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AP Calculus

Applications of Derivatives

2015-11-03

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Table of Contents

Related RatesLinear Motion

Linear Approximation & Differentials

click on the topic to go to that section

L'Hopital's Rule

Horizontal Tangents

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Related Rates

Return to Table of Contents

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Related Rates


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Related Rates is one of the topics which students struggle with more than any other. Take the necessary

time on each question for students to comprehend and visualize the

situation. Highly encourage them to draw pictures and work slowly but

efficiently through the problem.

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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

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(Answer) / 101

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


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Students can usually comprehend the balloon example, understanding

that although the air is being pumped in at a constant rate, the

radius changes very quickly at first and then slows down as the balloon

gets larger. There is no need to solve this example at this time, it is

just to give them an idea of the types of problems they will

experience.

Slide 6 / 101

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


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


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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


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


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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.

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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

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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


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


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rKnow:

Want: when r=2

3. Equation that relates volume of a sphere with radius of a sphere.

4. Differentiate with respect to t.

1. Picture 2. Identify Rates of Change

5. Substitute given values.

6. Solve for

7. Answer the question.

The radius is increasing at a rate of

when the radius is 2 feet.

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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.

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Hands-On Related Rates Lab(OPTIONAL)

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


Hands-On Related Rates Lab(OPTIONAL)

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


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It may take several attempts for students to observe the result. It helps is Student A does not watch Student B, but just walks at whatever pace needed

to keep the rope taut.

Desired observation: Student B is walking at a constant pace, Student A begins at a slow rate, but then the rate increases as they approach the corner.

URL for Lab: http://njctl.org/courses/math/ap-calculus-ab/application-of-derivatives/hands-on-related-rates/

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Hands-On 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.

STEP #1

http://njctl.org/courses/math/ap-calculus-ab/application-of-derivatives/hands-on-related-rates/

http://njctl.org/courses/math/ap-calculus-ab/application-of-derivatives/hands-on-related-rates/

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Hands-On 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.

STEP #1[This object is a pull tab]

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As said before, it may take several attempts for students to observe the result. It helps is Student A does not watch Student B, but just walks at

whatever pace needed to keep the rope taught.

Desired observation: Student B is walking at a constant pace, Student A begins at a slow rate, but then the rate increases as they approach the corner.

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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.

Hands-On Related Rates (OPTIONAL)

STEP #2

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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.

Hands-On Related Rates (OPTIONAL)

STEP #3

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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

Slide 15 (Answer) / 101 Slide 16 / 101

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?

Example

(Answer) / 101 Slide 17 / 101

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!Example


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.




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.

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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.


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Know:

Want: when r = 5

3. Find an appropriate equation.




6. Solve for


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

r

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Slide 20 (Answer) / 101 Slide 21 / 1014 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.


Slide 21 (Answer) / 1014 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 0.3 m/sec when the water is 120cm high.



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Know:

Want: when h=120cm

use 1.2m





6. Solve for

7. Answer the question. Choice C: The height of the water is rising at a rate of 0.25 m/s when the water is 120cm high.

w

h

8

2

5

*use similar triangles to express w in terms of h

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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

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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


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Know:

Want: when x=4 & y=3





6. Solve for


Choice B. 1

zy

x

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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?

AB

C

DE

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.


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?

AB

C

DE

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.[This object is a pull tab]

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Know:

Want:





6. Solve for


Choice D: The area is decreasing only when b>h.

b

h

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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


Linear Motion


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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

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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


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

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Have students share thoughts and definitions for each term. Be certain to listen for students referring to velocity as speed. The next slide will clarify the difference.

Position - location of an object in regards to its starting location

Velocity - how fast AND in what direction an object is moving

Acceleration - how fast AND in what direction the velocity is changing

Slide 28 / 101

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).


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).


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Note: The positive or negative direction is

determined by the object's initial position and what is

determined to be a positive/negative direction.

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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

/ 101

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

Slide 31 / 101

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


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


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

traveling 52mph the entire time.

distance

time

Slide 32 / 101

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

Slide 33 / 101 Slide 33 (Answer) / 101

/ 101

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


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



Example




constant acceleration

Slide 35 / 101

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


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



Example




Slide 36 / 101

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?

Example


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





Example


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*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.

/ 101

One More Reminder!

What is the difference between:

Average Velocity Instantaneous Velocity


One More Reminder!

What is the difference between:

Average Velocity Instantaneous Velocity


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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.

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8

A

BC

DE

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


8

A

BC

DE

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


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10

ABCDE

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?


