__._ .......... _ 'I ___________ J'"" -__._-._.- You may use your calculator on the following three questions. 11. Consider a curve given implicitly by the equation (1 + x)y3+ x'y 85 = 0. A. Calculate 51'? at a general point (x, y). B. Write the equation of the tangent line to the curve at the point (3, 1). C. At (3, 1 ), y(x) is defined implicitly as a function of x. Let g(x) be the inverse function of y(x). Compute g'(1). 12. A race car is running practice laps in preparation for an upcoming race. To judge how the car is performing, the crew takes measurements of the car's speed S(t) (in miles per hour, or mph) every minute. The measurements are given in the table below. 1' (minutes) 8(t) (mph) 0 201 205 208 214 218 212 219 223 220 221 10 217 218 216 \\OOOHJO'anth-JM ._.._. Mr A. Use the trapezoid rule with 4 equal subdivisions to approximate the total distance the car traveled (in miles) over the first 12 minutes. B. Find one approximation for 3(6), including the units. Explain what this quantity means in the context of the problem. C. What was the car's average speed in mph over the first 12 minutes? If the car needs to have an average speed of 210 mph to qualify for the race, is it currently running fast enough to qualify? 13. Let M be the region under the graph of f(x) = 83; from X = 0 to x = 5. A. Find the area of M. B. Find a value of c so that the line x = c divides the region M into two pieces with equal area. C. M is the base of a solid whose cross sections are semicircles whose diameter lies in the xy plane. The cross sections are perpendicular to the xaxis. Find the volume of this solid. You may NOT use your calculator on the following three questions. 14. it\"? ..\\\\/ ./ 113/ 234 Humhmc' Oil-0 1 l 1 1 S 6 I" 2 '3 Let g be a function whose domain is the closed interval [0,8]. The graph of g is shown above. The graph is made up of straight line segments and a semicircle. 2x-1 Let x] = j g(t)o't. 0 A. Computef(%). B. What is the domain of f? C. Find f'(3). D. Find the x-coordinate of the absolute maximum of f on its domain. 15. Water is draining from a small cylindrical tank into a larger one below it. The small cylindrical tank has a radius of 4 feet and a height of 6 feet; the large cylindrical tank has a radius of 8 feet and a height of 16 feet. The small tank is initially full of water, and the water drains out at a rate of % cubic feet per second. Note: The volume ofa cylinder is V = men. A. Find the volume Vs of the water remaining in the small tank as a function of time. B. How long does it take for the small tank to completely empty? 0. Let 2 be the depth of the water in the large tank, which is initially empty. Compute :72. D. What fraction of the total amount of water is in the large tank at time t = 6? 16. Suppose that fhas a continuous second derivative for all x, and that :10) = 1, f'(0) = 2, and f"(0) = 0. A. Does fhave an inflection point at x = 0? Explain your answer. B. Let g'(x] = (3x2 + 2)f(x) + (X3 + 2x + 5)f'(x). The point (0, 5] is on the graph of 9. Write the equation of the tangent line to g at this point. C. Use your tangent line to approximate 9(0. 3). D. Find g"(0]. 17. Consider the differential equation given by 3: = %. A. On the axes provided below, sketch a slope field for the given differential equation at the nine points indicated. B. Let y =x) be the particular solution to the given differential equation with the initial condition . Based on your slope field, how does the value ofl) compare tof(U)? Justify your answer. 0. Find the particular solution 32 =f(x} to the given differential equation with the initial conditionfm) = 3. Use your solution to findf(0.2). r t 3 o I 2 O t 1 0