Mathematics 134 – Intensive Calculus for Science 2 Makeup Exam I – February 20, 2006

Directions. Do all work in the blue exam booklet. You may use a calculator on any of the following problems. There are 100 possible points.

I. The given graph shows y = f(x).

A) (10) Use the graph to determine ∫_{0}^{4} f(x) dx. B) (5) Let F (x) be the antiderivative of f(x) with F (0) = 1. What is F (4)? C) (15) Give a qualitative sketch of y = F (x) from part B. Your graph should show where

F is increasing/decreasing, and where F is concave up/concave down. D) (5) The function G(x) is G(x) = ∫_{0}^{x} f(t) dt using the function f in the graph above.

What is G′(1)?

II. A car travels down a straightline road with acceleration a(t) = 4 + t/2 m/sec^{2}.

A) (10) Solve the equation dv/dt = 4 + t/2 to find the car’s velocity v(t), assuming

v(0) = 2 m/sec. B) (10) Use your answer v(t) from part A. If the car’s initial position is s(0) = 0, find

the position as a function of time by solving \frac{ds}{dt} = v(t) C) (10) What is the car’s average velocity over the time t = 0 to t = 3 sec?

III. Compute each of the integrals below using basic rules and/or substitution. You must show all work for full credit.

A) (10)

∫ 2x^{3} − \frac{4}{\sqrt{x}} + 5x\sqrt{x} + \frac{2}{x} dx

B) (15)

∫ sin(3x) + cos(5x) dx

C) (10)

∫_{0}^{1} xe^{2x^{2}−3} dx