1

2

AP Calculus Applications of Derivatives

2015­11­03

www.njctl.org

3

Table of Contents

Related RatesLinear MotionLinear Approximation & Differentials

click on the topic to go to that section

L'Hopital's Rule

Horizontal Tangents

4

Related Rates

Return to Table of Contents

Teacher N

otes

5

Related Rates is the application of implicit differentiation (which we learned in the previous unit) to real life situations.

In simplest terms, related rates are problems in which you need to figure out how fast one variable is changing when given the rate of change of another variable at a specific point in time.

For example, if a spherical balloon is being filled with air at a rate of 20 ft3/min, how fast is the radius changing when the

radius is 2 feet?

Related Rates

Teacher N

otes

6

Before we attempt a Related Rates example, let's practice a few implicit differentiation examples first.

Differentiate each equation with respect to time, t.

Recall: Implicit Differentiation

Answer

7

1) Draw a picture. Label the picture with numbers if constant or variables if changing.

2) Identify which rate of change is given and which rate of change you are being asked to find.

3) Find a formula/equation that relates the variables whose rate of change you seek with one or more variables whose rate of change you know.

4) Implicitly differentiate with respect to time, t.

5) Plug in values you know.

6) Solve for rate of change you are being asked for.

7) Answer the question. Try to write your answer in a sentence to eliminate confusion.

Helpful Steps for Solving Related Rates Problems

Teacher N

otes

Emphasize to students:WARNING! Most mistakes are made by subsituting the given

values too early. You must wait until after you differentiate!

*Note on Step 4: Occasionally, students may see a question where they need to differentiate with respect to a different variable; however, most often it will be time.

8

Step 3 requires you to think of an equation to relate variables. Some questions on the AP Exam will provide the equation for you, but if not, think of:

– trigonometry– similar triangles– Pythagorean theorem– common Geometry equations

Step 3

9

Let's take a look back at this example...

If a spherical balloon is being filled with air at a rate of 20 ft3/min, how fast is the radius changing when the

radius is 2 feet?

1) Draw and label a picture. 2) Identify the rates of change you know and seek.3) Find a formula/equation.4) Implicitly differentiate with respect to time, t.5) Plug in values you know. 6) Solve for rate of change you are being asked for.7) Answer the question.

Example

Answer

10

In the last question we answered the following:

The radius is increasing at a rate of when the radius is 2 feet.

Why is it important to write a sentence for an answer?

On the AP Exam, Related Rates questions are graded very critically. Graders will not award points without proper vocabulary usage (i.e. increasing or decreasing rate of change), appropriate units, and the actual correct answer. Take time when formulating your answer to

make sure it makes logical sense and includes all needed information.

11

Hands­On Related Rates Lab(OPTIONAL)

Click here to go to the lab titled "Related Rates"

Teacher N

otes

12

Hands­On Related Rates (OPTIONAL)Items needed: • 2 students • 1 long rope/cord/string (at least 15 feet for best display)• masking tape

Set up masking tape in a right angle classroom with enough room for each student to walk along the tape line.

Teacher N

otes

STEP #1

13

A B

Student A begins at the end of one piece of tape, and Student B begins in the corner. Each student holds one end of the rope

until it is taught.

Hands­On Related Rates (OPTIONAL)

STEP #2

14

B

A

It is imperative that student B walks at a CONSTANT and slow pace forward while student A simple walks at whatever pace needed to keep the rope taught. The class should watch Student A's rate of change over the course of his/her path. It may take several attempts to observe the result.

Hands­On Related Rates (OPTIONAL)

STEP #3

15

A balloon is rising straight up from a level ground and tracked by a range finder 500 feet from lift off point. At the moment the range finder's elevation reads the angle is increasing at a rate of 0.14 radians/minute. How fast is the balloon rising at that moment?

Example

Answer

16

A bag is tied to the top of a 5m ladder resting against a vertical wall. Supposed the ladder begins sliding down the wall in such a way that the foot of the ladder is moving away from the wall at a constant rate of 2m/s. How fast is the bag descending at the instant the foot of the ladder is 4m from the wall?

Answer 3. Find an appropriate equation.

4. Differentiate with respect to t.

5. Substitute given values.

6. Solve for

7. Answer the question. *Note what the negative rate of change means in this example.

Example

17

Water is pouring into an inverted conical tank at 2 cubic meters per minute. The tank is a right circular cone with height 16 meters and base radius of 4 meters. How fast is the water level rising when the water in the tank is 5 meters deep?

