 Find the derivative of each of the following functions using the power rule. (If necessary, first put the functions in the form x()

a) f(x) = x3

b)

x

x

f

=

)

(

c)

5

1

)

(

x

x

f

=

d)

)

(

)

(

3

x

x

x

f

=

Exercise 4 Use the definition of derivative to show that the derivative of a constant is equal to zero.

Exercise 5 Find the derivative of each of the following functions

a) f(x) = 3x2 x + 5

b) f(x) =

2

2

1

4

x

x

-

c) g(z) =

7

3

1

2

1

2

+

-

z

z

d)

2

2

3

5

4

3

2

)

(

x

x

x

x

x

f

-

+

-

=

e)

2

10

)

(

w

w

w

g

+

=

Exercise 6 The graph of

2

6

13

6

1

)

(

3

-

-

=

x

x

x

f

is given below.

a) Find f(x)

b) Find the equation of the tangent line at the point (1, f(1) ). Sketch the tangent line.

c) Find the equation of the tangent line at the point (-4, f(-4) ). Sketch the tangent line.

d) Find the values of x at which the tangent line to f(x) is horizontal (slope zero)

Derivative of the Natural Exponential Function

We have seen that the number e ( 2.718 . . . , and e =

n

n

n

1

1

lim

+

.

Exercise

a) Use the definition of derivative to find an expression for the derivative of the function f(x) = ex

b) Use the fact that

e

n

1

1

n

@

+

(when n is large) to show that

1

h

)

1

e

(

h

@

-

(when h gets close to zero). Verify this numerically by completing the table below:

h

h

)

1

e

(

h

-

0.1

0.01

0.001

c) Conclude that if f(x) = ex, then f'(x) = ex. That is, ex is its own derivative!

Exercise. Differentiate each function:

a) f(x) = 6ex + 3

b) f(x) = 3x2 - 7ex + 13

c) f(x) = ex+2 + 7x

Suggested Exercises pg 196 - 197:

Odds from 3 21 pg.

#24, 28, 40 (will do in class)

#43,45

Section 3.2

The Product and Quotient Rule

The Product Rule:

If f and g are both differentiable, then

[

]

)]

x

(

f

[

dx

d

)

x

(

g

)]

x

(

g

[

dx

d

)

x

(

f

)]

x

(

g

)

x

(

f

dx

d

+

=

In another notations:

[

]

)]

x

(

g

)

x

(

f

dx

d

= gf+ fg

Exercise 1: Differential the following functions:

a) g(x) =

x

e

x

b) f(x) = x5ex

c) y = (x5 5x)ex

d) f(x) = (x5 7x2 +3x 2)(x2 x)

Exercise 2: (similar to #19 on page 204 of text)

Find an equation of the tangent line to the curve y = 2xex at the point (0, 0). Plot the line on the graph given.

The Quotient Rule

If f and g are differentiable functions, then

[

]

2

)

x

(

g

)

x

(

f

)

x

(

'

g

)

x

(

g

)

x

(

'

f

)

x

(

g

)

x

(

f

dx

d

-

=

Exercise 2: Differentiate the following functions:

a)

2

x

1

x

)

x

(

f

+

=

b)

4

x

x

e

)

x

(

f

=

c)

x

4

x

4

)

x

(

f

+

-

=

d)

1

x

1

x

)

x

(

f

+

-

=

Exercise 3: (similar to #21 on page 204 of text)

The curve

2

x

1

1

y

+

=

is called a witch of Maria Agnesi (graph shown below). Find an equation of the tangent line to this curve at the point (1, ). Plot the tangent line on the graph below.

Exercise 5: (similar to #22, pg 204 of text book)

The curve

2

x

1

x

y

+

=

is called a serpentine (graph shown below).

a) Find

dx

dy

.

b) Find the values of x for which

dx

dy

= 0. Compare to the graph.

c) Find the equation of the tangent line to this curve at the point (3, 0.3). Plot the tanget line.

Suggested exercisessection 3.2:

Pg 204 205

#1 17 odds, 23, 25, 27, 31

Section 3.3

Rates of Change in the Natural and Social Sciences

Example 1:

A particle is moving in a straight line in such a way that its displacement is given by the function s(t) = 4t3 9t2 + 6t + 2; t ( 0, where s(t) is in metres and t is in seconds. Find the following:

a) the average velocity in the first 2 seconds

b) the instantaneous velocity at t = 2 seconds

c) the times when the particle is at rest

d) the times when the particle is moving in the positive direction and in the negative direction

e) draw a diagram to illustrate the path of the particle

f) the time intervals when the particle is speeding up and slowing down

Example 2:

A pharmaceutical company is growing an organism to be used in a vaccine. The population grows according to the formula P(t) = (400)2t, where t is measured in hours.

a) What will be the number of organisms after 4 hours?

b) What is the rate of growth at 4 hours?

c) Find the equation of the tangent line at t = 4, and sketch it.

