Exam II Review - Worksheet Math 1131 Name: Section: The Chain Rule 1. Let y = Class #31 √ 2x + 1. Find dy dx . 2. Let y = (1 − x)200 . Find y 0 (x). 2 3. Let y = e1−x . Find dy dx . 4. Let f (x) = ln(x2 ). Find f 0 (x). 5. Let f (x) = 2 ln(x) . Find f 0 (x). 1 Exam II Review - Worksheet Math 1131 Name: 6. Let f (x) = Section: p ex + sin(x) + 5x3 . Find f 0 (x). 7. Let f (x) = tan(ex · x2 ). Find f 0 (x). 8. Compute the derivative of f (x) = cos(x2 ) · sin(x2 ) 9. Let f (x) = Class #31 p ln(x2 + x + 1). Find f 0 (x). 2 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: 10. Let f (x) = sin(cos(tan(x))). Find f 0 (x). 11. Let f (x) = cot(x · csc(2x)). Find f 0 (x). 12. Let f (x) = sec(cos(sin(x))). Find f 0 (x). 3 Exam II Review - Worksheet Math 1131 Name: Section: Implicit Differentiation 1. Suppose that xy = x2 . Find Class #31 dy . dx 2. Suppose that y 2 − 4xy + x2 = 1. Find 3. Suppose that ey = cos(x) + y. Find dy . dx dy . dx 4. Suppose that ey = y cos(y) + x. Find dy . dx 4 Exam II Review - Worksheet Math 1131 Name: Section: Logarithms and Differentiation 1. Let f (x) = ln Class #31 p ex · cos(x) · (x2 − 1) . Find f 0 (x) 2. Let f (x) = ln xsin(x) . Find f 0 (x) 5 Exam II Review - Worksheet Math 1131 Name: Section: ex 3. Let f (x) = x . Find f 0 (x). 4. Let f (x) = Class #31 sin(x) ln(x) . Find f 0 (x). 6 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: Related Rates 1. An airplane flies directly over a radar station, at a constant altitude of 3 mi above the ground. A little while later, the radar station measures that (a) the distance between the plane and radar station equals 5 mi and (b) that the distance between the plane and radar station is increasing at a rate of 500 mi/hr. What is the ground speed of the airplane at the time of the second measurement? 2. An ice cube melts, with its surface area decreasing at a rate of 3 in2 /s. How fast is the side length decreasing when the side length is 1 in? 7 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: 3. A streetlight is mounted at the top of a 6 meter pole, and a 2 meter tall person is walking toward it at 2 meters per second. How fast is the length of their shadow changing when they are 4 meters from the streetlight? What about when they are 1 meter from the light? 4. A police officer is walking down a city street, when they spot a wanted felon standing 200 ft away at the corner of the next block. The police officer takes off after the felon at 12 ft/s, and the felon immediately cuts around the corner and runs away at 9 ft/s. What is the rate of change of the distance between the officer and the felon after 10 seconds have passed? 8 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: 5. Suppose there is a 100 cm long water trough shaped as a triangular prism whose cross-section is an inverted triangle ∇ which is 20 cm across the top, and is 10 cm tall. If the tank is being being filled with water at a constant rate of 400 cm3 /s, how fast is the height changing when the water is half way up the tank? 6. Suppose the water trough above leaks (100 cm long, cross section is a ∇, top = 20 cm, and height = 1 cm). If water is being added to the tank at a rate of 400 cm3 /s, and is leaking out of the tank at 100 cm3 /s, how fast is the height changing when the water is half way up the tank? 9 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: L’Hospital’s Rule Use L’Hospital’s Rule to evaluate the following limits. Remember to show all work. This is the best way to learn to do these problems correctly! lim ln(x3 ) x3 + 3 2. Compute the limit lim ln(x3 ) x3 + 3 1. Compute the limit x→0+ x→∞ 3. Compute the limit lim 2x sin x→∞ 1 2x 10 Exam II Review - Worksheet Math 1131 Name: Section: 4. Compute the limit lim x→∞ 5. Compute the limit Class #31 lim x→0− 8 1+ x x 1 + 3 sin(x) 2/x 11 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: Linear Approximations 1. Approximate e−0.1 using the method of linear approximation. You must give your answer as a decimal without using a calculator. √ 2. Approximate 8 using a linear approximation. You must give your answer as a decimal without using a calculator. 12 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: 3. Find the linear approximation to f (x) = x1 + x at a = 2. Use your answer to approximate f (1.9). You must give your answer as a decimal without using a calculator. 4. Find the linearization of f (x) = x ln(x2 ) at a = 1. Use your answer to approximate f (1.2). You must give your answer as a decimal without using a calculator. 