Math 2211 3.5-3.10 Review Name _________________________ Just Relax and Think Positively!!!! I Know you CAN Do This!!! 1. Let () = 1 . Find the value of (1). 2. Let () = 1 2. Find the value of (0). 3. Let () = 1 ( 2 + 1). Find the value of (1). 4. 5. ( 1 (3)). (1 ()). 6. Let () = ( 2 3). Find the value of (2). 7. Find the Derivative: a. = 1 ( 3 ) b. () = 1 ( ) c. = ( 3 ) d. () = 1 (5) e. () = 3 + 5 f. () = (7) g. = 1 (5) h. = arctan() 8. Find an equation of the tangent line to the graph of = at the point (1, 0). 9. Differentiate the following functions: a. () = ()3 b. () = c. () = ( 2+1 ) 10. Find . a. = ( 2 + 1) b. = () 11. Find . 47 a. = 2 +2 b. = (ln())4 12. The population of a bacteria colony after t hours is given by () = 2000 0.087 . Find the growth rate of the colony when t = 16 hours. 13. A particle moves along a straight line with equation of motion = 2 2. Find the Instantaneous velocity of the particle at time t = 1. 14. Suppose the amount of a drug left in the body t hours after administration is () = In mg/h, find the rate of decrease of the drug 4 hours after administration. 20 +1 mg. 15. Suppose that a baseball is tossed straight upward and that its height (in feet) as a function of time (in seconds) is given by the formula () = 128 16 2 . a. Find the instantaneous velocity and acceleration of the baseball at time t. b. What is the maximum height attained by the ball? c. How long does it take before the ball lands? 16. A certain company has Cost function () = 2000 + 3 + 0.02 2 and Revenue function () = 86 0.04 4 . Find the production level that will maximize the profit. 17. The relationship between the rate of a certain chemical reaction and temperature under certain circumstances is given by () = 0.1(0.05 3 + 4 2 + 120) grams/sec, where R is the rate of reaction and T is the temperature (in C). a. Find the temperature T at which the reaction rate reaches its maximum. b. What is the maximum reaction rate? 18. The length of a rectangle is increasing at the rate of 2 feet per second, while the width is increasing at the rate of 1 foot per second. When the length is 4 feet and the width is 3 feet, how fast is the length of the diagonal of the rectangle increasing? 19. A ladder 10 feet long is leaning against a wall. If the foot of the ladder is being pulled away from the wall at 3 feet per second, how fast is the top of the ladder sliding down the wall when the foot of the ladder is 8 feet away from the wall? 20. At what rate is the surface area of a cube increasing if its edges are 2 inches and are increasing at a rate 3 inches per minute? 21. A particle moves along the curve = changing when x = 2? 1 . If x increases at a rate 8 units per minute, at what rate is y 22. Let V be the volume of a cylinder having height h and radius r, and assume that h and r vary with time. When the height is 5 in. and is increasing at 0.2 in./s, the radius is 3 in. and is decreasing at 0.1 in./s. How fast is the volume changing at that instant? Is the volume increasing or decreasing at that instant? 23. Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure P and Volume V satisfy the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 600 cm3, the pressure is 150 kPa, and the pressure is increasing at a rate of 20 kPa/min. At what rate is the volume decreasing at this instant? 24. If the radius of a sphere is increasing at 1 centimeter per second, find the rate of change of its volume when the radius is 6 centimeters. 25. A water tank has the shape of an inverted circular cone with base radius 6 feet and height of 12 feet. The tank is full of water. If the water level is falling at the rate of 2 ft/min, how fast is the tank losing water when the water is 8 feet deep? 26. A child throws a stone into a still millpond causing a circular ripple to spread. If the radius of the circle increases at the constant rate of 0.5 meter per second, how fast is the area of the ripple increasing when the radius of the ripple is 20 meters? 27. Use logarithmic differentiation to find the derivative of the function: a. b. c. d. = = = = e. = 2 ( () 2 +1)10 ( 2 +1)4 (2+1)3 (31)5 28. Find the linear approximation to () = 29. Find the linear approximation to () = 1 (2+)3 1 3 +1 at a = 0 at a = 2 SOLUTIONS 1. 1 2 2. 2 3. 0.4 4. 5. 3 19 2 1 2(1+) 6. 8 7. a. b. 3 2 1 6 1 2 3 c. d. +1 5 2(1+25 2 )1 (5) 1 (3+5)2 e. f. 5 g. 3 1 2(3+5)2 (7) 1+25 2 h. + + 1 (5) 1+2 8. = 1 9. a. 3()3 2+ b. 2 1 c. 10. a. 2 2+ 1 2 b. 2 +1 1 2 +1 11. a. 47 2 +2 b. 4(ln())3 12. 700 13. 0 4 14. 5 mg/h 15. a. () = 128 32 ft/s () 32 ft/s/s b. The maximum height occurs when () = 0 at t = 4. (4) = 256 feet. c. The ball strikes the ground after 8 seconds. 16. 692 17. a. = 53.33 b. 391.3 grams/second 18. 2.2 ft/s 19. 4 ft/s 20. 72 in2/min 21. 2 unit/min 22. 1.2 in3/s. It is decreasing 23. Volume is decreasing at a rate of 80 cm3/min 24. 144 cm3/s 25. 32 ft3/m 26. 20 62.8 m2/s 27. a. b. c. d. e. 1 2 ( 2 + 1)10 (2 + 2 + (1 + ) ( )) () ( ) ( 2 +1)4 8 ( (21)3 (31)5 2 +1 28. () = 3 29. () = 2 16 9 + 1 + 7 8 9 6 20 2 +1 15 ) 2+1 31)