Example 3

The length of a certain plant after t weeks is

t

5

.

e

5

2

.

6

)

t

(

f

-

+

=

centimeters. Estimate answers to the following questions by studying the graphs of f, f and f, given below:

a) Identify each graph

b) When is the plant 10 centimeters long?

c) How fast is the plant growing after 10 weeks?

d) When is the plant growing at a rate of 2 cm/week?

e) What is the maximum rate of growth?

f) Find

)

t

(

f

lim

t

. Relate your answer to the graph and to the mature length of the weed.

Suggested Exercises Section 3.3 Pages 215218 1 15 (odds)

Section 3.4

The Derivatives of Trigonometric Functions

The graph of the trigonometric functions y = sin(x) and y = cos(x) are given below. On each graph, sketch an approximation of the graph of the derivative of the function.

Cos(x)

Sin(x)

In each case, use basic trig identities to put each function in terms of sin(x) and cos(x), then find the indicated derivative:

a) h(x) = tan(x)

b) h(x) = sec(x)

c) h(x) = cot(x)

d) h(x) = csc(x)

The results of the previous exercises are summarized in the following table:

In each case find

dx

dy

:

a) y = tan(x) - 2cos(x)

b) y = x2sin(x)

c) y = exsin(x)

d)

2

x

)

x

sin(

y

=

Exercise: The graph of h(x) = xcos(x) is given. Find the equation of the tangent line to the curve at the point ((, -(). Plot the tangent line.

Suggested Exercises: pp. 223-224: 1 11 odd, 17, 21 a)

Section 3.5

The Chain Rule

The chain rule is used for differentiating composite functions:

Exercise 1: In each case identify f and g so that h(x) = f(g(x)), then find h(x).

a) h(x) = sin(cos(x))

b) h(x) = (x4-2x + 7)10

c) h(x) = cos(x2)

d) h(x) = cos2(x)

e) h(x) = tan(ex)

f) h(x) =

2

x

e

g) h(x) =

5

2

1

x

+

Exercise 2: Find the derivative of each function

a) y = e-2xsin(4x)

b) y = sin( cos( tan(x ) ) )

c) y = sec(x2)tan(x5)

d) y =

2

x

)

x

sin(

e) y = sin5(x3)

f) y = tan(x2) + tan2(x)

Exercise 3 The graph of

x

3

4

8

y

+

=

is given. Find the equation of the tangent line to the curve at the point ( 4, 2 ). Plot the line.

Exercise 4: The graph of y = x2e-x is given.

a) Find

dx

dy

b) Find the critical points of the function and mark them on the graph.

c) Find the equation of the tangent line to the curve when x = 6. Plot the line.

Suggested Exercises: pg 233: #1 31 odd

Section 3.6

Implicit Differentiation

All the curves we have differentiated so far have been function in which it was possible to express y explicitly in terms of x. That is, the y (or f(x) ) was all by itself on one side of the equal sign, while all the xs were on the other side.

Some curves cannot be expressed in this fashion. For example, the circle x2 + y2 = 1 cannot be expressed in the form y = f(x). In such a case, we say that the relation between x and y is expressed implicitly by the equation x2 + y2 = 1.

The word implicit means implied or hidden. To understand how implicit differentiation works, imagine that an unknown function of x is hidden inside the y. So when we differentiate a term containing y, we have to use the chain rule.

Example: Find

dx

dy

for the implicit relation x2 + y2 = 25 by following the steps given:

a) Replace y with y(x) in the equation. (to remind yourself that y has a function of x hidden inside it)

b) Take the derivative of both sides with respect to x.

c) Solve the equation for y(x). (remember y(x) and

dx

dy

are the same thing).

d) Find the equation of the tangent line at the point (3,4). Sketch the line.

Exercise: In each case, find

dx

dy

by implicit differentiation.

a) y4 + y = 4x3

b) xy + 2x = 4y2 + 2

c) xy2 + y2 = 2

d) (x + y)2 = 2x

Example:

Find an equation of the tangent line to the curve 2(x2 + y2)2 = 25(x2 - y2) at

the point (3,1). Plot the tangent line.

