AP Calculus. Slide 1 / 101. Slide 2 / 101. Slide 3 / 101. Applications of Derivatives. Table of Contents

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1 Slide 1 / 101 Slide 2 / 101 AP Calculus Applications of Derivatives Table of Contents click on the topic to go to that section Slide 3 / 101 Related Rates Linear Motion Linear Approximation & Differentials L'Hopital's Rule Horizontal Tangents

2 Slide 4 / 101 Related Rates Return to Table of Contents Slide 4 () / 101 Teacher Notes 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. Related Rates Return to Table of Contents Related Rates Slide 5 / 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 ft 3 /min, how fast is the radius changing when the radius is 2 feet?

3 Related Rates Slide 5 () / 101 Students can usually comprehend Related Rates is the application the balloon of example, implicit differentiation understanding (which we learned in the previous that although unit) to the real air life is situations. being pumped in at a constant rate, the radius changes very quickly at first and then slows down as the balloon In simplest terms, related rates are problems in which you need gets larger. There is no need to to figure out how fast one solve variable this example is changing at this time, when it given is the rate of change of another just variable to give them at a specific an idea point of the in time. types of problems they will [This experience. object is a pull tab] 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? Teacher Notes Recall: Implicit Differentiation 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 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.

4 Helpful Steps for Solving Related Rates Problems 1) Draw a picture. Label the picture with numbers if constant or variables if changing. Slide 7 / 101 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) the question. Try to write your answer in a sentence to eliminate confusion. Helpful Steps for Solving Emphasize to students: WARNING! Most mistakes are Related Rates made by Problems subsituting the given values too early. You must wait until 1) Draw a picture. Label the picture with after numbers you differentiate! constant or variables if changing. *Note on Step 4: Occasionally, students may see a question where 2) Identify which rate of change is they given need and to differentiate which rate with of respect change you are being asked to find. to a different variable; however, most often it will be time. 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. Teacher Notes Slide 7 () / 101 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. Try to write your answer in a sentence to eliminate confusion. Step 3 Slide 8 / 101 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

5 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. Slide 9 / 101 Let's take a look back at this example... Example If a spherical balloon r is being filled with air at a rate of 20 ft 3 Know: /min, how fast is the radius changing when the radius is 2 feet? 1. Picture 2. Identify Rates of Change 3. Equation that relates volume of a sphere with radius of a sphere. 4. Differentiate with respect to t. 5. Substitute given values. Want: when r=2 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. Slide 9 () / Solve for 7. the question. 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? Slide 10 / 101 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.

6 Hands-On Related Rates Lab (OPTIONAL) Slide 11 / 101 Click here to go to the lab titled "Related Rates" students to observe the result. It helps Hands-On is Student A does Related not watch Student Rates B, Lab but just walks at whatever pace needed to (OPTIONAL) keep the rope taut. Teacher Notes It may take several attempts for 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. Click URL here for Lab: to go to the lab titled "Related math/ap-calculus-ab/application-of- Rates" derivatives/hands-on-related-rates/ Slide 11 () / 101 Hands-On Related Rates (OPTIONAL) Items needed: 2 students 1 long rope/cord/string (at least 15 feet for best display) masking tape Slide 12 / 101 STEP #1 Set up masking tape in a right angle classroom with enough room for each student to walk along the tape line.

7 Hands-On Related Rates (OPTIONAL) Items needed: 2 students 1 long rope/cord/string (at As least said before, 15 feet it may for take best several display) masking tape STEP #1 Teacher Notes 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. Set up masking tape in a right angle classroom with enough room for each student to walk along the tape line. 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. Slide 12 () / 101 Hands-On Related Rates (OPTIONAL) Slide 13 / 101 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 Hands-On Related Rates (OPTIONAL) Slide 14 / 101 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

8 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? Slide 15 / 101 Slide 15 () / 101 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? Slide 16 / 101

9 Slide 16 () / 101 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? Slide 17 / 101 Slide 17 () / 101

10 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. Slide 18 / 101 Slide 18 () / 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? Slide 19 / 101 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

11 2 Water leaking onto a floor r forms Know: a circular pool. The radius of the pool increases at a rate of 4 cm/min. How Want: when r = 5 fast is the area of the pool increasing when the radius is 3. Find an appropriate equation. 5 cm? A B C D E 1. Picture 2. Identify Rates of Change 4. Differentiate with respect to t. The area of 5. Substitute the circle given is values. increasing at a rate of cm 2 /min when the radius is 5cm. 6. Solve for The area of the circle is increasing at a rate of cm 2 /min 7. the question. when the radius is 5cm. The area of the Choice circle E: The is area increasing of the circle at is increasing a rate at of a cm 2 /min rate of [This cm when the radius is 5cm. 2 /min object when is a the pull radius tab] is 5cm. The area of the circle is increasing at a rate of cm 2 /min when the radius is 5cm. The area of the circle is increasing at a rate of cm 2 /min when the radius is 5cm. Slide 19 () / 101 Slide 20 / 101 Slide 20 () / 101