CHALLENGE!

Answer

3. Find an appropriate equation.

4. Differentiate with respect to t.

5. Substitute given values.

6. Solve for

7. Answer the question.

Example

18

1

A

B

C

D

E

A person 6 feet tall is walking away from a streetlight 20 feet high at the rate of 7 ft/sec. At what rate is the length of the person's shadow increasing?

The shadow is increasing at a rate of 3 ft/sec.

The shadow is increasing at a rate of 3/7 ft/sec.The shadow is increasing at a rate of 7/3 ft/sec.

The shadow is increasing at a rate of 14 ft/sec.

The shadow is increasing at a rate of 7 ft/sec.

Answer

19

2

A

B

C

D

E

Water leaking onto a floor forms a circular pool. The radius of the pool increases at a rate of 4 cm/min. How fast is the area of the pool increasing when the radius is 5 cm?

The area of the circle is increasing at a rate of cm2/min when the radius is 5cm.

The area of the circle is increasing at a rate of cm2/min when the radius is 5cm.The area of the circle is increasing at a rate of cm2/min when the radius is 5cm.The area of the circle is increasing at a rate of cm2/min when the radius is 5cm.The area of the circle is increasing at a rate of cm2/min when the radius is 5cm.

Answer

20

3

A

B

C

D

E

Two people are 50 feet apart. One of them starts walking north at a rate so that the angle formed between them is changing at a constant rate of 0.01 rad/min. At what rate is the distance between the two people changing when

radians ?

Calculator OK

The distance between the people is increasing at a rate of 0.311 ft/min when radians

The distance between the people is increasing at a rate of 0.004 ft/min when radians

The distance between the people is increasing at a rate of 0.01 ft/min when radians The distance between the people is increasing at a rate of 0.006 ft/min when radians The distance between the people is increasing at a rate of 0.05 ft/min when radians

Answer

21

4 A trough of water is 8 meters long and its ends are in the shape of isosceles triangles whose width is 5 meters and height is 2 meters. If water is being pumped in at a constant rate of 6 m3/sec. At what rate is the height of the water changing when the water has a height of 120 cm?

A

B

C

D

E

The height of the water is increasing at a rate of 0.25 m/sec when the water is 120cm high.The height of the water is increasing at a rate of 40 m/sec when the water is 120cm high.

The height of the water is increasing at a rate of 6 m/sec when the water is 120cm high.

The height of the water is increasing at a rate of 0.3 m/sec when the water is 120cm high.

The height of the water is increasing at a rate of 20 m/sec when the water is 120cm high.

Answer

22

5 The sides of the rectangle pictured increase in such a way that and . At the instant where x=4 and y=3, what is the value of

A B C D E

zy

x Answer

23

6 If the base, b, of a triangle is increasing at a rate of 3 inches per minute while it's height, h, is decreasing at a rate of 3 inches per minute, which of the following must be true about the area, A, of the triangle?

A B

C

D E

A is always increasing.A is always decreasing.

A is decreasing only when b < h.A is decreasing only when b > h.A remains constant.

Answer 3. Find an appropriate equation.

4. Differentiate with respect to t.

5. Substitute given values.

6. Solve for

7. Answer the question.

24

7 The minute hand of a certain clock is 4 in. long. Starting from the moment that the hand is pointing straight up, how fast is the area of the sector that is swept out by the hand increasing at any instant during the next revolution of the hand? Note: Area of a sector

Answer

25

Linear Motion

Return to Table of Contents

26

Another useful application of derivatives is to describe the linear motion of an object in two dimensions, either left and right, or up and down. This is a concept where calculus is

extremely applicable. We will revisit this topic again in the next unit involving graphing, and again in the unit about integrals!

Linear Motion

27

A remarkable relationship exists among the position of an object, the velocity of an object and the acceleration of an object.

First... let's review what each of these words mean.

Position

Velocity

Acceleration

Position, Velocity & Acceleration

Teacher N

otes

28

Are Velocity and Speed the Same Thing?

Although you may hear velocity and speed interchanged often in common conversation, they are, in fact, 2 distinct quantities. Sometimes they are equivalent to each other, but this depends on the direction of the object.

Velocity is a vector quantity meaning it has both magnitude and direction.