Example:

a) The curve with the equation y2 = x3 + 3x2 is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point (1, -2). Plot the tangent line.

b) At what points does this curve have a horizontal tangent?

Suggested Exercises: pp 243 1 15 odd

Exercise: Recall that y = arctan(x), or equivalently, y = tan-1(x), means that tan(y) = x. In other words, arctan(x) is the angle whose tangent is x. For example, arctan(1) = 45( = (/2 radians, since tan(45() = 1. Find

(

)

)

x

arctan(

dx

d

by following the steps given:

a) Let y = arctan(x), then tan(y) = x. Find y by differentiating tan(y) = x implicitly.

b) Use the trig identity 1 + tan2(y) = sec2(y) and the fact that tan(y) = x to express y in terms of x only.

Exercise: Recall that y = arcsin(x), or equivalently, y = sin-1(x), means that sin(y) = x. In other words, arcsin(x) is the angle whose sine is x. For example, arcsin() = 30( = (/6 radians, since sin(30() = . Find

(

)

)

x

arcsin(

dx

d

by following the steps given:

c) Let y = arcsin(x), then sin(y) = x. Find y by differentiating sin(y) = x implicitly.

d) Use the trig identity sin2(y) + cos2(y) = 1 and the fact that sin(y) = x to express y in terms of x only.

1. In each case, find f(x)

a)

)

x

6

x

arctan(

)

x

(

f

2

-

=

b) f(x) = arctan(ex)

c)

)

x

(

sin

x

)

x

(

f

1

5

-

=

4. The graph of f(x) = arctan(x) is given below.

a) What is f(x)?

b) Find

)

x

(

'

f

lim

x

. Does your answer agree with the graph? Explain.

c) Use the graph to select the best answers to fill in the blanks below:

=

)

x

arctan(

lim

x

_______

=

-

)

x

arctan(

lim

x

_______

Select your answers from the numbers in the box:

Suggested exercises: pp 244. 27, 29, 31, 33

Section 3.7

Derivatives of Logarithmic Functions

Recall that if Loga(x) is a base-a logarithm, then: Loga(x) = y ( ay = x

In particular, if Loge(x) = ln(x) = y, then ey = x.

Exercise: Make up some examples using numbers for a, x and y to verify that Loga(x) = y ( ay = x

Exercise:

a) Differentiate both sides of ey = x using implicit differentiation.

b) Solve for y in terms of x only.

c) since y = ln(x), conclude that y =

(

)

y

dx

d

=

(

)

)

x

ln(

dx

d

= ____________

Exercise: Differentiate each function:

a) f(x) = ln(x2 + 5) b) f(x) = ln( sin(x) )

b) f(x) = sin( ln(x) )

c) f(x) = sin(x)ln(x)

Exercise: The graph of

x

)

x

ln(

)

x

(

f

=

is given.

a) Find f(x)

b) Find the equation of the tangent line to the curve at the point (1,0) and plot the line.

c) Find the equation of the tangent line to the curve at the point (e, 1/e) and plot the line.

Exercise: Find the derivative of each function:

a)

EMBED Equation.3

+

=

3

x

x

ln

y

b)

-

+

=

1

x

x

ln

y

2

c)

)

x

ln(

y

=

d)

))

x

ln(ln(

y

=

e)

)))

x

ln(ln(ln(

y

=

Suggested exercises: pp. 250 3 19 odd

Logarithmic Differentiation

This technique of logarithmic differentiation is used to simplify derivatives of powers, products and quotients. The basic idea is to use the laws of logarithms to turn products and quotients into sums and differences, and powers into products:

Exercise: (Review of the Laws of Logarithms)

For each of the three laws, write the equivalent law of exponents and write an example of how the law works:

Logarithmic Form

Exponential Form

1. Loga(NM) = Loga(N) + Loga(M) _________________________________

Example:

2. Loga(M/N) = Loga(M) - Loga(N) __________________________________

Example:

3. Loga(Mc) = c Loga(M) __________________________________

Example:

Exercise: Differentiate the function

8

2

3

2

)

7

x

5

(

1

x

3

x

y

-

-

=

using logarithmic differentiation. Follow the steps given:

a) Take the natural log of both sides of the equation

8

2

3

2

)

7

x

5

(

1

x

3

x

y

-

-

=

and use the laws of logarithms to eliminate products, quotients and powers (where possible) on the right-hand side of the equation.

b) Differentiate both sides implicitly with respect to x.

c) Solve the resulting equation for y.