12 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? Slide 21 / 101 A The height of the water is increasing at a rate of 0.3 m/sec when the water is 120cm high. B The height of the water is increasing at a rate of 6 m/sec when the water is 120cm high. C The height of the water is increasing at a rate of 0.25 m/sec when the water is 120cm high. D The height of the water is increasing at a rate of 40 m/sec when the water is 120cm high. E The height of the water is increasing at a rate of 20 m/sec when the water is 120cm high. 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 1. Picture 5 2. Identify Rates of Change height is 2 meters. 8 If water is being pumped in at a constant rate of 6 m 3 Know: /sec. At what rate is the height of the w water changing when the water Want: has a height when h=120cm of 120 cm? 2 h use 1.2m A The height 3. of Find the an appropriate water is equation. increasing at a rate of 0.3 m/sec *use similar triangles to when the water is express 120cm w in terms high. of h B The height 4. of Differentiate the water with respect is increasing to t. at a rate of 6 m/sec when the water 5. Substitute is 120cm given values. high. C The height 6. of Solve the for water is increasing at a rate of 0.25 m/sec when the water 7. is the 120cm question. high. D The height of the Choice water C: The is height increasing of the water is at rising a rate a rate of 40 m/sec of 0.25 m/s when [This object the water is a is pull 120cm tab] high. when the water is 120cm high. E The height of the water is increasing at a rate of 20 m/sec when the water is 120cm high. Slide 21 () / 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 Slide 22 / 101 y x z A B C D E

13 5 The sides of the rectangle pictured increase in such a way 1. Picture 2. Identify Rates of Change that and. At the instant where x=4 and y=3, z Know: y what is the value of x Want: when x=4 & y=3 y x z 3. Find an appropriate equation. 4. Differentiate with respect to t. 5. Substitute given values. Slide 22 () / Solve for A 7. the question. B C D E Choice B. 1 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? Slide 23 / 101 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. 6 If the base, b, of a triangle is increasing at a rate of 3 inches per minute 1. Picturewhile it's height, 2. Identify h, Rates is decreasing of Change at a rate of 3 inches per minute, which Know: of the following must be true about the area, A, h of the triangle? A B C D E b A is always increasing. 3. Find an appropriate equation. A is always decreasing. 4. Differentiate with respect to t. A is decreasing 5. Substitute given only values. when b < h. 6. Solve for A is decreasing only when b > h. 7. the question. A remains constant. Want: Choice D: The [This area object is decreasing is a pull tab] only when b>h. Slide 23 () / 101

14 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 / 101 Slide 24 () / 101 Slide 25 / 101 Linear Motion Return to Table of Contents

15 Linear Motion 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! Position, Velocity & Acceleration A remarkable relationship exists among the position of an object, the velocity of an object and the acceleration of an object. Slide 27 / 101 First... let's review what each of these words mean. Position Velocity Acceleration Position, Velocity & Acceleration Have students share thoughts and definitions A remarkable relationship for exists each term. among Be certain the to position listen for students of an object, the velocity of an object and referring the acceleration to velocity as speed. of an The object. next slide will clarify the difference. Teacher Notes Position - location of an object in regards to First... let's review what each its starting of these locationwords mean. Velocity - how fast AND in what direction an Position object is moving Acceleration - how fast AND in what direction Velocity the velocity is changing Acceleration [This object is a teacher notes pull tab] Slide 27 () / 101

16 Are Velocity and Speed the Same Thing? Slide 28 / 101 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? Slide 28 () / 101 Although you may hear velocity and Note: speed The interchanged positive or often in common conversation, they are, in fact, negative 2 distinct direction quantities. is Sometimes they are equivalent to determined each other, by but the this object's depends on the direction of the object. initial position and what is Velocity is a vector quantity meaning determined it has both to magnitude be a and direction. positive/negative 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 Notes Distance vs. Position Slide 29 / 101 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.

17 Typical Notation for Linear Motion Problems Slide 30 / 101 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? Slide 31 / 101 Example distance Consider driving your car along the highway. time 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? Important: This is the average velocity. It does not necessarily mean you were traveling 52mph the entire time. Slide 31 () / 101

18 Average Velocity vs. Instantaneous Velocity 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. Slide 33 / 101 Slide 33 () / 101

19 Example Slide 34 / 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 Slide 34 () / 101 d the instantaneous velocity seconds. 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. at is the car's acceleration? b) What is the car's acceleration? constant [This object acceleration is a pull tab] Example 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.