10

ABCDE

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?


Ans

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Slide 41 / 101

11

A

BCD

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.


11

A

BCD

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.


Ans

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12

A

B

C

D

E

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


12

A

B

C

D

E

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


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*Note: question asks for speed, not

velocity.

/ 101

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.


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.


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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.


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.


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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.


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.


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Linear Approximation

& DifferentialsReturn to

Table of Contents

Slide 47 / 101

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.


Slide 48 / 101 Slide 49 / 101

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.



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/ 101

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

Example


Slide 52 / 101

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

Example, Continued


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

Example, Continued

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[This object is a teacher notes pull tab]

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.


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Given , approximate .

Example


Given , approximate .

Example


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Slide 55 / 101

16 Given Approximate


16 Given Approximate


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Slide 56 / 101

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


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


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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.

/ 101 Slide 57 (Answer) / 101


Slide 59 / 101

20 Find the approximate value of using linear approximation.


20 Find the approximate value of using linear approximation.


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Let

Then

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21 Given and approximate the value of



Slide 62 / 101

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.

Slide 63 / 101

Let's try an example: Find the differential .

Differentials

(Answer) / 101

Let's try an example: Find the differential .

Differentials


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Slide 64 / 101


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

vs.

Slide 66 / 101

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


/ 101

23 Find the differential if




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Slide 68 / 101





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26

A

B

C

D

E

F

Find and evaluate for the given values of and .


26

A

B

C

D

E

F

Find and evaluate for the given values of and .


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C



/ 101 Slide 73 (Answer) / 101

Slide 74 / 101

L'Hopital's RuleReturn to

Table of Contents

Slide 75 / 101

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

L'Hopital's Rule

Slide 76 / 101

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 "Lho-pee-talls")

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.

Slide 77 / 101

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

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

or

Then,


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

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

or

Then, [This object is a pull tab]

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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.

Slide 79 / 101

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

Slide 80 / 101

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

Example


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

Example


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take derivative

take derivative

Our answers match!

Slide 81 / 101

Evaluate the following limit:

Example

(Answer) / 101


Example

[This object is a pull tab]A

nsw

er

Applying L'Hopital's Rule...

Slide 82 / 101


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

Example



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

Example


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Applying L'Hopital's Rule...

still indeterminate!

We can apply L'Hopital's Rule again!

Slide 83 / 101

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.

Slide 84 / 101

30

A

B

C

D

E



30

A

B

C

D

E



Ans

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D

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31A

B

C

D

E



31A

B

C

D

E



Ans

wer

C

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32

A

B

C

D

E



32

A

B

C

D

E



Ans

wer A

Discuss with students why they cannot apply

L'Hopital's Rule on this problem.

Slide 87 / 101

33

A

B

C

D

E


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


33

A

B

C

D

E


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


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wer B

Rewrite as to apply L'Hopital's Rule.

/ 101

34

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B

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34

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wer C

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

Slide 89 / 101

Horizontal TangentsReturn to

Table of Contents

Slide 90 / 101

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


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


Teac

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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.

Slide 91 / 101

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

Horizontal Tangents

page3svg

(Answer) / 101

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

Horizontal Tangents


Teac

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Often students will confuse this idea with finding the derivative and evaluating at 0, rather

than setting it equal to 0. Be sure to clear up any confusion between the two ideas.

It is critical to allow students time to think and discuss this idea. Some may not come to the answer on their own, so you may ask leading questions:· What is another way to describe horizontal?

> slope of zero· What is another word for slope?

> derivative

Desired response, set the derivative = 0.

Slide 92 / 101

Let's try an example...

At what x-value(s) does the following function have a horizontal tangent line?

Example


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 x-value, but if you are asked at what point(s) the function has horizontal tangent lines, you need both the x- and y-coordinates.

Example


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 x-value, but if you are asked at what point(s) the function has horizontal tangent lines, you need both the x- and y-coordinates.

Example


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Slide 94 / 101


Example

(Answer) / 101


Example


Ans

wer

Allow students to struggle with the meaning when they don't get a real solution for this problem, and ask them what they think that means about this particular function.

no real solutions...

Therefore, no horizontal tangentsIt may also be helpful to have students

check the graph of this function to visualize their answer.

Slide 95 / 101



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

(Answer) / 101



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Slide 98 / 101




(Answer) / 101



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Slide 101 / 101


ap calculus - njctlcontent.njctl.org/courses/math/ap-calculus-ab/... · 11/4/2015 · ap calculus...

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