Exercise: Find the derivative of each function using logarithmic differentiation:

a)

2

x

4

x

y

+

=

b)

)

3

x

)(

2

x

)(

1

x

(

y

-

-

-

=

c)

1

x

)

1

x

3

(

x

y

2

3

+

-

=

d) y = 3x

e) y = xtan(x)

f) y = (sin(x))x

Exercise: The graph of y = xx is given.

a) Use logarithmic differentiation to find y.

b) Find the equation of the tangent line at the point on the curve where x = 1

c) Find the value of x at which the curve has a horizontal tangent line.

Suggested exercises: pp 251 27 35 odd

Section 3.8

Linear Approximations

Imagine your phone is ringing. You answer it, and are told that you have just won a car in a raffle! Youre very excited. But before you can receive the car, you must answer the following skill-testing question:

Find

7

.

3

, accurate to 3 decimal places.

No problem! Clutching your portable phone, you dash around your house looking for a calculator. Your own calculator seems to be missing from of your backpack, but undaunted, you keep searching. Finally you locate your brothers old calculator in the back of the junk drawer in the kitchen. Breathing a sigh of relief, you turn it on and begin to search for the

button, or maybe a yx button. But there isnt one! In fact, this crummy old calculator can only add, subtract, multiply and divideit has no other functions!! What are you going to do now?

Well, youre in luck, because in this section we are going to learn how to approximate complicated functions such as square roots, logarithms, sines, cosines, etc, by using linear functions (straight lines). The advantage of linear functions is that they can be evaluated by just multiplying, dividing, adding and subtractingthe very operations that your brothers old calculator can do!

The straight lines we will be using are tangent lines, which are already familiar to you. You want to find

7

.

3

. Well, if we can find the equation of the tangent line to the curve

x

)

x

(

f

=

at some point close by to x = 3.7, then the tangent line should be able to help us solve the problem. Below we have the graph of

x

)

x

(

f

=

(solid line) together with its tangent line at the point x = 4 (dotted line). Notice how the two lines seem to be touching for values of x close to 4? Follow the steps below and you will be able approximate

7

.

3

using only that poor old calculator:

1. Find the equation of the tangent line to

x

)

x

(

f

=

at the point x = 4.

2. Write the equation in the form

y = mx + b (if you havent already)

Now replace the y with g(x). So now you have a new function, g(x), with the property that g(x) ( f(x) for values of x close to 4. In particular, g(3.7) ( f(3.7).

3. Got the idea? Now just evaluate g(3.7) using those +, -, (, and ( keys on that old calculator, and the car is yours!

Example

a) Find the equation of the linear approximation (tangent line) for the function f(x) = ln(3x 5) at the point x = 2, and write it in the form

g(x) = mx + b

b) Use the linear approximation, g(x), to estimate f(x) for the following values of x, and compare your estimates to the actual values of f(x) (produced with the help the ln function on your calculator).

i) x = 1.94

Linear estimate g(x) :___________ Actual value of f(x):__________

ii) x = 2.05

Linear estimate g(x) :___________ Actual value of f(x):__________

iii) x = 2.1

Linear estimate g(x) :___________ Actual value of f(x):__________

iv) x = 2.2

Linear estimate g(x) :___________ Actual value of f(x):__________

How far away from x = 2 is the linear approximation accurate to within 0.1? (use the graph to answer this)

2 Example

Let

3

x

)

x

(

f

=

.

a) Find the linear approximation (tangent line) of f(x) at x = 8 and write it in the form g(x) = mx + b

b) Use the linear approximation, g(x), to estimate f(x) for the following values of x, and compare your estimates to the actual values of f(x) (produced with the help of a cube root or power function on your calculator).

i) x = 5

Linear estimate g(x) :___________ Actual value of f(x):__________

ii) x = 6

Linear estimate g(x) :___________ Actual value of f(x):__________

iii) x = 9

Linear estimate g(x) :___________ Actual value of f(x):__________

iv) x = 14

Linear estimate g(x) :___________ Actual value of f(x):__________

Example

Let f(x) = sin(x). Find the linear approximation (tangent line) of f(x) at

x = 0 and use it to approximate sin(-1.2), sin(0.3), sin(-0.2) and sin(1). Compare the linear approximations to values given by a calculator that has trigonometric functions. (Angles are in radians).

Use the graph to estimate the values of x for which the linear approximation is accurate to within 0.1.

Suggested exercises: pg. 256 1, 3, 5

Section 1.7

Parametric Curves

So far we have used an equation with two variables, x and y, to describe a curve. Sometimes it is convenient to use a third variable t, and express both x and y as functions of t.

