AP Calculus 2015 11 03 www.njctl.org Table of Contents click on the topic to go to that section Related Rates Linear Motion Linear Approximation & Differentials L'Hopital's Rule Horizontal Tangents 1
Related Rates Teacher Notes Return to Table of Contents Related Rates 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. Teacher Notes For example, if a spherical balloon is being filled with air at a rate of 20 ft 3 /min, how fast is the radius changing when the radius is 2 feet? 2
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. 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. Helpful Steps for Solving Related Rates Problems 6) Solve for rate of change you are being asked for. 7) the question. Try to write your answer in a sentence to eliminate confusion. Teacher Notes 3
Step 3 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 Let's take a look back at this example... Example If a spherical balloon is being filled with air at a rate of 20 ft 3 /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) the question. 4
Why is it important to write a sentence for an answer? In the last question we answered the following: The radius is increasing at a rate of when the radius is 2 feet. 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. Hands On Related Rates Lab (OPTIONAL) Click here to go to the lab titled "Related Rates" Teacher Notes 5
Hands On Related Rates (OPTIONAL) Items needed: 2 students 1 long rope/cord/string (at least 15 feet for best display) masking tape STEP #1 Set up masking tape in a right angle classroom with enough room for each student to walk along the tape line. Teacher Notes Hands On Related Rates (OPTIONAL) STEP #2 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. A B 6
Hands On Related Rates (OPTIONAL) STEP #3 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. B A Example 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? 7
Example 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 CHALLENGE! 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? 8
1 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? A The shadow is increasing at a rate of 3/7 ft/sec. B C D E 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 3 ft/sec. The shadow is increasing at a rate of 7 ft/sec. 2 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? A B C D E The area of the circle is increasing at a rate of when the radius is 5cm. The area of the circle is increasing at a rate of when the radius is 5cm. The area of the circle is increasing at a rate of when the radius is 5cm. The area of the circle is increasing at a rate of when the radius is 5cm. The area of the circle is increasing at a rate of when the radius is 5cm. cm 2 /min cm 2 /min cm 2 /min cm 2 /min cm 2 /min 9
3 Calculator OK 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? A B C D E 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 0.311 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 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 m 3 /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 0.3 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 0.25 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 20 m/sec when the water is 120cm high. 10
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 y x z A B C D E 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. 11
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 Return to Table of Contents 12
Linear Motion 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! 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 Teacher Notes 13
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. Teacher Notes 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). Distance vs. Position 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. 14
Typical Notation for Linear Motion Problems is the notation for our position function is the notation for our velocity function is the notation for our acceleration function 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? 15
Average Velocity vs. Instantaneous Velocity 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. Position, Velocity & Acceleration Therefore, if given a position function, x, as a function of t. The velocity of the object is given by: Furthermore, the acceleration of the object is: is also commonly used for position Teacher Notes 16
Example 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 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. 17
Example 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? One More Reminder! What is the difference between: Average Velocity Instantaneous Velocity Teacher Notes 18
7 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? A B C D E 8 A particle moves along the x axis so that at any time t>0 minutes its position is given by. For what values of t is the particle at rest? A B C D E No values only only only 19
9 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? A B C D E 10 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. A B C D E 20
11 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. A B C D E 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. 21
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 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. 22
Linear Approximation & Differentials Return to Table of Contents Linear Approximation 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. 23
What is the Purpose of Linear Approximation? Let's consider the graph of Zoomed in 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. Linear Approximation 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. 24
To Find the Linear Approximation: not whole whole whole whole not whole Teacher Notes 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) Example Practice: Use linear approximation to approximate the value of f(8.9). 25
Example, Continued Is our approximation greater than or less than the actual value of f(8.9)? Why or why not? Teacher Notes Example Given, approximate. 26
Example Given, approximate. Given Approximate 27
12 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. Recall A Greater than B Less than Given and approximate the value of 28
Given Approximate Find the approximate value of using linear approximation. 29
Given and approximate the value of Approximate the value of 30
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. Differentials Let's try an example: Find the differential. 31
Practice Now let's practice with given values to substitute... Given find when and vs. Note the difference between and. If we calculate both, we can then compare the values to calculate the percentage change or approximation error. 32
Example 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. Find the differential if 33
Find the differential if 13 Find and evaluate for the given values of and. A B C D E F 34
14 Find and evaluate for the given values of and. A B C D E F Given,, and calculate the estimated change. 35
Given,, and calculate the true change. Given,, and calculate the approximation error. 36
L'Hopital's Rule Return to Table of Contents L'Hopital's Rule One additional application of derivatives actually applies to solving limit questions! 37
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. 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. Indeterminate Form 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. 38
L'Hopital discovered an alternative way of dealing with these limits! L'HOPITAL'S RULE Suppose you have one of the following cases: or Teacher Notes Then, L'Hopital's Rule 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! 39
Example Let's try L'Hopital's Rule on our previous example: Example Evaluate the following limit: 40
Evaluate the following limit: Example Note: L'Hopital's Rule can be applied more than one time, if needed. 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. 41
15 Evaluate the following limit: A B C D E 16 Evaluate the following limit: A B C D E 42
17 Evaluate the following limit: A B C D E 18 Evaluate the following limit: A B C D E Hint: Sometimes it is helpful to rewrite before applying L'Hopital's Rule. 43
19 Evaluate the following limit: A B C D E Horizontal Tangents Return to Table of Contents 44
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? Teacher Notes Horizontal Tangents Do you think there is a way to find out where the horizontal tangents are occurring aside from just estimating? Teacher Notes 45
Let's try an example... 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. 46
Example At what x value(s) does the following function have a horizontal tangent line? 20 At what x value(s) does the following function have a horizontal tangent line? A D B C E F G No Horizontal Tangents 47
21 At what x value(s) does the following function have a horizontal tangent line? A D B C E F G No Horizontal Tangents At what point(s) does the following function have a horizontal tangent line? 48
22 At what x value(s) does the following function have a horizontal tangent line? A D B C E F G No Horizontal Tangents 23 At what x value(s) does the following function have a horizontal tangent line? A D B C E F G No Horizontal Tangents 49
At what point(s) does the following function have a horizontal tangent line? 24 At what x value(s) does the following function have a horizontal tangent line? A D B C E F G No Horizontal Tangents 50