20 Example Slide 35 () / 101 A spring is pulled to 6 inches below its resting state and bounces up and ty and acceleration at time t. down. Its position is modeled by. a) Find its velocity and acceleration at time t. g's velocity and acceleration. b) Find the spring's velocity and acceleration after seconds. s object is a pull tab] 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 Slide 36 () / 101 A dynamite of this blast problem, shoots which is a more rock like questions straight seen up on into the AP the air. Its Exam. Work slowly and check for understanding frequently. height at any given time is feet after t seconds. *Students may struggle with comprehension and visualization a) How high does the rock travel? When the rock reaches its peak, the velocity will be equal to 0, then we can find the position at that time. a) How high does the rock travel? b) What is the velocity and speed of the rock when it is 256 feet above ground? b) What We first is must the find velocity at what time the and speed of the rock when it is 256 feet position is 256ft, and then find v(time). above ground? c) What is the acceleration at any time, t? Acceleration is the 2nd derivative of position, and the 1st derivative of velocity. d) When does the rock hit the ground? c) What When the is rock the hits acceleration the ground its position at any time, t? will be equal to 0 feet. At t=0, that is it's starting position. d) When does the rock hit the ground?

21 One More Reminder! Slide 37 / 101 What is the difference between: Average Velocity Instantaneous Velocity One More Reminder! Slide 37 () / 101 What Sometimes is the difference when students between: begin practicing questions involving instantaneous velocity they forget how to calculate average velocity. Take a minute to reiterate the difference. Average Velocity Instantaneous Velocity In simple terms: Teacher Notes Average Velocity - slope formula with 2 points Instant. Velocity - derivative evaluated at 1 point. 8 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? Slide 38 / 101 A B C D E

22 8 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? Slide 38 () / 101 A B C D E B Slide 39 / 101 Slide 39 () / 101

23 10 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 / 101 A B C D E 10 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 D Slide 40 () / 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. Slide 41 / 101 A B C D E

24 11 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 A Slide 41 () / 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 / 101 A B C D E 12 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 E *Note: question asks for speed, not velocity. Slide 42 () / 101

25 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 / 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 () / 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 / 101

26 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 () / 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 / 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 () / 101

27 Slide 46 / 101 Linear Approximation & Differentials Return to Table of Contents Linear Approximation 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

28 Linear Approximation 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. Slide 50 / 101 Slide 50 () / 101

29 Example Slide 51 / 101 Practice: Use linear approximation to approximate the value of f(8.9). Slide 51 () / 101 Example, Continued Slide 52 / 101 Is our approximation greater than or less than the actual value of f(8.9)? Why or why not?

30 Example, Continued Slide 52 () / 101 Is our approximation greater than or less than the actual value of f(8.9)? Why or why not? 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. Teacher Notes [This object is a teacher notes pull tab] Slide 53 / 101 Slide 53 () / 101

31 Example Slide 54 / 101 Given, approximate. Example Slide 54 () / 101 Given, approximate. 16 Given Approximate Slide 55 / 101

32 16 Given Approximate Slide 55 () / 101 Slide 56 / 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 Slide 56 () / 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 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.

33 Slide 57 / 101 Slide 57 () / 101 Slide 58 / 101

34 Slide 58 () / Find the approximate value of using linear approximation. Slide 59 / Find the approximate value Let of using linear approximation. Then Slide 59 () / 101

35 21 Given and approximate the value of Slide 60 / 101 Slide 60 () / 101 Slide 61 / 101

36 Slide 61 () / 101 Differentials Slide 62 / 101 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 Slide 63 / 101 Let's try an example: Find the differential.

37 Differentials Slide 63 () / 101 Let's try an example: Find the differential. Slide 64 / 101 Slide 64 () / 101

38 vs. Slide 65 / 101 Note the difference between and. If we calculate both, we can then compare the values to calculate the percentage change or approximation error. Example 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. Slide 66 () / 101

39 23 Find the differential if Slide 67 / Find the differential if Slide 67 () / Find the differential if Slide 68 / 101

40 24 Find the differential if Slide 68 () / 101 Slide 69 / 101 Slide 69 () / 101

41 26 Find and evaluate for the given values of and. Slide 70 / 101 A B C D E F 26 Find and evaluate for the given values of and. A D C Slide 70 () / 101 B C E F Slide 71 / 101

42 Slide 71 () / 101 Slide 72 / 101 Slide 72 () / 101

43 Slide 73 / 101 Slide 73 () / 101 Slide 74 / 101 L'Hopital's Rule Return to Table of Contents

44 Slide 75 / 101 L'Hopital's Rule One additional application of derivatives actually applies to solving limit questions! L'Hopital's Rule (pronounced "Lho-pee-talls") Slide 76 / 101 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. Slide 77 / 101

45 L'Hopital discovered an alternative way of dealing with these limits! Slide 78 / 101 L'HOPITAL'S RULE Suppose you have one of the following cases: or Then, L'Hopital discovered an alternative way of dealing with these limits! Sometimes students will attempt to L'HOPITAL'S use the quotient RULE rule on these problems. Emphasize that the original question is asking for a limit, Suppose you have one of the following cases: and L'hopital's rule deals with the numerator and denominator as two distinct orfunctions and differentiates each separately. Teacher Notes Slide 78 () / 101 Then, L'Hopital's Rule 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!

