Calculus
TEST 6 This is your test 6. You will need to show all your work to get full credit. 1. Find the following antiderivatives: (a) Z √ x5 + 3 x − 2x−1 ; dx Z x−3 + 4×2 + 8 dx x5 Z √ x x + 4×4 + 3ex dx (b) (c) 2. Use the Fundamental Theorem of Calculus to evaluate Z 2 x2 + 1 dx 0 3. The picture below represents the graph of the function f (x) = x2 − 9. In the positive axis, the function is a always increasing and it intersects the x-axis at x = 3, in the interval of interest. The area between the function and the x-axis is shadowed. Use the fundamental theorem of calculus to compute the area between the function and the x-axis, between x = 0 and x = 5 4. Consider the graph below of the first derivative function f 0 (x): The graph represents a function that intersects the x-axis and x = −0.8, x = 1.6 and x = 2.6. The function goes to infinity as x goes to negative infinity and it is decreasing in the interval (−∞, 0) intersecting the x-axis at x = −0.81. It is increasing in the interval (0, 2) intersecting the x-axis at x = 1.612.. It is decreasing again in the interval (2, ∞) intersecting the x-axis at x = 2.6. (a) What are the critical points of the function f (x). (b) Indicate the intervals in which f (x) is increasing. (c) Indicate the intervals in which f (x) is decreasing. 5. Consider the cost function C(x) = 2×3 − 12x + 17 where x is the number of items produced. (a) Find the marginal cost to produce 5 items. (b) Find the rate of change of cost with respect of time (measured in hours)when dx dt = 4 and x = 5. (c) What is the difference between the first value and the second value. 6. The marginal profit is P 0 (x) = x2 (30x+30) on Mascarpone cheese sold at the cheese store, where x is the amount of cheese sold in hundreds of pounds. The ”profit” is −20$ when no cheese is sold. (a) Find the profit function. (b) Find the profit on selling 200lb of cheese. 7. Pollution from a factory is entering a lake. The rate of concentration of the pollutant at time t is given by P 0 (t) = 140t5/2 What is the level of pollution after 4 years? (P (0) = 0 as at the beginning the factory has not contributed to any pollution.) Biologists estimate that the quantity that the lake can support is 4850 units before some fish species will die. Can the factory operate without killing any fish? 8. Consider the function f (x) = 3×3 + x2 . (a) Compute Z 1 3×3 + x2 dx −1 (b) What is the area between the result you found in the previous question and the area between the function and the x axis between x = −1 and x = 1? (Consider that the definite integral of a function that is negative is a negative number, while the area is always positive).
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