Example: suppose a particle in the plane moves in such a way that its coordinates x and y after t seconds are given by:

1

t

t

3

x

3

+

=

1

t

t

3

y

3

2

+

=

, 0 t 4 seconds

Complete the table, plot the points, and sketch the path along which the particle moves. Indicate what direction the particle is moving along the path.

t

x

y

0

.2

.4

.6

.8

1

1.4

1.6

2

3

4

Example: A particle moves in such a way that its position at time t seconds is given by:

t

x

=

, y = 3t 2, 0 t 4

Complete the table, plot the points, and sketch the path along which the particle moves. Indicate what direction the particle is moving along the path.

t

x

y

0

.2

.4

.6

.8

1

1.4

1.6

2

3

4

b) Eliminate the parameter t to express the curve as using x and y only.

Example: A particle moves in such a way that its position at time t seconds is given by:

)

t

cos(

2

x

=

, y = 2sin(t), 0 t

2

3

p

Complete the table, plot the points, and sketch the path along which the particle moves. Indicate what direction the particle is moving along the path.

t

x

y

0

.5

1

1.5

2.0

2.5

3

3.5

4

4.5

4.712

b) Eliminate the parameter t to express the curve as using x and y only. (hint: use the fact that sin2(t) + cos2(t) = 1

Example: In each case eliminate the parameter t to find an equation of the curve in terms of x and y.

a) x = 3t 1, y = 2t + 1

b) x = t + 1, y = t2

c) x = t3, y = t2

d) x = t 1,

1

t

t

y

-

=

e) x = 3cos(t), y = 3 sin(t)

f) x = 4cos(t), y = 2sin(t)

g) x = 2 + cos(t), y = 1 sin(t)

Tangents to Parametric Curves

Suppose a curve is defined by the parametric equations x = f(t) and y = g(t).

By the chain rule,

dt

dx

dx

dy

dt

dy

=

. Divide both sides by

dt

dx

to obtain:

=

dt

dx

dt

dy

dx

dy

Example: The graph of the parametric curve x = t sin(t), y = 1 cos(t) is given. This curve, called a cycloid, is traced by a fixed point on the edge of a rolling circle.

Find the equation of the tangent line to the curve at the point where t =

4

3

p

and plot the line.

Example: The spiral of Archimedes has parametric equations x = tcos(t), y = tsin(t). Find the equation of the tangent line at the point where t = 11.5 and plot the line.

Suggested exercises: pg. 81, 1 13 odd

Section 4.1

Related Rates

Related rates problems typically involve two quantities that vary with time and an equation that expresses some relationship between them. Usually the value of these quantities at some instant in time are given, together with all of their time rate of changes but one.

Example: A rocket is launched vertically and is tracked by a radar station, which is located on the ground 3 miles from the launch site. What is the vertical speed of the rocket at the instant when its distance from the radar station is 5 miles and this distance is increasing at the rate of 5000 miles per hour?

Example: A man 6 feet tall walks with a speed of 8 feet per second away from a street light atop an 18-foot pole. How fast is the tip of his shadow moving along the ground when he is 100 feet from the light pole?

Section 4.2

Maximum and Minimum Values

Definition: A function f has an absolute maximum (or global maximum) at c if f(c) ( f(x) for all x in the domain of f. The number f(c) is the maximum value of f.

Similarly, f has an absolute minimum (or global minimum) at c if f(c) ( f(x) for all x in the domain of f. The number f(c) is the minimum value of f.

Not every function has an absolute maximum or an absolute minimum. If a function does have an absolute maximum, it is the highest point of the graph of f. Similarly, an absolute minimum is the lowest point on the graph of f.

Exercise: Use the graph to find the absolute maximum or absolute minimum of each function on the interval given. If none exist, say so.

a) f(x) = x2 6x + 13 on the interval (-(,()

b) f(x) = -x2 + 6x 5 [0,6]?

c)

x

4

x

3

1

)

x

(

f

3

-

=

on the interval [-4, ()

Definition: A function f has a local maximum (or relative maximum) at c if f(c) ( f(x) when x is near c. Similarly, f has a local minimum at c if

f(c) ( f(x) when x is near c.

Exercise:

Use the graph to find the local maximum and minimum of the function

x

4

x

3

1

)

x

(

f

3

-

=

on the interval (-(,()

Note that the function has no absolute maximum or minimum on (-(,().

