Related Rates Return to Table of Contents Slide 5 / 101 Related - PDF document

Slide 1 / 101 Slide 2 / 101 AP Calculus Applications of Derivatives 2015-11-03 www.njctl.org Slide 3 / 101 click on the topic to go to that section Table of Contents Related Rates Linear Motion Linear Approximation & Differentials L'Hopital's Rule Horizontal Tangents

Slide 4 / 101 Related Rates Return to Table of Contents Slide 5 / 101 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. 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? Slide 6 / 101 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.

Slide 7 / 101 Helpful Steps for Solving Related Rates Problems 1) Draw a picture. Label the picture with numbers if constant or variables if changing. 2) Identify which rate of change is given and which rate of change you are being asked to find. 3) Find a formula/equation that relates the variables whose rate of change you seek with one or more variables whose rate of change you know. 4) Implicitly differentiate with respect to time, t. 5) Plug in values you know. 6) Solve for rate of change you are being asked for. 7) Answer the question. Try to write your answer in a sentence to eliminate confusion. Slide 8 / 101 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 Slide 9 / 101 Example Let's take a look back at this 1) Draw and label a picture. example... 2) Identify the rates of change you know and seek. 3) Find a formula/equation. 4) Implicitly differentiate with respect to time, t. If a spherical balloon is being filled 5) Plug in values you know. with air at a rate of 20 ft 3 /min, how 6) Solve for rate of change you are being asked for. 7) Answer the question. fast is the radius changing when the radius is 2 feet?

Slide 10 / 101 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. Slide 11 / 101 Hands-On Related Rates Lab (OPTIONAL) Click here to go to the lab titled "Related Rates" Slide 12 / 101 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 STEP #1 each student to walk along the tape line.

Slide 13 / 101 Hands-On Related Rates (OPTIONAL) Student A begins at the end of one piece of tape, and Student B begins in STEP #2 the corner. Each student holds one end of the rope until it is taught. A B Slide 14 / 101 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 B should watch Student A's rate of change over the course of his/her path. It may take several attempts to observe the result. A Slide 15 / 101 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 16 / 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 17 / 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 18 / 101 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 The shadow is increasing at a rate of 7/3 ft/sec. C The shadow is increasing at a rate of 14 ft/sec. D The shadow is increasing at a rate of 3 ft/sec. E The shadow is increasing at a rate of 7 ft/sec.

Slide 19 / 101 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 The area of the circle is increasing at a rate of cm 2 /min when the radius is 5cm. B The area of the circle is increasing at a rate of cm 2 /min when the radius is 5cm. C The area of the circle is increasing at a rate of cm 2 /min when the radius is 5cm. D The area of the circle is increasing at a rate of cm 2 /min when the radius is 5cm. E The area of the circle is increasing at a rate of cm 2 /min when the radius is 5cm. Slide 20 / 101 Slide 21 / 101 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 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 B 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.

Slide 22 / 101 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 z y x A C D E B Slide 23 / 101 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 A is always increasing. B A is always decreasing. C A is decreasing only when b < h. D A is decreasing only when b > h. E A remains constant. Slide 24 / 101 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 25 / 101 Linear Motion Return to Table of Contents Slide 26 / 101 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! Slide 27 / 101 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

Slide 28 / 101 Are Velocity and Speed the Same Thing? Although you may hear velocity and speed interchanged often in common conversation, they are, in fact, 2 distinct quantities. Sometimes they are equivalent to each other, but this depends on the direction of the object. Velocity is a vector quantity meaning it has both magnitude and direction. For example, if the velocity of an object is -3 feet per second, then that object is moving backwards or to the left (direction) at a rate of 3 feet per second (magnitude). Slide 29 / 101 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. Slide 30 / 101 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