For example, if the velocity of an object is ­3 feet per second, then that object is moving backwards or to the left (direction) at a rate of 3 feet per second (magnitude).

Teacher N

otes

Note: The positive or negative direction is

determined by the object's initial position and what is

determined to be a positive/negative direction.

29

Similarly, there is a difference between distance and position.

Distance is how far something has traveled in total; distance is a quantity.

Whereas position is the location of an object compared to a reference point; position is a distance with a direction.

Distance vs. Position

30

is the notation for our position function

is the notation for our velocity function

is the notation for our acceleration function

Typical Notation for Linear Motion Problems

31

Consider driving your car along the highway. The time it takes you to travel from mile marker 27 to mile marker 105 is an hour and a half. How fast were you driving?

Example

Answer

Important: This is the average velocity. It does not necessarily mean you were

traveling 52mph the entire time.

32

We know that the average velocity can be found by dividing the distance traveled by the time; however, how can we find the instantaneous velocity (how fast you are traveling at a specific

moment in time)?

Because we are interested in the instantaneous rate of change of a position, we are able to take the derivative of the position function and find the instantaneous velocity.

Note: This requires a position function to be given.

Average Velocity vs. Instantaneous Velocity

33

The velocity of the object is given by:

Therefore, if given a position function, x, as a function of t.

Furthermore, the acceleration of the object is:

is also commonly used for position

Position, Velocity & Acceleration

Teacher N

otes Discuss units with students:

position: m, ft, cm, yds., in., milesvelocity: m/s, ft/s, miles/hracceleration: m/s2, ft/s2, miles/hr2

34

A race car is driven down a straight road such that after seconds it is feet from its origin.

a) Find the instantaneous velocity after 8 seconds.

b) What is the car's acceleration?

Example

Answer

35

A spring is pulled to 6 inches below its resting state and bounces up and down. Its position is modeled by .

a) Find its velocity and acceleration at time t.

b) Find the spring's velocity and acceleration after seconds.

Example

Answer

36

A dynamite blast shoots a rock straight up into the air. Its height at any given time is feet after t seconds.

a) How high does the rock travel?

b) What is the velocity and speed of the rock when it is 256 feet above ground?

c) What is the acceleration at any time, t?

d) When does the rock hit the ground?

Answer

a) How high does the rock travel?

b) What is the velocity and speed of the rock when it is 256 feet above ground?

c) What is the acceleration at any time, t?

d) When does the rock hit the ground?

*Students may struggle with comprehension and visualization of this problem, which is more like questions seen on the AP Exam. Work slowly and check for understanding frequently.

When the rock reaches its peak, the velocity will be equal to 0, then we can find the position at that time.

We first must find at what time the position is 256ft, and then find v(time).

Acceleration is the 2nd derivative of position, and the 1st derivative of velocity.

When the rock hits the ground its position will be equal to 0 feet. At t=0, that is it's starting position.

Example

37

One More Reminder!

What is the difference between:

Average Velocity Instantaneous Velocity

Teacher N

otes

Sometimes when students begin practicing questions involving instantaneous velocity they forget how to calculate average velocity. Take a minute to reiterate the difference.

In simple terms:

Average Velocity ­ slope formula with 2 points

Instant. Velocity ­ derivative evaluated at 1 point.

38

8

A

B

C

D

E

A particle moves along the x­axis so that at any time t>0 seconds its velocity is given by m/s. What is the acceleration of the particle at time ?

Answer B

39

9

A

B

C

D

E

A particle moves along the x­axis so that at any time t>0 minutes its position is given by . For what values of t is the particle at rest?

No values

only

only

only

Answer

Note: Discuss why t=­1 is an extraneous answer (no negative time)

40

10

A

B

C

D

E

The position of a particle moving along a straight line at any time t is given by . What is the acceleration of the particle when t=4?

Answer D

41

11

A

B

C

D

E

A mouse runs through a straight pipe such that his position at any time is inches. Find the average velocity during the first 5 seconds.

Answer A

42

12

A

B

C

D

E

An object moves along the x­axis so that at time t>0 its position is given by meters. Find the speed of the object at t=3 seconds.

Answer

E*Note: question

asks for speed, not velocity.

43

13 A rock thrown vertically upward from the surface of the moon at a velocity of 24m/s reaches a height of

meters in t seconds. Find the rock's acceleration as a function of time.

