AP Calculus AB: Part 1 - Module 14

AP Calculus AB: Part 1 Module 14 Content Guide Related Rates and Linearization Learning Outcomes After completing this module students will be able to do the following: 1. Find a rate at which one quantity is changing by relating it to other quantities whose rates of change are known. 2. Approximate a value on a curve using the equation of a tangent line. My Notes AP Calculus AB: Part 1 Module 14 Content Guide Topic 14.1 Solving for Related Rates Part A Fill in the blanks. Process for Solving Related Rate Problems 1. Draw a picture 2. Make a list of all known and unknown rates and quantities. 3. Write an equation that relates all the variables in the problem. 4. Differentiate with respect to time . 5. Substitute the known quantities and solve. 6. Include the correct units . AP Calculus AB: Part 1 Module 14 Content Guide Topic 14.1 Solving for Related Rates Part B Use the following context to answer the questions below: A spotlight on the ground shines on a wall 12 meters away. If a 2-meter man walks from the spotlight to the wall at a speed of 1.6 meters per second, how fast is the length of his shadow on the wall decreasing when he is 4 meters from the wall? Modeled Practice 1: Draw and label a picture to represent the context. Modeled Practice 2: Write an equation that relates the variables in the problem. xy=24 Modeled Practice 3: Differentiate with respect to time and solve for the rate the shadow is decreasing. -.6m/sec AP Calculus AB: Part 1 Module 14 Content Guide Topic 14.1 Solving for Related Rates Part C Use the following context to answer questions 1, 2, and 3. (Round to 4 decimal places as needed). A 1.4-meter-tall person walks away from a 6-meter lamp post at the rate of 1.5 meters per second. The light at the top of the post casts a shadow in front of the person. How fast is the top of the shadow moving along the ground? Guided Practice 1: Draw and label a picture. Guided Practice 2: Write an equation that relates the variables in the problem. y=1.3043x Guided Practice 3: Solve and use the correct units for your answer. 1.9565 m/sec AP Calculus AB: Part 1 Module 14 Content Guide Topic 14.2 Trigonometry and Related Rates Part A Fill in the blanks. Angle of Elevation Suppose a rocket is launched vertically and tracked by a radar station located on the ground 5 km from the launch pad. Suppose that the elevation angle 𝜃𝜃 of the line of sight to the rocket is increasing at 3 degrees per second when 𝜃𝜃 = 60 degrees. What is the velocity of the rocket at that instant? Which variables are changing with respect to time? 1,3,4 1. The velocity of the rocket 2. The distance between the radar station and the launch pad 3. The angle of elevation 4. The height of the rocket AP Calculus AB: Part 1 Module 14 Content Guide Topic 14.2 Trigonometry and Related Rates Part B Modeled Practice 1: A balloon rises into the air from a point on the ground 15 feet from an observer. Relate the rate of change of the angle of elevation of the balloon from the observer and the distance between the observer and the balloon. -sinx dx/dt=-15/c^2 dc/dt Modeled Practice 2: Suppose you are stationary on the ground and are watching a bird fly horizontally at a rate of 15 m/sec. If the bird is located 40 feet above your head, how fast does the angle of elevation change when the horizontal distance between you and the bird is 9 m? -.258 radians/sec Modeled Practice 3: A boat is being pulled toward a dock by a rope attached to its bow through a pulley on the dock. The pulley is 7 feet higher than the boat's bow. If the rope is hauled in at a rate of 4 ft/sec, how fast is the boat approaching the dock when 25 feet of rope is out? -25/6 ft/sec AP Calculus AB: Part 1 Module 14 Content Guide Topic 14.2 Trigonometry and Related Rates Part C Guided Practice 1: An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other. One plane is 100 miles from the point moving at 350 miles per hour and the other plane is 240 miles from the point moving at 500 miles per hour. At what rate is the distance between the planes decreasing? 