46 Example Let's try L'Hopital's Rule on our previous example: Slide 80 / 101 Example Let's try L'Hopital's Rule on our previous example: take derivative Slide 80 () / 101 take derivative Our answers match! Example Slide 81 / 101 Evaluate the following limit:

47 Example Slide 81 () / 101 Evaluate the following limit: Applying L'Hopital's Rule... Evaluate the following limit: Example Note: L'Hopital's Rule can be applied more than one time, if needed. Slide 82 / 101 Evaluate the following limit: Example Applying L'Hopital's Rule... Note: L'Hopital's Rule can be applied more than one time, if needed. still indeterminate! We can apply L'Hopital's Rule again! Slide 82 () / 101

48 Important Fact to Remember: Slide 83 / 101 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. 30 Evaluate the following limit: A B C D E Slide 84 / Evaluate the following limit: Slide 84 () / 101 A B C D E D

49 31 Evaluate the following limit: A Slide 85 / 101 B C D E 31 Evaluate the following limit: A B C D E C Slide 85 () / Evaluate the following limit: A B C D E Slide 86 / 101

50 32 Evaluate the following limit: A B C A Discuss with students why they cannot apply L'Hopital's Rule on this problem. Slide 86 () / 101 D E 33 Evaluate the following limit: A B C Slide 87 / 101 D E Hint: Sometimes it is helpful to rewrite before applying L'Hopital's Rule. 33 Evaluate the following limit: A B C B Rewrite as to apply L'Hopital's Rule. Slide 87 () / 101 D E Hint: Sometimes it is helpful to rewrite before applying L'Hopital's Rule.

51 34 Evaluate the following limit: A Slide 88 / 101 B C D E 34 Evaluate the following limit: A B C D E C Students may recall the shortcut of using the highest power's coefficients, or may apply L'Hopital's rule twice. Slide 88 () / 101 Slide 89 / 101 Horizontal Tangents Return to Table of Contents

52 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? Slide 90 / 101 Tangent Lines Recall what it means to be tangent Allow to students a function. to We make could draw an infinite amount of tangent lines observations, below; however, and discuss looking with at the ones given what observations can you classmates. make about The the desired black tangent lines? observation is that they recognize the black tangent lines are the only ones that are horizontal, or have a slope of zero. Teacher Notes Slide 90 () / 101 Horizontal Tangents Do you think there is a way to find out where the horizontal tangents are occurring aside from just estimating? Slide 91 / 101

53 Horizontal Tangents It is critical to allow students time to think and Do you think there discuss a way this to idea. find Some out where may not the come horizontal to the tangents are occurring aside from answer just estimating? 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 Teacher Notes Desired response, set the derivative = 0. 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 [This object between is a pull the tab] two ideas. Slide 91 () / 101 Let's try an example... Example At what x-value(s) does the following function have a horizontal tangent line? Slide 92 / 101 Slide 92 () / 101

54 Example At what point(s) does the following function have a horizontal tangent line? Slide 93 / 101 ***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. 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 Slide 93 () / 101 Example At what x-value(s) does the following function have a horizontal tangent line? Slide 94 / 101

55 Example Allow students to struggle with the meaning when they don't get a real solution for this At what x-value(s) does problem, the following and ask function them what have they a think horizontal that tangent line? means about this particular function. Slide 94 () / 101 no real solutions... Therefore, no horizontal tangents It may also be helpful to have students check the graph of this function to visualize their answer. Slide 95 / 101 Slide 95 () / 101

56 Slide 96 / 101 Slide 96 () / At what point(s) does the following function have a horizontal tangent line? Slide 97 / 101

57 37 At what point(s) does the following function have a horizontal tangent line? Slide 97 () / 101 Slide 98 / 101 Slide 98 () / 101

58 Slide 99 / 101 Slide 99 () / At what point(s) does the following function have a horizontal tangent line? Slide 100 / 101

59 40 At what point(s) does the following function have a horizontal tangent line? Slide 100 () / 101 Slide 101 / 101 Slide 101 () / 101