Answer

44

14 A rock thrown vertically upward from the surface of the moon at a velocity of 24m/s reaches a height of

meters in t seconds. Find the rock's average velocity during the first 3 sec. A

nswer

45

15 A rock thrown vertically upward from the surface of the moon at a velocity of 24m/s reaches a height of

meters in t seconds. Find the rock's instantaneous velocity at t=3 sec.

Answer

46

Linear Approximation& Differentials

Return to Table of Contents

47

In the last unit we explored what it meant for a differentiable function to be "locally linear". Also in the previous unit, we

discussed how to find the equation of a tangent line to a function. In this section, we will expand on those ideas and how they become

useful in a topic called Linear Approximation.

Linear Approximation

48

Let's consider the graph of

If asked to evaluate we could quickly conclude the answer is 3.However, if asked to evaluate , we know the answer is near 3 but don't have a very accurate estimate. Linear approximation allows us to better estimate this value using a tangent line.

Zoomed in

What is the Purpose of Linear Approximation?

49

Observe the black tangent line to the function at x=9.

If we write the equation of the tangent line at x=9, we can then use this line and substitute 8.9 into our equation to find an approximation for f(8.9). Again, it won't be exact, but will be much closer than just saying 3.

Linear Approximation

50

To Find the Linear Approximation:

not wholewhole whole

not wholewhole

For the sake of understanding we will refer to as the "whole" number (i.e. x=9), and as the "not whole" number (i.e. x=8.9)

Teacher N

otes

This formula can sometimes confuse students as they forget which values are x & a.

It may be more meaningful for students to write the tangent line equation for the nice "whole" number, then simply substitute the other value into their tangent line equation.

Show students that technically, they are the same method and same equation; however, they are free to use whichever "method" makes more sense to them.

51

Practice: Use linear approximation to approximate the value of f(8.9).

Example

Answer

52

Is our approximation greater than or less than the actual value of f(8.9)? Why or why not?

Example, Continued

Teacher N

otes

Often students have a misconception about why the approx. is high or low. Remind them that it depends on whether or not the tangent line lies above or below the curve at the point of interest, not simply whether one number is larger or smaller than

the other.

53

Given , approximate .

Example

Answer

54

Given , approximate .

Example

Answer

55

16 Given Approximate

Answer

56

Recall

17 For the previous question, is the approximation of greater than or less than the actual value? You may look at a graph of the function to decide.

A Greater than

B Less than Answer

The approximation is greater than the actual value in this case, because at the point in consideration, the tangent line would lie above the curve, thus producing a high approximation. Note: Students haven't yet learned the concept of concavity, however you can mention it to them to foreshadow.

57

18 Given and approximate the value of

Answer

58

19 Given Approximate

Answer

59

20 Find the approximate value of using linear approximation.

Answer

Let

Then

60

21 Given and approximate the value of

Answer

61

22 Approximate the value of

Answer Let

Then

62

Differentials

So far we have been discussing and , but sometimes in

calculus we are interested in only . We call this the differential.

The process is fairly simple given we already know how to find .

This is called differential form.

63

Let's try an example: Find the differential .

Differentials

Answer

64

Now let's practice with given values to substitute...

Given find when and

Practice

Answer Substitute given values...

65

Note the difference between and . If we calculate both, we can then compare the values to calculate the percentage change or approximation error.

vs.

66

The radius of a circle increases from 10 cm to 10.1 cm. Use to estimate the increase in the circle's Area, . Compare this estimate with the true change, , and find the approximation error.

Example

Answer

Therefore, the approximation error is:

67

23 Find the differential if

Answer

68

24 Find the differential if

Answer

69

25

A

B

C

D

E

F

Find and evaluate for the given values of and .

Answ

er

70

26

A

B

C

D

E

F

Find and evaluate for the given values of and .

Answ

er

71

27 Given , , and calculate the estimated change .

Answer

72

28 Given , , and calculate the true change .

Answer

73

29 Given , , and calculate the approximation error.

Answer

Approximation error is

74

L'Hopital's RuleReturn to

Table of Contents

75

One additional application of derivatives actually applies to solving limit questions!

L'Hopital's Rule

76

Cool

Fact!

In the 17th and 18th centuries, the name was commonly spelled "L'Hospital",

however, French spellings have been altered and the silent 's' has been dropped.