Exercise:

a) What is the maximum value of the function f(x) = x3 on the interval [0,2)?

b) What is the maximum value of the function f(x) = x3 on the closed interval [0,2]?

The Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers cand d in [a,b].

Definition: A critical number of a function f(x) is a number c in the domain of f such that either f(x) = 0 (flat spot) or f(x) does not exist (cusp).

Closed Interval Method:

To find the absolute maximum and minimum values of a continuous function f on the closed interval [a,b]:

1. Find the values of f at the critical numbers of f (if any exist) in (a,b)

2. Find the values of f at the endpoints a and b.

3. The largest of the values from steps 1 and 2 is the absolute maximum value and the smallest is the absolute minimum.

Exercise: For each function:

a) find the critical numbers (if any) on the given closed interval

b) find the absolute maximum and minimum values of f(x) on the given closed interval.

f(x) = x3 12x + 1 on [-3, 5]

2

x

9

)

x

(

f

-

=

on [-1, 2]

f(x) = 2cos(x) - x on [-(, (]

x

x

3

e

)

x

(

f

-

=

on [-1 , 0]

3

5

3

2

x

x

5

)

x

(

f

-

=

on [-1,4]

Suggested exercises: pg 276 277: 1, 3, 5, 15 43 odd

Section 4.3

Derivatives and the Shape of Curves

Example 1:

The graph of f(x) = 3x5 160x3 is given. (graph of f, too)

a) Use f to find the exact values of the critical points and mark them on the graph.

b) Find the intervals on which f'(x) > 0, and mark them on the graph. Verify that the Increasing/decreasing Test holds.

c) Find the intervals on which f'(x) < 0, and mark them on the graph. Verify that the Increasing/decreasing Test holds.

d) Verify that the First Derivative Test holds for this function.

Exercise: The graph of f(x) = esin(x) is given. (f and f are also given)

a) Find f, and use it to find the critical points on the interval for x = 0 to x = 7. Verify that the first derivative test holds.

b) Find f, and use it to find the inflection points and mark them on the graph. Mark the intervals where f is concave up and concave down.

e) Verify that the two test given in the boxes below hold for f(x) = esin(x)

Exercise: The graph of f(x) = 3x5 160x3 is given.

a) Find f and use it to find the inflection points of f

b) Find the intervals on which f''(x) > 0, and mark them on the graph.

f(x) should be concave up on these intervals.

c) Find the intervals on which f''(x) < 0, and mark them on the graph.

d) Use this example to verify the Second Derivative Test.

Example 2:

Let f(x) = x3 3x2 45x

a) find the zeros of f (where f(x) = 0)

b) find the critical numbers of f (where f'(x) = 0)

c) find the intervals on which f is increasing or decreasing

d) find the local maximum and minimum values of f

e) find the intervals of concavity and the inflection points (where f''(x) = 0)

f) use the information from parts a) to d) to sketch the graph of f

Suggested exercises: pg 288: 1,3, 5, 7 13 odd, 17, 19

Section 4.5

Indeterminate Form and LHospitals Rule

Recall in chapter 2 that we evaluated limits such as

6

x

5

x

10

x

3

x

lim

2

2

2

x

+

-

-

+

. Notice that when x is replaced by 2, we obtain

0

0

, which makes no sense and is undefined. Such a limit is called an indeterminate form of the type

0

0

. Recall that we also considered limits such as

3

x

7

x

2

3

x

2

x

3

lim

2

2

x

-

+

+

-

. When we allow x to approach infinity, the expression approaches

, which also makes no sense. Such a limit is called an indeterminate form of the type

.

LHopitals Rule

Suppose that f and g are differentiable functions and that g(x) 0 nearby the point a (although g(a) may be zero). Suppose also that:

)

x

(

g

lim

0

)

x

(

f

lim

a

x

a

x

=

=

or that:

=

)

x

(

f

lim

a

x

and

=

)

x

(

g

lim

a

x

Then:

EMBED Equation.3

)

x

(

'

g

)

x

(

'

f

lim

)

x

(

g

)

x

(

f

lim

a

x

a

x

=

provided the limit on the right exists (or is or -).

