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Practice Problems 1. If the total profit, in thousands of dollars, for a product is given by √ P (x) = 10 x + 1 − 3x − 14, what is the marginal profit at a production level of 8 units? 2. The revenue from the sale of x tables is represented by the following formula. R = 20(3x + 1)−1 + 60x − 15 dollars Find the marginal revenue when 3 tables are sold. 3. Given y = x3 −6x2 +3x−5. Find the critical values of the function. Make a sign diagram and determine the relative maxima and minima. Determine absolute maxima and minima on interval [2, 4]. 4. Is the graph of f (x) = x3 − 4x2 + 3 concave up or down at the point (1, 0)? 5. Use properties of limits and algebraic methods to find the limit, if it exists. a) lim 3x25x+2 +2x−4 x→2 2 2 b) lim 4(x+h)h −4x x→0 6. Find the derivative of the function. a) y = ln(x10 ) b) y = (ln x)7 7. Find the derivative of the function. a) y = ln(5x3 + 3x) + 5x2 + 18 3 2 4 b) y = 5e(x +5x −7x) + 10x2 − 3 8. Find the derivative of the function a) y = e2x (x − 2)3 − 57x + 3 2 −1 b) y = x3x+4x−5 9. Find the first four derivatives of f (x) = x5 − 2x4 + 3x2 − 11x + 2 − e5x 10. Find dy/dx for the function. x2 + 12y 2 − 9 = 0 11. Let y = x5 − 6x + 1. Find dy/dt given that x = 2 and dx/dt = 3. 12. Evaluate the integral. Check your answer by differentiating. (Use C for the constant of integration.) Z 1 3 x7 − 6 + √ + e5x dx 4 x x−1 1 13. Use algebra to rewrite the integrand; then integrate and simplify. (Use C for the constant of integration.) Z (3x2 + 2)2 x4 dx 14. Evaluate the integral. Check your result by differentiation. (Use C for the constant of integration.) Z 3x4 dx (7x5 − 4)4 15. Cost is in dollars and x is the number of units. If the marginal cost for a product is M C = 6x + 3, and the production of 10 units results in a total cost of $520, find the total cost function. 16. In this problem, use properties of definite integrals. R0 R 11 3 R 11 3 14641 x If −11 x3 dx = −14641 and dx = , what does 0 −11 x dx equal? 4 4 17. Suppose that the cost in dollars for x chairs is given by C(x) = 400+x+0.3x2 . What is the average value of C(x) for 10 to 20 units? 18. Equations are given whose graphs enclose a region. Find the area of the region. f (x) = x3 − x2 , g(x) = 2x. R∞ dt 19. Evaluate the improper integral t3/2 20. Evaluate the improper integral 1 −1 R −∞ 4 x dx 21. In this problem, p is in dollars and x is the number of units. Find the producer’s surplus for a product if its demand function is p = 169 − x2 and its supply function is p = x2 + 8x + 127. 22. Evaluate the function as indicated. z(x, y) = (5x − 1)ex−y ; find z(0, −1) 23. Give the √ domain of the function. 3x5y − y+11x z= x+2y ∂z ∂z 24. If z = x3 + 5x2 − 3x + 7y 3 − 2y − 11, find ∂x and ∂y . 2 y 2 25. If z = xy − 3xe + y , find the following. a) zxx = b) zxy = c) zyx = d) zyy = 2