L'Hopital's Rule(pronounced "Lho­pee­talls")

Guillaume de L'Hopital was a french mathematicion from the 17th century. He is known most commonly for his work calculating limits involving indeterminate forms and . L'Hopital was the first to publish this notion, but gives credit to the Bernoulli brothers for their work in this area.

77

Recall from the limits unit when we woud try to substitute our value into the limit expression and it would result in either or , known as indeterminate forms.

As we know, upon substitution, this results in and indeterminate form, and our previous method was to factor, reduce and substitute again.

Indeterminate Form

78

L'Hopital discovered an alternative way of dealing with these limits!

L'HOPITAL'S RULESuppose you have one of the following cases:

or

Then,

Teacher N

otes

Sometimes students will attempt to use the quotient rule on these problems. Emphasize that the original question is asking for a limit, and L'hopital's rule deals with the numerator and denominator as two distinct functions and differentiates each separately.

79

What does this mean?• You now have an alternative method for calculating these indeterminate limits.

Why didn't you learn this method earlier? • You didn't know how to find a derivative yet!

L'Hopital's Rule

80

Let's try L'Hopital's Rule on our previous example:

Example

Answer

81

Evaluate the following limit:

Example

Answer

82

Evaluate the following limit:

Note: L'Hopital's Rule can be applied more than one time, if needed.

Example

Answer

Applying L'Hopital's Rule...

We can apply L'Hopital's Rule again!

83

Important Fact to Remember:

ONLY use L'Hopital's Rule on quotients that result in an indeterminate form upon substitution.

Using the rule on other limits may, and often will, result in incorrect answers.

84

30

A

B

C

D

E

Evaluate the following limit:

Answ

er

85

31

A

B

C

D

E

Evaluate the following limit:

Answ

er

86

32

A

B

C

D

E

Evaluate the following limit:

Answ

er Discuss with students why they cannot apply L'Hopital's Rule on

this problem.

87

33

A

B

C

D

E

Evaluate the following limit:

Hint: Sometimes it is helpful to rewrite before applying L'Hopital's Rule.

Answ

er

88

34

A

B

C

D

E

Evaluate the following limit:

Answ

er Students may recall the shortcut of using the highest power's coefficients, or may apply L'Hopital's rule twice.

89

Horizontal TangentsReturn to

Table of Contents

90

Recall what it means to be tangent to a function. We could draw an infinite amount of tangent lines below; however, looking at the ones given what observations can you make about the black tangent lines?

Tangent Lines

Teacher N

otes

Allow students to make observations, and discuss with

classmates. The desired observation is that they recognize the black tangent lines are the only ones that are horizontal, or have a

slope of zero.

91

Do you think there is a way to find out where the horizontal tangents are occurring aside from just estimating?

Horizontal Tangents

Teacher N

otes

92

Let's try an example...

At what x­value(s) does the following function have a horizontal tangent line?

Example

Answer

Students may also recognize the correlation between the location of the horizontal tangent and the vertex of the parabola.

93

At what point(s) does the following function have a horizontal tangent line?

***Note the alternative wording. Pay attention on the AP Exam! Some questions will only ask for the x­value, but if you are asked at what point(s) the function has horizontal tangent lines, you need both the x­ and y­coordinates.

Example

Answer

94

At what x­value(s) does the following function have a horizontal tangent line?

Example

Answer

It may also be helpful to have students check the graph of this function to visualize

their answer.

95

35 At what x­value(s) does the following function have a horizontal tangent line?

A

B

C

D

E

F

G No Horizontal Tangents

Answ

er

96

36 At what x­value(s) does the following function have a horizontal tangent line?

A

B

C

D

E

F

G No Horizontal Tangents

Answer Some students may struggle with

solving for x, you may remind them that for the fraction to equal zero, the numerator must equal zero.

97

37 At what point(s) does the following function have a horizontal tangent line?

Answ

er

98

38 At what x­value(s) does the following function have a horizontal tangent line?

A

B

C

D

E

F

G No Horizontal Tangents

Answ

er

Students may view on graphing calculators for confirmation.

99

39 At what x­value(s) does the following function have a horizontal tangent line?

A

B

C

D

E

F

G No Horizontal Tangents

Answ

er

100

40 At what point(s) does the following function have a horizontal tangent line?

Answ

er

101

41 At what x­value(s) does the following function have a horizontal tangent line?

A

B

C

D

E

F

G No Horizontal Tangents

Answ

er

**Derivative is given rather than function!