596.154 mph Guided Practice 2: You stand 40 feet from a bottle rocket on the ground and watch as it takes off vertically into the air at a rate of 20 ft/sec. Find the rate at which the angle of elevation changes when the rocket is 30 feet in the air. .32 radians/sec Guided Practice 3: A ladder 25 feet long is leaning against the wall of a house. If the base of the ladder is pulled away from the wall at a rate of 2 ft/sec, how fast is the top moving down the wall when the base of the ladder is 7 feet from the wall? 7/12 ft/sec AP Calculus AB: Part 1 Module 14 Content Guide Topic 14.3 Geometric Figures and Related Rates Part A Fill in the blanks. Translate Phrases to Mathematics Statements dh The height of the water is increasing 2 feet per hour → 𝑑𝑑𝑑𝑑 = 2𝑓𝑓 hr Water is pouring into the cone at a steady rate of 7 𝑖𝑖𝑛𝑛3 /min → dV 𝑑𝑑𝑑𝑑 = 7 𝑖𝑖𝑛𝑛3 min The radius of the surface of the water is decreasing 1.5 inches per minute → dr 𝑑𝑑𝑑𝑑 = -1.5 𝑖𝑖𝑛𝑛 min The height of the soda is a can is decreasing 2 centimeters per minute → dh 𝑑𝑑𝑑𝑑 = 2𝑐𝑐𝑚𝑚 min AP Calculus AB: Part 1 Module 14 Content Guide Topic 14.3 Geometric Figures and Related Rates Part B Modeled Practice 1: An ice cube is melting at a rate of 6 cubic centimeters per hour. At what rate is the side of the cube changing when the side of the cube is 4 cm. -1/8 cm/hr Modeled Practice 2: The base of a triangle is increasing at a constant rate of 0.7 cm/sec and the height is decreasing at 0.3 cm/sec. When the height is 3 cm and the base is 10 cm, at what rate is the area increasing or decreasing? -.45 cm^2/sec Modeled Practice 3: Water is draining from the bottom of a cone-shaped funnel at a rate of 16 𝑖𝑖𝑛𝑛3 /sec. The height of the funnel is 14 inches and the radius at the top of the funnel is 7 inches. At what rate is the level of the water in the funnel decreasing when the height of the water is 8 inches? Round your answer to the nearest thousandth. decreasing at a rate of .318 in per sec AP Calculus AB: Part 1 Module 14 Content Guide Topic 14.3 Geometric Figures and Related Rates Part C Guided Practice 1: Suppose a 25-foot-long ladder is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 ft/sec and the top of the ladder is moving down the wall at a rate of 0.5 ft/sec. Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. 1 ft^2/sec Guided Practice 2: A circular pool of water is expanding at the rate of 16𝜋𝜋 𝑖𝑖𝑛𝑛2 /sec. At what rate is the radius expanding when the radius is 4 inches? 2 in/sec Guided Practice 3: Suppose a vertical cylinder is leaking water at a rate of 2 𝑓𝑓𝑑𝑑 3 /sec. If the cylinder has a height of 12 feet and a radius of 2 feet, at what rate is the height of the water changing when the height is 6 feet? -1/2pi in per sec AP Calculus AB: Part 1 Module 14 Content Guide Topic 14.4 Approximation Using Linearization Part A Fill in the blanks. Overestimate or Underestimate We can use the concavity of the function to determine if the linear approximation is an overestimate or an underestimate. Remember a section of the curve is concave up where the slope is increasing and a section is concave down when the slope is decreasing . If 𝑓𝑓(𝑥𝑥) is concave up, 𝐿𝐿(𝑥𝑥) gives an below underestimate the curve. If 𝑓𝑓(𝑥𝑥) is concave down, 𝐿𝐿(𝑥𝑥) gives an is above or the tangent line is the curve. overestimate or the tangent line AP Calculus AB: Part 1 Module 14 Content Guide Topic 14.4 Approximation Using Linearization Part B Modeled Practice 1: Let 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 =− 5𝑑𝑑 𝑑𝑑 and 𝑓𝑓(2) = 3. Write an equation for the line tangent to the graph of 𝑓𝑓 at the point (2,3) and use the tangent line to approximate 𝑓𝑓(1.9). 3.267 𝑒𝑒 2𝑥𝑥 Modeled Practice 2: Suppose 𝑓𝑓 𝑥𝑥 = is concave up on the interval (−1, ∞). Find an 𝑑𝑑+1 estimate for 𝑓𝑓(−0.1) and determine whether it is an underestimate or an overestimate. underestimate since the function is concave up 𝜋𝜋 Modeled Practice 3: Let 𝑔𝑔 𝑥𝑥 = 3 sin 𝑥𝑥 + 1. Use the tangent line to 𝑔𝑔(𝑥𝑥) at 𝑥𝑥 = to 2 approximate 𝑔𝑔(1.5). Then use a calculator to evaluate 𝑔𝑔(1.5) and find a difference between the actual value and the estimate. approx 4 the difference is .008 AP Calculus AB: Part 1 Module 14 Content Guide Topic 14.4 Approximation Using Linearization Part C Guided Practice 1: Let 𝑓𝑓(𝑥𝑥) be a function with 𝑓𝑓 2 = −5 and the slope of 𝑓𝑓 at any point 4𝑑𝑑 (𝑥𝑥, 𝑦𝑦) is given by 2 . Write the equation for the line tangent to 𝑓𝑓 at 𝑥𝑥 = 2 and use it to estimate 𝑓𝑓 2.2 . 𝑑𝑑 8/25 approx -4.936 Guided Practice 2: Suppose 𝑓𝑓 is concave up 𝑓𝑓 3 = −1 and 𝑓𝑓′ 3 = 2. What is the estimate for 𝑓𝑓(2.8) using the local linear approximate for 𝑓𝑓 at 𝑥𝑥 = 3? Explain whether it is an underestimate or overestimate. underestimate since it is concave up at 3 𝜋𝜋 Guided Practice 3: Suppose 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 cos 𝑥𝑥 is concave down at 𝑥𝑥 = . Find an estimate for 𝜋𝜋 2 𝑓𝑓(1.5) using the local linearization for 𝑓𝑓 at 𝑥𝑥 = . Explain whether it is an underestimate or 2 overestimate. .111 overestimate since concave down