Calculus Function Key

Question 4 Let f be the function defined for x > 0, with f(e) = 2 and f', the first derivative of f, given by f'(x) = x²ln x. (a) Write an equation for the line tangent to the graph of f at the point (e, 2). (b) Is the graph of f concave up or concave down on the interval 1 < x < 3? Give a reason for your answer. a) f'(x) = x²ln x f'(e) = e²lne = e² line tangent thru (e, 2) y - 2 = e²(x - e) or y = e²x - e³ + 2 b) f''(x) = x + 2xln x For 1 < x < 3, x and 2xln x are > 0, so f''(x) is > 0. f(x) is concave up on (1,3) Question 4 (seconds) 0 10 20 30 40 50 60 70 80 v(t) (feet per second) 5 14 22 29 35 40 44 47 49 Rocket A has positive velocity v(t) after being launched upward from an initial height of   0 feet at time t = 0 seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 ≤ t ≤ 80 seconds, as shown in the table above.   (a) Find the average acceleration of rocket A over the time interval 0 ≤ t ≤ 80 seconds. Indicate units of measure.   average rate of change of f(x) = (f(b) - f(a)) / (b - a) on [a, b] (b) Using correct units, explain the meaning of ∫(10 to 70) v(t)dt. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate ∫(0 to 70) v(t)dt. a) v(80) - v(0) / 80 - 0 = 49 - 5 / 80 = 11/20 ft/sec² b) ∫(10 to 70) v(t)dt is the distance traveled, in feet, by the rocket from t = 10 sec to t = 70 sec. MRAM = 20(v(20) + v(40) + v(60)) = 20(22 + 35 + 44) = 2020 ft. [correct units in a) and b] Question 3 Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per minute. The oil slick takes the form of a right circular cylinder with both its mdius and height changing with time. (Note: The volume of a right circular cylinder with radius and height & is given by V= pi * r ^ 2 * h .) (a) At the instant when the radius of the oil slick is 100 centimeters and the height is 0.5 centimeter, the radius is increasing at the rate of 2.5 centimeters per minute. At this instant, what is the rate of change of the height of the oil slick with respect to time, in centimeters per minute? (b) A recovery device arrives on the scene and begins removing oil. The rate at which oil is removed is R(t) = 400sqrt(t) cubic centimeters per minute, where is the time in minutes since the device began working. Oil continues to leak at the rate of 2000 cubic centimeters per minute. Find the time I when the oil slick reaches its maximum volume. Justify your answer. (c) By the time the recovery device began removing oil, 60,000 cubic centimeters of oil had already leaked. Write, but do not evaluate, an expression involving an integral that gives the volume of oil at the time found in part (b).