Exam Review Sheets Combined

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1 Exam Review Sheets Combined Fall Fall 2007 Exam 1 1. For each part, if the statement is always true, circle the printed capital T. If the statement is sometimes false, circle the printed capital F. For each T/F question, write a careful and clear justification or describe a counterexample. d (a) dx ( x2 4 ) = 2x. (b) The function f(x) = x 7 8x has a zero in the interval (0, 1). (c) If lim x 2 f(x) g(x) exists, then it must be f(2) g(2). (d) If θ is the angle illustrated in the figure (where the lengths of two sides of a right triangle are shown) then θ = arctan(a/ 1 a 2 ). x (e) lim x 0 sin(6x) = The height h (in feet) of a ball t seconds after being thrown up into the air from a hot air balloon is given by h(t) = 16t t (a) Find the velocity of the ball at time t and give units. (b) What is the height of the ball when it stops ascending and starts descending? 3. Use the properties of limits to calculate Show all steps. ( ) 3 x + 3 lim x 3 x x + 3 1

2 4. Use the chain rule to find the derivative of the function y = 4(sin(2x)) If f(π/8) = 0 and f (x) = 3/2 + (cos(2x)) 2, find the equation of the tangent line to the graph of this function at the point (π/8, 0). 6. Sketch the graph of f (x) for the function f(x) given: 7. (a) Write the limit definition for the derivative f (x). (b) Use the limit definition of the derivative and algebra to find the derivative of the function f(x) = 4x Find the horizontal asymptote(s) for the function f(x) = 8x4 + 5x x 3 x 4 by computing the appropriate limit(s). Show all work. 9. Let f(x) = { 5x x 3 + k, x 2, 5x 2 10x x 2, x < 2. (a) Find lim f(x). x 2 (b) Find lim f(x). x 2 + (c) There is a constant k that makes the function f(x) above continuous at x = 2. Find the number k. (d) Use limits and the definition of continuity to explain why this choice of k makes f(x) continuous everywhere. (e) With this choice of k, will f(x) be differentiable at x = 2? Explain your reasoning. 10. Given g(2) = 3 and g (2) = 4. (a) If f(x) = x2 + g(x), find f (2). g(x) (b) If f(x) = e g(x), find f (2). 2

3 2 Fall 2007 Exam 2 1. (a) If f and g are increasing on (a, b), then fg is increasing on (a, b). d (b) dx (ln(10)) = (c) If f has an absolute minimum value at x = c, then f (c) = 0. (d) If f, f and f are continuous, then the inflection points of f are the local extrema of f. (e) A function defined on all points of a closed interval [a, b] always has a global maximum and a global minimum on [a, b]. 2. For the function f(x) = x4 4 2x3 + 9x , use the techniques of calculus to answer the following. Show your work and proper justification for your answers. (a) Find f (x). (b) Determine all critical points of f. Classify each as a local maximum, a local minimum, or neither. (c) Find the intervals on which the function is increasing. (d) Determine any global maxima or minima (if any). (e) Find any inflection points and the intervals on which the graph of f is concave down. 3. (a) Find the tangent line approximation to f(x) = x 2/3 at a = 8. (b) Use your answers to (a) to obtain a linear approximation estimating the value of (7.98) 2/3. (c) Will this be an overestimate or an underestimate? reasoning. [Hint: Look at the shape of the graph.] 4. Use l Hôpital s rule to evaluate lim x 0 e 3x 3x 1 x 2. Explain your 5. The graph of f is shown below. Use the graph of f to answer the following questions. (a) On which intervals, if any, is f increasing? (b) At which values of x, if any, does f have a local max? A local min? (c) On which intervals, if any, is f concave up? (d) Which values of x, if any, correspond to inflection points on the graph of f? (e) Assume that f(0) = 0. Sketch the graph of f. 6. A man 6 feet tall is walking away from a street light 12 feet high. If the man is walking at a rate of 1.8 ft/min, how fast is the length of his shadow increasing? At what rate is the top end of the shadow moving? 3

