APPLICATIONS OF DERIVATIVES UNIT PROBLEM SETS
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1 APPLICATIONS OF DERIVATIVES UNIT PROBLEM SETS PROBLEM SET #1 Related Rates ***Calculators Allowed*** 1. An oil tanker spills oil that spreads in a circular pattern whose radius increases at the rate of 15 ft/min. a) How fast is the circumference increasing when the radius is 20 feet? b) How fast is the area increasing when the radius is 30 feet? 2. A 15 ft. ladder leans against a wall. The lower end of the ladder is being pulled away from the wall at the rate of 1.5 ft/sec. How fast is the top of the ladder moving down the wall at the instant it is 9 feet above the ground? 3. A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second. At how many meters per second is the distance between the train and the observer changing 4 seconds after it passes through the intersection?
2 4. A balloon is rising vertically above a level field and tracked by a range finder 300 feet from the lift off point. At the moment the range finder s elevation is π 3, the angle is increasing at the rate of 0.26 radians/min. How fast is the balloon rising? 5. The radius of a 6-inch long cylinder is increasing at a rate of one-thousandth of an inch every 3 minutes. How fast is the cylinder s volume increasing when the diameter is 3.8 inches? 6. ** The volume of an expanding sphere is increasing at a rate of 12 cubic feet per second. When the volume of the sphere is 36π cubic feet, how fast in square feet per second is the surface area increasing? Note: (V = 4 3 πr3 and S = 4πr 2 )
3 PROBLEM SET #2 Linear Motion ***Calculators Allowed*** 1. An object moves along the x-axis; its position is given by x(t) = t 3 4t What is the object s acceleration at = 2? 2. The position of a particle moving on a straight line after t seconds is s(t) = costsint meters. What is the velocity of the particle at t = 11π 6 seconds? 3. A rock thrown straight up from a planet s surface reaches a height of s = 832t 2.6t 2 feet after t seconds. How long does it take the rock to hit the ground? 4. A car travels along a straight path and its position is x(t) = 9t t miles in t hours. What is the car s average velocity during the first 2 hours? 5. An object s velocity is given by v(t) = 2t 3 9t t 7. Find the object s speed when the acceleration is zero.
4 6. A bullet is fired from 6 feet above ground and its position is given by x(t) = t 2 + 5t + 6 feet after t seconds. a) Find the bullet s position after 2 seconds. b) Calculate the average velocity over the first 4 seconds. c) What is the instantaneous velocity of the bullet after 3 seconds? d) At what time does the bullet hit the ground? e) What is the speed of the bullet when it hits the ground? 7. Describe the difference between speed and velocity: 8. Create a position function that will result in a positive velocity at t = 1 but a negative acceleration at that time.
5 PROBLEM SET #3 Linear Approximation ***Calculators Allowed*** 1. Use linear approximation to estimate Given f(x) = e x, use linear approximation to estimate f( 0.1). 3. a) Estimate 1 using linear approximation b) Is your approximation from part (a) greater than or less than the actual value? Explain your answer. 4. Given f(2) = 3 and f (2) = 1, approximate the value of f(2.3) Estimate tan47 using linear approximation.
6 3 6. Estimate 129 using linear approximation. 7. Find the differential, dy, for each of the following: a) y = lnx c) y = sin 2 x b) y = 2x 3 4x 2 + 5x 1 d) y = 3x4 2x 1 8. Given y = 3 sec (x + π ), evaluate dy if x = 0 and dx = After being painted, the side of a cube increases from 4 inches to 4.07 inches. Use ds to estimate the increase in the cube s Surface Area, S. Compare this estimate to the true change, S, and find the approximation error.
7 PROBLEM SET #4 L Hopital s Rule ***Calculators Not Allowed*** Evaluate each of the following limits, using L Hopital s Rule when necessary. 1. lim x x 1 x 7. lim x 3 x 2 9 x + 3 tanx 2. lim x 0 x 8. lim x 2 x 3 8 x lim x 6x 2 3x 7x lim t 1 ln (t 2 ) t lim ( 1 t 0 sint 1 t ) 10. lim x x 1 1 x 2 5. lim x 0 e x 1 3x + x lim x log 2 x log 3 x 3e x 6. lim x x 12. lim x 0 e 2x 1 tanx
8 PROBLEM SET #5 Horizontal Tangents ***Calculators Not Allowed*** Determine the x-values (if any) at which the function has a horizontal tangent line. 1. y = 3x 2 12x f(x) = x 5 15x f(x) = 3x 1 6. y = cosx on [ π 2, 3π 2 ] 3. y = x 4 8x f(x) = (x 2)(x 2 x 11) 4. f(x) = 1 3 x3 + 5x x 8. y = lnx Determine the point(s) (if any) at which the function has a horizontal tangent line. 9. y = 2x 3 3x y = 3 4x x y = 3 2x 2 4x y = tanx 13. There are exactly two horizontal lines which are tangent to = sinx. What are the equations of these lines?
9 Applications of Derivatives - Answer Keys Problem Set #1 Related Rates 1. a) b) dy dt dr dt dh dt dv dt dc dt da = 30π ft/min = 900π dt ft2 /min = 2 ft/s = 57.6 m/s = 312 ft/min = or in3 min 6. ds dt = 8 ft2 /sec Problem Set #2 Linear Motion 1. a(2) = 4 2. v ( 11π ) = 0.5 m/s 6 3. t = 320 sec mph 5. at t = 1: 2 m/s at t = 2: 3 m/s 6. a) x(2) = 12 feet b) 1 ft/s c) 1 ft/s d) t = 6 sec e) 7 ft/sec 7. answers vary; should mention that speed is how fast an object is moving; whereas velocity is not only how fast, but in what direction 8. answers vary Problem Set #3 Linear Approximation 1. f(102.9) f( 0.1) a) f(4.22) b) Estimate is less than the actual value because the tangent line lies below the curve at this value. 4. f(2.3) tan or a) dy = 1 x dx b) dy = (6x 2 8x 5)dx c) dy = (2sinxcosx)dx d) dy = 16x4 12x 3 4x 2 4x+1 dx 8. dy = or ds = 3.36 in 2 S = in 2 S ds = in 2 Problem Set #4 L Hopital s Rule ln3 ln
10 Problem Set #5 Horizontal Tangents 1. x = 2 2. No horizontal tangent 3. x = 0 x = 2 x = 2 4. x = 7 x = 3 5. x = 0 x = 3 x = 3 6. x = π 7. x = 3 x = 1 8. No horizontal tangent 9. (0,0) (1, 1) 10. (1, 3) 11. ( 2,7) 12. No horizontal tangent 13. y = 1 y = 1
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