ap calculus - njctlcontent.njctl.org/courses/math/ap-calculus-ab/application-of... · ap calculus...
TRANSCRIPT
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!