4 7. A bacteria culture contains 300 cells initially and grows at a rate proportional to its size. After two hours the population has increased to 420 cells. (a) Find the number of bacteria after t hours. (b) Find the rate of growth after 4 hours. (c) When will the population reach 15,000? 8. A rectangular building is to cover 20,000 square feet. Building lots are rectangular. Zoning regulations require 20 foot borders in front and back of the building and a 10 foot border on each side. Use the optimization techniques of calculus to find the dimensions of the smallest piece of property on which the building can be legally built. 9. A worker is drawing cider from a storage vat at the rate of R(t) = 25 t 2 liters per minute. Use a Riemann sum with 4 subintervals, taking the sample points to be left endpoints, to estimate the amount of cider that flows from the tank during the first four minutes. Show all work and illustrate the Riemann sum with a diagram. 10. A potato is launched with an initial upward velocity of 112 ft/sec from the roof of a dorm 128 ft above the ground. Throughout its flight, the acceleration of the potato is a constant 32 ft/sec 2. (a) Find the formulas for both the velocity of the potato at time t sec and the height of the potato at time t sec. (b) Find the maximum height of the potato. 3 Fall 2007 Final Exam 1. For each part, if the statement is always true, circle the printed capital T. If the statement is sometimes false, circle the printed capital F. (a) If the radius of a sphere is increasing at 3 ft/sec then its volume is increasing at 27 cubic feet per second. (Volume = 4 3 πr3.) (b) The derivative of a product is the product of the derivatives. (c) If f(x) is differentiable and f( 1) = f(1), then there exists a number c such that f (c) = 0. (d) According to l Hôpital s rule, cos(2x) 2 sin(2x) 4 cos(2x) lim x 0 3x 2 = lim = lim = 4 x 0 6x x (e) If f (x) > 0 and g (x) < 0, then f(x) g(x) is concave up. 2. Use the fundamental theorem to find the area between the curve y = x x 3 and the x-axis. 4

5 3. Find the derivative of the following functions. 4. Let (a) x t2 dt. (b) 3 cos x 1 + t2 dt. (a) Find x 2 4 x 2, x < 2, f(x) = 0, x = 2, k 3x, x > 2. lim x 2 f(x) and lim x 2 + f(x). (b) Is it possible to choose k so that the function f(x) is continuous at x = 2? Explain your reasoning in such a way that you demonstrate an understanding of the limit definition of continuity. 5. Given the function f(x) = 1 3 x x. (a) Find f (x). (b) Determine where f is increasing and where it is decreasing. (c) Find where the local extrema of f occur. (d) Find where the graph of f(x) is concave down. (e) Does f(x) have an absolute maximum value on the interval [ 5, 5]? Explain. 6. Let f(x) be a continuous function given by the following graph. (a) Find 9 f(x) dx. 1 (b) If F (x) = f(x) and F (1) = 2, determine the value of F (9). (c) If F (x) = f(x), for which x-values does F (x) have a local maximum value? 7. Given the graph of the function f(x) below, sketch the graph of its derivative f (x). 5

6 8. A landscape architect is planning borders for a 3000 square foot botanical garden. She will build a stone wall costing $50 per foot along all four sides of the outer border. The garden will then be divided into three equal sized rectangular plots (as shown) by two parallel rows of shrubs costing $25 per foot. Find the dimensions that minimize cost. 9. A boy on a dock is pulling a rope fastened to the bow of a small boat. If the boy s hands are 10 feet higher than the point where the rope is attached to the boat and if he is retrieving the rope a rate of 2 feet per second, how fast is the boat approaching the dock when 25 feet of rope are still out? (Assume that the rope is taut and lies along a straight line.) 10. Consider the function f(x) = ln(x) + 3, x > 0. x 6

7 (a) Find the equation of the tangent line at x = 1. (b) Use the tangent line approximation of f(x) to approximate f(1.02). 11. An unknown amount of a radioactive substance is being studied. After two days, the mass is grams. After eight days, the mass is grams. (a) How much was there initially? (b) What is the half-life of this substance? 7