11. Consider a curve given implicitly by the equation (1 + x)y3 + xy - 85 = 0. A. Calculate " 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. t (minutes) S(t) (mph) 0 201 205 N - 208 214 218 212 219 223 220 221 10 217 11 218 12 216 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 S'(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 rst 12 minutes? If the car needs to have an average speed of 2m mph to qualify for the race. is it currently running fast enough to qualify? 13. Let M be the region under the graph of x} = 31, from x = D to x = 5. A. Find the area of M. E. Find a value of c so that the line .1: = 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 th roo questions. 14. no} unwoun- 123456?3 Let g be a function whose domain is the closed interval [-13.8]. The graph of g is shown above. The graph is made up of straight line segments and a semicircle. Ext Let x} = I grant. a A. Compute :%) B. What is the dcrnain of i"? c. Find r13}. D. Find the x-coordinate of the absolute maximum of f on its domain. 15. Water is draining from a small cytindrical tank into a larger one below it. The small cylindrical tank has a radius of 4 feet and a height of E feet; the large cylindrical tank has a radius of 8 feet and a height of 16 feet. The small tank is initiallyr full of water, and the water drains out at a rate of % cubic feet per second. Note: The volume of a cylinder is V = malt. A. Find the volume V: 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? G. Let 2 be the depth of the water in the large tank. which is initially empty. Compute 3%. D. What fraction of the total amount of water is in the large tank at time t = E? 16. Suppose that f has a continuous second derivative for all x, and that } = 1. f'f} = 2. and f'ii = u. A. Does if have an inection point at x = ? Explain your answer. B. Let 91):] = {3):2 + 2}x} + {x3 + 2x + ERIK]. The point [(1.5) is on the graph of g. Write the equation of the tangent line to g at this point. C. Use your tangent line to approximate 9110.3). [1. Find gm]. 17. Consider the differential equation given by dx xy 2 A. On the axes provided below, sketch a slope field for the given differential equation at the nine points indicated B. Lety = ((x) be the particular solution to the given differential equation with the initial condition . Based on your slope field, how does the value of f(0.2) compare to f(0)? Justify your answer. C. Find the particular solution y = (x) to the given differential equation with the initial condition f(0) = 3. Use your solution to find f(0. 2). 3 2 1 -1