Exercise: Use LHopitals rule to evaluate the following limits. In each case check that it is appropriate to use LHopitals rule.

a)

6

x

5

x

10

x

3

x

lim

2

2

2

x

+

-

-

+

b)

3

x

7

x

2

3

x

2

x

3

lim

2

2

x

-

+

+

-

c)

x

)

x

sin(

lim

0

x

d)

)

x

2

sin(

1

e

lim

x

0

x

-

e)

2

0

x

x

x

)

x

sin(

lim

+

f)

2

x

x

x

x

e

lim

+

g)

x

)

x

ln(

lim

x

h)

)

x

cos(

x

3

)

x

sin(

x

lim

x

+

+

i)

4

x

)

x

cos(

2

x

lim

2

2

x

-

p

-

j)

)

x

cos(

1

)

x

ln(

x

1

lim

1

x

p

+

+

-

Exercise: The following are indeterminate products of the form 0. Convert them to quotients and evaluate:

a)

)

x

cot(

x

lim

0

x

b)

x

x

xe

lim

-

c)

+

-

1

x

1

x

ln

x

lim

x

Suggested exercises: pg 305, 1, 5 17 odd

Section 4.6

Optimization Problems

Example 1:

Find the point on the line 2x + y = 3 that is closest to the point (3,2).

a) What quantity is to be optimized? Is it to minimized or maximized?

b) Let (x, y) be the point you are looking for. Express the quantity that is to be optimized as a function f of the x coordinate of the point (x, y)

c) Sketch a graph of f using the techniques of sections 4.3., and use it to determine: i) the value of x that optimizes the function

ii) the optimum value of the function

Example 2:

Suppose 11 metres of fencing are available to build a rectangular pen against an existing wall. What are the dimensions of the rectangle that will maximize the area of the pen?

Example 3:

You must construct a closed rectangular box with volume 576 in.3 and with its base twice as long as it is wide. Find the dimensions of the box that will minimize its total surface area.

Example 4:

Each page of a book will contain 30 in.2 of print, and each page must have 2-in. margins at top and bottom and 1-in Margins at each side. What is the minimum possible area of such a page?

Example 5:

An open-topped cylindrical pot is to have volume 250 cm3. The material for the bottom of the pot costs 4 cents per cm2 and that for the its curved sides costs 2 cents per cm2. What dimensions will minimize the total cost of this pot?

h

P

Q

s distance in feet

t time in seconds

Definition of Limit

We write: EMBED Equation.3

and we say: the limit as x approaches a,of g(x), is L

provided g(x) gets closer and closer to L as x gets closer and closer to a, but not necessarily equal to a.

P

Q

Q

P

h

a

X((

f(x)

2

5

11

101

1001

10001

All limits may be divided into two basic types:

1. Limits at a finite number: x ( a

2. Limits at infinity: x (( or x (-(

Limits at a finite number: EMBED Equation.3

To begin, evaluate f(a). There are three possibilities:

f(a) is a real number

In this case you are done.

f(a) = EMBED Equation.3 . In this case, simplify f(x) and try evaluating f(a) again. If a real number results, then you are done

f(a) = EMBED Equation.3 , where b ( 0.

In this case the value of the limit is either ( or -(. To determine which, evaluate f(x) at a number very close to a (on the correct side, for one-sided limits). Either a large positive or a large negative number should result. The value of the limit is ((, accordingly

Limits at Infinity: EMBED Equation.3

We wish to know how the function behaves at values of x that are a long way away from zero. This can be accomplished by the following steps:

Divide every term in the numerator and denominator by the highest power of x that appears in the denominator.

Simplify each of the little sub-fraction you created in part 1.

Note that any fraction of the form EMBED Equation.3 , where b is any number and n is a positive number, will approach zero as x ( ((. So replace all such fractions with zero and evaluate the remaining expression.

1

3

2

A note about notations:

There are a variety of different notations to represent a derivative. All of the following denote the derivative of a function y = f(x):

f(x) EMBED Equation.3 EMBED Equation.3 y f EMBED Equation.3

Power Rule

If n is any number and f(x) = xn,

then f(x) = nxn-1

Derivative of a Linear Combination

If f and g are functions and a and b are constant numbers, then

EMBED Equation.3

Derivative of Sin(x) and Cos(x):

EMBED Equation.3 EMBED Equation.3

Derivatives of the Trigonometric Functions

EMBED Equation.3 EMBED Equation.3

EMBED Equation.3 EMBED Equation.3

EMBED Equation.3 EMBED Equation.3

The Chain Rule

If f and g are both differentiable functions, then EMBED Equation.3

From the previous two exercises, conclude that:

EMBED Equation.3 and EMBED Equation.3

(-(1-1(/2 -(/2 (/3 -(/30

Radar station

18 ft

EMBED Word.Picture.8

Increasing/Decreasing Test

a) if f(x) > 0 on an interval, then f is increasing on that interval

b) if f(x) < 0 on an interval, then f is decreasing on that interval

First Derivative Test:

Suppose that c is a critical number of a continuous function f.

a) if f changes from positive to negative at c, then f has a local maximum at c

b) If f changes from negative to positive at c, then f has a local minimum at c.

c) If f does not change sign at c (that is, f is positive on both sides of c or negative on both sides), then f has no local maximum or minimum at c.

