AP Calculus - NJCTLcontent.njctl.org/courses/math/ap-calculus-ab/... · 11/4/2015 · AP Calculus...
Transcript of AP Calculus - NJCTLcontent.njctl.org/courses/math/ap-calculus-ab/... · 11/4/2015 · AP Calculus...
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AP Calculus
Applications of Derivatives
2015-11-03
www.njctl.org
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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
Return to Table of Contents
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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
Slide 5 (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.
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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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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
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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
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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
Slide 9 (Answer) / 101
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"
Slide 11 (Answer) / 101
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
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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
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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
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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
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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.
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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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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.
4. Differentiate with respect to t.
1. Picture 2. Identify Rates of Change
5. Substitute given values.
6. Solve for
7. Answer the question.
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.
The height of the water is increasing at a rate of 20 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 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.
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Know:
Want: when h=120cm
use 1.2m
3. Find an appropriate equation.
4. Differentiate with respect to t.
1. Picture 2. Identify Rates of Change
5. Substitute given values.
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
3. Find an appropriate equation.
4. Differentiate with respect to t.
1. Picture 2. Identify Rates of Change
5. Substitute given values.
6. Solve for
7. Answer the question.
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.
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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.
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Know:
Want:
3. Find an appropriate equation.
4. Differentiate with respect to t.
1. Picture 2. Identify Rates of Change
5. Substitute given values.
6. Solve for
7. Answer the question.
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
Slide 24 (Answer) / 101 Slide 25 / 101
Linear Motion
Return to Table of Contents
Slide 26 / 101
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
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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
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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
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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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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
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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
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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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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
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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
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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
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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
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a) Find the instantaneous velocity after 8 seconds.
b) What is the car's acceleration?
constant acceleration
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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
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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
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a) Find its velocity and acceleration at time t.
b) Find the spring's velocity and acceleration after seconds.
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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
Slide 36 (Answer) / 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
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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.
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One More Reminder!
What is the difference between:
Average Velocity Instantaneous Velocity
Slide 37 (Answer) / 101
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 ?
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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 ?
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Slide 39 / 101 Slide 39 (Answer) / 101
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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?
Slide 40 (Answer) / 101
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?
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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.
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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.
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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.
Slide 42 (Answer) / 101
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.
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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.
Slide 43 (Answer) / 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.
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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.
Slide 44 (Answer) / 101
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.
Slide 45 (Answer) / 101
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.
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.
Linear Approximation
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Slide 51 / 101
Practice: Use linear approximation to approximate the value of f(8.9).
Example
Slide 51 (Answer) / 101
Slide 52 / 101
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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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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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.
Slide 53 / 101 Slide 53 (Answer) / 101
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Given , approximate .
Example
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Given , approximate .
Example
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16 Given Approximate
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16 Given Approximate
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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
Slide 56 (Answer) / 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
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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.
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20 Find the approximate value of using linear approximation.
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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
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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.
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Let's try an example: Find the differential .
Differentials
Slide 63 (Answer) / 101
Let's try an example: Find the differential .
Differentials
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Note the difference between and . If we calculate both, we can then compare the values to calculate the percentage change or approximation error.
vs.
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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
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23 Find the differential if
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23 Find the differential if
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24 Find the differential if
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24 Find the differential if
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26
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B
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F
Find and evaluate for the given values of and .
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26
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F
Find and evaluate for the given values of and .
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C
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L'Hopital's RuleReturn to
Table of Contents
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One additional application of derivatives actually applies to solving limit questions!
L'Hopital's Rule
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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.
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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,
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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, [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.
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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
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Let's try L'Hopital's Rule on our previous example:
Example
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Let's try L'Hopital's Rule on our previous example:
Example
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take derivative
take derivative
Our answers match!
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Evaluate the following limit:
Example
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Evaluate the following limit:
Example
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nsw
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Applying L'Hopital's Rule...
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Evaluate the following limit:
Note: L'Hopital's Rule can be applied more than one time, if needed.
Example
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Evaluate the following limit:
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!
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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.
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30
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Evaluate the following limit:
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30
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Evaluate the following limit:
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31A
B
C
D
E
Evaluate the following limit:
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31A
B
C
D
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Evaluate the following limit:
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C
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32
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B
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D
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Evaluate the following limit:
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32
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Evaluate the following limit:
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Discuss with students why they cannot apply
L'Hopital's Rule on this problem.
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33
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B
C
D
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Evaluate the following limit:
Hint: Sometimes it is helpful to rewrite before applying L'Hopital's Rule.
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33
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B
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D
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Evaluate the following limit:
Hint: Sometimes it is helpful to rewrite before applying L'Hopital's Rule.
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Rewrite as to apply L'Hopital's Rule.
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34
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Evaluate the following limit:
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34
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Evaluate the following limit:
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Students may recall the shortcut of using the highest power's coefficients, or may apply L'Hopital's rule twice.
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Horizontal TangentsReturn to
Table of Contents
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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
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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
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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.
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Do you think there is a way to find out where the horizontal tangents are occurring aside from just estimating?
Horizontal Tangents
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Do you think there is a way to find out where the horizontal tangents are occurring aside from just estimating?
Horizontal Tangents
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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.
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Let's try an example...
At what x-value(s) does the following function have a horizontal tangent line?
Example
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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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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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At what x-value(s) does the following function have a horizontal tangent line?
Example
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At what x-value(s) does the following function have a horizontal tangent line?
Example
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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.
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37 At what point(s) does the following function have a horizontal tangent line?
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37 At what point(s) does the following function have a horizontal tangent line?
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40 At what point(s) does the following function have a horizontal tangent line?
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40 At what point(s) does the following function have a horizontal tangent line?
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