Exam 3 Review (Sections Covered: , 6.7topic and )

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1 Exam 3 Review (Sections Covered: , 6.7topic and ) 1. Find the most general antiderivative of the following functions. (Use C for the constant of integration. Remember to use absolute values where appropriate.) (a) ( 5 x 5 + 4e x 2x 5 + 3x 2 1 ) x dx (b) ( ) x 2 + 7x 4 x 3 dx (c) ( ) 7e x + 13 e x dx (d) ( 2 x + 3 5x 1 ) 4 x 7 dx (e) ( ) 48 + u 2 du 8u

2 2. For the following functions, evaluate the integral. (Use C for the constant of integration. Remember to use absolute values where appropriate.) (a) (x 4 + 2)(10x + x 5 + 1) 3 dx (b) x 4 e x5 dx (c) (3x 3 9)e (3x4 36x) dx (d) (ln x) 36 x dx (e) 30e 6/x x 2 dx 2 Fall 2016, Maya Johnson

3 3. The speed of a runner increased steadily during the first twelve seconds of a race. Her speed at two-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these twelve seconds using a left-hand sum and a right-hand sum with n = 6. t(s) v(f t/s) Speedometer readings for a motorcycle at 12-second intervals are given in the table. t(s) v(f t/s) (a) Estimate the distance traveled by the motorcycle during this time period using a left-hand sum with n = 5. (b) Estimate the distance traveled by the motorcycle during this time period using a right-hand sum with n = 5. 3 Fall 2016, Maya Johnson

4 5. Use a left-hand sum and a right-hand sum with rectangles of equal width for the given value of n to approximate the integral. Round the answers to two decimal places (2x 2 + 1) dx, n = 3 6. Use a left-hand sum and a right-hand sum with rectangles of equal width for the given value of n to approximate the integral. Round the answers to four decimal places x 2 ln(x) dx, n = 3 7. Use a left-hand sum and a right-hand sum with rectangles of equal width for the given value of n to approximate the integral. Round the answers to two decimal places (2x 3 + x) dx, n = 4 4 Fall 2016, Maya Johnson

5 8. Evaluate the following definite integrals: (a) 1 A 6 x dx. Assume A < 1 (b) B 2 (3x 2 7x 3 + 7x 2) dx. Assume B > 2 (c) B 0 (10e x 6x 4 + 2) dx. Assume B > 0 (d) A x x dx. Assume A > 1 9. If f(4) = 18, f is continuous, and 6 4 f (x) dx = 30, what is the value of f(6)? 5 Fall 2016, Maya Johnson

6 10. Suppose the marginal cost function for a certain commodity is given by C (x) = 0.5x and C(0) = 200, find the cost to make 12 units of this commodity. 11. Suppose the marginal revenue function for a certain commodity is given by R (x) = 10x 6 and R(1) = 100, find the revenue when 10 units of this commodity are sold. 12. Find the average value of the following functions on the given interval. (Round answers to two decimal places as needed.) (a) f(x) = 6x + 9x 2, [0, 5] (b) f(x) = 12e x, [5, 7] (c) f(x) = 12x 3 10x 2, [2, 6] 6 Fall 2016, Maya Johnson

7 13. The rate of sales of a certain brand of bicycle by a retailer in thousands of dollars per month is given by d S(t) = 15t 0.57t2 dt (a) Find the amount of sales, in thousands of dollars, for the first six months after the start of the advertising campaign. Give answer to three decimal places. (b) Find the average sales per month for the second six month period of the advertising campaign. Give answer to three decimal places. 14. Suppose that copper is being projected to be extracted from a certain mine at a rate given by P (t) = 320e 0.08t where P (t) is measured in tons of copper and t is measured in years. (a) How many tons of copper is projected to be extracted during the second four year period? Give answer to three decimal places. (b) How many tons of copper is projected to be extracted during the third four year period? Give answer to three decimal places. 7 Fall 2016, Maya Johnson