Concavity Test

a) If f(x) > 0 for all x in an interval I, then the graph of f is concave upward on I

b) If f(x) < 0 for all x in an interval I, then the graph of f is concave downward in I.

Second Derivative Test

Suppose f is continuous near c.

a) If f(c) = 0 and f(c) > 0, then f has a local minimum at c.

b) If f(c) = 0 and f(c) < 0, then f has a local maximum at c.

( Suggested Exercise: #1 page 99

(

(

( Suggested Exercises: Section 2.10 page 178-180: 1, 3, 11, 13, 15, 17, 23

-1-

_1046177922.unknown

_1093333310.unknown

_1099467497.unknown

_1129719561.unknown

_1130562777.unknown

_1185618827.unknown

_1185618838.unknown

_1185620311.unknown

_1185620326.unknown

_1185619442.unknown

_1185618835.unknown

_1130562901.unknown

_1130564396.unknown

_1130562824.unknown

_1129720527.unknown

_1129721956.unknown

_1129720497.unknown

_1099468839.unknown

_1099470388.unknown

_1129718245.unknown

_1129718247.unknown

_1099470679.unknown

_1099469038.unknown

_1099469370.unknown

_1099469960.unknown

_1099470053.unknown

_1099469859.unknown

_1099469245.unknown

_1099468905.unknown

_1099468346.unknown

_1099468638.unknown

_1099468712.unknown

_1099468373.unknown

_1099468224.unknown

_1099468284.unknown

_1099468098.unknown

_1099468109.unknown

_1099468093.unknown

_1099467646.unknown

_1093421423.unknown

_1093421523.unknown

_1099467290.unknown

_1099467348.unknown

_1093421529.unknown

_1093421459.unknown

_1093421520.unknown

_1093421429.unknown

_1093421257.unknown

_1093421412.unknown

_1093421417.unknown

_1093421397.unknown

_1093421243.unknown

_1093421255.unknown

_1093333330.unknown

_1046436125.unknown

_1093333180.unknown

_1093333256.unknown

_1093333286.unknown

_1093333238.unknown

_1047306513.unknown

_1093333061.unknown

_1093333178.unknown

_1047307054.unknown

_1093333047.unknown

_1046436466.unknown

_1047306318.unknown

_1046436220.unknown

_1046432901.unknown

_1046432971.unknown

_1046433656.unknown

_1046432952.unknown

_1046432766.unknown

_1046432861.unknown

_1046432720.unknown

_1041173697.unknown

_1044796395.unknown

_1046106302.unknown

_1046177129.unknown

_1046177192.unknown

_1046107430.unknown

_1046107439.unknown

_1046107003.unknown

_1044799202.unknown

_1045574572.unknown

_1044798682.unknown

_1044441671.unknown

_1044795831.unknown

_1044795857.unknown

_1044445097.unknown

_1044795622.unknown

_1044420278.unknown

_1044441429.unknown

_1044441457.unknown

_1044441467.unknown

_1044441546.unknown

_1044441439.unknown

_1044441447.unknown

_1044420312.unknown

_1043583096.unknown

_1043583977.unknown

_1041173698.unknown

_1040733453.unknown

_1040738453.unknown

_1041172071.unknown

_1041173695.unknown

_1041173696.unknown

_1041172517.unknown

_1041173694.unknown

_1041172686.unknown

_1041172156.unknown

_1040738710.unknown

_1041171940.unknown

_1040738558.unknown

_1040738019.unknown

_1040738342.unknown

_1040737875.unknown

_988050549.unknown

_1020061811.unknown

_1021652991.unknown

_1040628960.unknown

_1040731774.unknown

_1040733150.unknown

_1040731736.unknown

_1021656367.unknown

_1021238795.unknown

_1021648925.unknown

_1021649412.doc

_1021238652.unknown

_1020061435.unknown

_1020061602.unknown

_1019934048.unknown

_1020061418.unknown

_1019934021.unknown

_1019933981.unknown

_987786525.unknown

_987788803.unknown

_987919740.unknown

_987786859.unknown

_987783970.unknown

_987784372.unknown

_987786445.unknown

_987401686.unknown