13 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: 5. Find a linearization of the function f (x) = sin(x) at a = π4 . Use your answer to approximate sin 5π 16 . 6. Find a linear approximation for the function f (x) = cos(x) at a = Use your answer to approximate cos 5π 16 . 7. Find a linearization of the function f (x) = tan(x) at a = Use your answer to approximate tan 5π 16 . 14 π 4 π 4 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: Maxima, Minima, and Graph Shape 1. Let f (x) = x3 − 6x2 + 9x + 1 Find the following. If a requested quantity doesn’t exist, answer “DNE”. (a) Find the intervals where f is increasing and decreasing (b) Find the intervals where f is concave up and concave down (c) Find the numbers x where the are local maxima and minima (d) Find the numbers x where the is a point of inflection 15 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: 2. Let f (x) = x4 − 8x2 + 16. Find the following. If a requested quantity doesn’t exist, answer “DNE”. (a) Find the intervals where f is increasing and decreasing (b) Find the intervals where f is concave up and concave down (c) Find the numbers x where the are local maxima and minima (d) Find the numbers x where the is a point of inflection 16 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: Optimization 1. In optimization problems, it is important to verify that you have the correct answer. (a) Find two positive numbers whose sum is 25 and whose product is as large as possible. (b) Find two positive numbers whose sum is 25 and whose product is as small as possible. 17 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: 2. Find two numbers x and y such that xy 2 = 54 and which minimizes F = x2 + y 2 . 18 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: 3. You are building a rectangular garden. You are tired of your neighbors stealing your produce and you have looked into several types of electric fence. The material for the side of the fence near your house is $1 per foot. The other three sides will cost $2 per foot. If you have $100, what is the maximum amount of area you can protect? You must show all work, including verifying that area is maximized. 19 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: 4. You want to turn part of your yard into a dog park. For zoning reasons, you want the park to be exactly 100 ft2 . Suppose that your house will form the west boundary of the yard, that the fence for the north boundary will cost $3 per foot and that the other two sides will cost $1 per foot. What are the dimensions of the yard with the cheapest surrounding fence? You must show all work, including verifying that cost is minimized. 20 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: 5. You are designing a rectangular display box with an open top. The box should have a volume of 2 m3 , and the length of the base should equal twice the width of the base. The material for the base costs $5 per square meter, and the material for the sides costs $10 per square meter. Find the cost of the materials for the cheapest container. You must show all work, including verifying that cost is minimized. 21 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: 6. You are designing a rectangular display box with an open top. The width of the base must equal twice its length. The material for the base costs $6/in2 and the material for the sides costs $2/in2 . What is the largest volume box that you can make for $120? You must show all work, including verifying that volume is maximized. 22 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: 7. You are crossing a 2 mile wide river. Your camp is on the other side, 5 miles downstream. You are hungry, and you want to get there as quickly as possible. If you can swim at 3 mi/hr and hike at 5 mi/hr, where should you land to get there the fastest? You must show all work, but this time you do not need to verify that time is minimized. 23 Exam II Review - Worksheet Math 1131 Name: Class #31 Section: Anti-Derivatives 1. Find the (general) anti-derivative of f (x) = sin(x) + π cos(x) + 2. Find the (general) anti-derivative of f (x) = 3π 2 sec2 (x). 1 x2 + x + 1 +√ . x3 x 3. Suppose f 0 (x) = x2 + 2x + 5 and that f (0) = 3. Find f (x). 4. Suppose f 0 (x) = 1 x + x + ex and that f (1) = 0. Find f (x). 5. Suppose f 00 (x) = 12x2 − ex , that f (0) = 1, and that f 0 (0) = 2. Find f (x). 24
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