8 15. Use properties of the definite integral and information listed below to solve the following problems: (Assume a and b are two real numbers such that a < b.) b a b a (a) Evaluate f(x) dx = 20 g(x) dx = 12 u(x) dx = 16 u(x) dx = 50 2a 2a f(x) dx. (b) Evaluate a b g(x) dx. (c) Evaluate b a [f(x) 3 g(x)] dx. 2 (d) Evaluate 3 3 u(x) dx. 16. Determine the area that is bounded by the graphs of the following equations. y = 64x, y = x 3 8 Fall 2016, Maya Johnson

9 17. Determine the area that is bounded by the graphs of the following equations. (Round answer to three decimal places.) y = 3x, y = 9x x Determine the area that is bounded by the graphs of the following equations on the interval below. (Round answer to three decimal places.) y = x 2 + 7x, y = 8x The graph of f is shown. Use the graph to evaluate each integral. (a) f(x) dx (b) 36 0 f(x) dx (c) 12 0 f(x) dx 9 Fall 2016, Maya Johnson

10 20. Calculate the producers surplus at the indicated price level for the supply equation below. (Round answer to the nearest cent.) p = S(x) = x 2, p o = $ Calculate the consumers surplus for the demand equation at the given number of units demanded. (Round answer to the nearest cent.) p = D(x) = 27 2x 1/3, x o = Fall 2016, Maya Johnson

11 22. Determine the consumers surplus for the demand function below at the indicated price level. p = D(x) = x, p o = $ Determine the indicated values of the following functions. f(x, y) = 4x 2 xy + y 9 4 g(x, y) = x 5 2y 2 (a) f( 5, 2) (b) g(1, 9) (c) f(1, 3) 3g(7, 1) 24. Determine the indicated value of the function. (Round answer to one decimal place.) W (0.9, 6, 1, 5) for W (a, b, c, d) = a(1 + b) d2 2c 25. Determine the indicated values of the function. 3xy + 6z f(x, y, z) = 3xz 6y (a) f(1, 0, 1) (b) f(0, 1, 1) (c) f( 1, 1, 0) 11 Fall 2016, Maya Johnson

12 26. Macrosoft produces two versions of its popular gaming console: the Elite and the Casual. The weekly demand and cost functions for the consoles are p = 300 4x + 2y q = 225 x + 9y C(x, y) = x + 120y where x represents the weekly demand for the Elite version; y represents the weekly demand for the Casual version; p and q represent the price (in dollars) of an Elite console and a Casual console, respectively; and C(x, y) is the cost function. (a) Determine R(x, y), the weekly revenue function. (b) Determine P (x, y), the weekly profit function. (c) Find P (4, 2). 27. HeadsRock produces two versions of its popular headphones: the RockUrWorld and the SilentRocker. The weekly demand and cost functions for the headphones are p = 300 7x + 2y q = 225 x + 8y C(x, y) = x + 120y where x represents the weekly demand for the RockUrWorld version; y represents the weekly demand for the SilentRocker version; p and q represent the price (in dollars) for a pair of Rock- UrWorld headphones and a pair of SilentRocker headphones, respectively; and C(x, y) is the cost function. (a) Determine R(x, y), the weekly revenue function. 12 Fall 2016, Maya Johnson

13 (b) Determine P (x, y), the weekly profit function. (c) Find P (7, 2). 28. Find the first partial derivatives of the function. w = 9z + 10e xyz (a) w x (b) w y (c) w z 29. Find the first partial derivatives of the function. f(x, y) = x 6 y 5 + 7x 4 y (a) f x (x, y) (b) f y (x, y) 13 Fall 2016, Maya Johnson

14 30. Find all the second partial derivatives. f(x, y) = x 9 y 5 + 3x 9 y + x y 2 (a) f xx (b) f yy (c) f yx = f xy 31. Find all the second partial derivatives. f(x, y) = x 5 y 3 + 2x 10 y 2x y 3 (a) f xx (b) f yy (c) f yx = f xy 14 Fall 2016, Maya Johnson