Math 115 Final Review 5 December 15, 2015

Transcription

1 Math 115 Final Review 5 December 15, 2015 Name: 1. (From Team Homework 7) A 12-foot tree is 8 feet from a light that can be raised up and down, as shown in the figure below. 12 ft h shadow 8 ft The height h of the light (in feet) is a function h = w(t) of the time t in hours since noon. A table of values of w(t) and its derivative w (t) is given below. t w(t) w (t) a. How long is the tree s shadow at 1:00 PM? b. How fast is the length of the shadow changing at 2:00 PM? Is the length increasing or decreasing at that time?

2 Math 115 / Final Review 5 (December 15, 2015) page 2 2. Sand is falling from the upper chamber of an hourglass into the lower chamber. The two chambers of the hourglass are cones with base diameter 6 cm and height 8 cm, as shown in the digram to the right. Let U be the height of the sand in the upper chamber and let L be the height of the sand in the lower chamber. Recall that the volume V of a cone with height h and base radius r is 8 cm 6 cm U V = 1 3 πr2 h. L a. 20 seconds after the hourglass is turned, U = 4 and L = 2. What is the total volume of sand in the hourglass? b. 40 seconds after the hourglass is turned, U = 3 and U is decreasing at a rate of 0.15 cm per second. How quickly is the volume of sand in the upper chamber decreasing at this time? c. 10 seconds after the hourglass is turned, L = 0.5 and the volume of sand in the bottom chamber is increasing at a rate of 0.3 cm 3 per second. How quickly is the height L increasing at this time?

3 Math 115 / Final Review 5 (December 15, 2015) page 3 3. A spherical snowball rolls down a hill and gathers more snow. The volume V of the snowball grows at a rate proportional to the speed s of the snowball, so dv dt = ks for some constant k. The graph below shows the speed s(t) of the snowball t seconds after it begins rolling down the hill. (meters/second) s(t) t (seconds) a. Use a right Riemann sum with 4 subintervals of equal length to estimate the distance traveled by the snowball during the first 12 seconds. b. Is your estimate in part a an underestimate or an overestimate for the distance traveled? underestimate overestimate c. After 3 seconds, the volume of the snowball is increasing at a rate of 5 cm 3 per second. Use this information to find the value of the constant k. d. After 8 seconds, the snowball has a volume of 40 cm 3. At what rate is the volume of the snowball increasing after 9 seconds? Include units. e. At what rate is the radius of the snowball increasing after 9 seconds? Include units. f. At what rate is the surface area of the snowball increasing after 9 seconds? Include units.

4 Math 115 / Final Review 5 (December 15, 2015) page 4 4. Red and Blue are racing from Cinnabar Island to the Seafoam Islands, a total distance of 10 kilometers. Red is surfing on his Poliwrath with velocity r(t) meters per minute and Blue is surfing on his Golduck with velocity b(t) meters per minute t minutes after their race began. Below is a graph of the functions r and b. Unfortunately, the graph was drawn by the same forgetful scribe who drew the graph for the team homework, so there are no labels on the vertical axis. b(t) r(t) t (minutes) As shown in the graph, Blue won the race after 12 minutes, and Red finished 3 minutes later. a. When was Red traveling faster than Blue? b. When was Red ahead of Blue? c. What was Red s maximum speed? d. What was Red s average speed during the race?

5 Math 115 / Final Review 5 (December 15, 2015) page 5 5. Consider the curve C in the xy-plane defined by the equation a. Find a formula for dy dx 3(x 2 + y 2 ) 2 = 25(x 2 y 2 ). for the curve C. b. Which of the following points is on the curve C? (1, 2) (2, 1) (3, 2) (2, 3) c. Find the tangent line to the curve C at the point you circled in part b. d. Find the x- and y-coordinates of all points on the curve at which the tangent line is horizontal. Note that the tangent line is not horizontal at the point (0, 0).

6 Math 115 / Final Review 5 (December 15, 2015) page 6 6. Consider the function f(x) = (x 2 a)e x where a is a positive constant. a. Find the minimum and maximum values of f(x) on the interval x 0. Your answers may involve the constant a. b. Find the minimum and maximum values of f(x) on the interval (a, ). Your answers may involve the constant a. c. Find the minimum and maximum values of f(x) on the interval (, 0]. Your answers may involve the constant a.

7 Math 115 / Final Review 5 (December 15, 2015) page 7 7. Consider the function L(t) = et ae t + b where a and b are positive constants. The derivative and second derivative of L are given by L (t) = Find the values of the constants a and b if and L has an inflection point at t = 20. be t (ae t + b) 2 and L (t) = bet (ae t b) (ae t + b) 3. lim L(t) = 100 t