Cascadia Community College Calculus

1. (6 Points) Taco Time reported $1376 million in total revenue in 1986. In 2012, the total revenue was $52.2 billion. (You may use a calculator here when asked to report decimal approximations.) a.) Assuming an exponential model, find the growth rate k, to four decimal places. b.) Determine the revenue function R, with R(t) in billions of dollars. c.) Predict the company’s total revenue for 2027. (Round your answer to the nearest whole number. Make sure to include units.) 2. (12 Points) The lizard population, P (t), in a square kilometer of the Mojave Desert after t years can be approximated by the logistic equation P (t) = 3000 20 + 130e−0.214t a.) Find the tortoise population after 0 years, 10 years, and 15 years. (Round your answer to the nearest whole number. Make sure to include units.) 2 b.) Find the rate of change in the population, P 0 (t). c.) Find the rate of change in the population after 0 years, 10 years, and 15 years. (Again, make sure to include units.) d.) What is the limiting value for the population of lizards in a square kilometer of the Mojave Desert? 3 3. (14 points) Let f (x) = −3(x − 2)2/3 + 3. (a) What are the critical points of f ? (Notice that you are asked for a point, so you should provide the x- and y- coordinates for any critical points you found). (b) On what intervals is f increasing or decreasing? (Use interval notation) (c) At what values of x, if any, does f assume local maximum and minimum values? Be sure to specify what type of extremum you find (label as being a maximum or a minimum). 4 (d) What are the possible inflection points of f ? (e) On what intervals is f concave up or concave down? (f) Are the points you found in part (d) actually points of inflection? Why or why not? 5 4. (8 Points) Recently, while digging outside of Albuquerque, New Mexico, archaeologists found maize pollen that had lost 39% of its carbon-14. The age of the maize pollen was evidence that Native Americans had been cultivating crops in the southwest centuries earlier than scientists had thought. What was the age of the pollen, if carbon-14 has a decay rate of 0.012%? (Round your answer to the nearest whole number. Make sure to include units.) 5. (10 points) Sketch the curve y = f (x) that fits the following conditions, given that f is continuous on (−∞, ∞). You must justify your work with some work or brief written explanation: f (−1) = −2, f 0 (−1) = 0, f 00 (−1) > 0 f (0) = 0, f 0 (0) = 0, f 00 (0) < 0 f (1) = −2, f 0 (1) = 0, f 00 (1) > 0 6