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The final exam for a college-level mathematics course, math 105. The exam covers various topics including calculus, integration, differentiation, and logarithmic functions. Students are required to solve problems related to finding limits, derivatives, antiderivatives, and solving differential equations.
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MATH 105 FINAL EXAM December 12, 2002
Name:
Remember that final answers are not as important as how you get there. Show all your steps clearly so you will be eligible for the most partial credit. Simplify arithmetic quantities completely. Good luck!
1.) (10 pts.) The graph shown below is that of the velocity of an object (in meters/second).
a.) Find an upper and a lower estimate of the total distance traveled from t = 0 to t = 5 seconds.
b.) At what times is the acceleration zero?
2.) (5 pts.) If f is even and
∫ (^3)
0
f (x)dx = 16, what does
∫ (^3)
0
f (x)dx =?
3.) (5 pts.) True or False: An antiderivative of 2x cos x is x^2 sin x. Whether you answer true or false, explain your answer.
4.) (5 pts.) True or False: If a function is continuous, then it must be differentiable. If true, explain. If false, give a counterexample.
7.) (10 pts.) Find the general solution of the differential equation dy dx
x
8.) (10 pts.) Compute
dy dx if
a.) y =
√ (ln x)^2 + 5
b.) xy^2 + sin y + x^3 = 8
9.) (10 pts.) Find a pair of parametric equations for the line passing through the points (4, 1) and (− 2 , −6).
10.) (10 pts.) Find the derivative of g(x) = 3x^2 + 2x − 4 algebraically. That is: use the definition of the derivative, which involves limits. Shortcut-only answers will receive NO CREDIT.
13.) (5 pts.) The graph of f ′^ is shown below. (The graph of f is not shown.) Use the graph of f ′ to answer the following questions.
a.) On which intervals, if any, is f increasing?
b.) At which values of x does f have a local maximum? A local minimum?
c.) On which intervals, if any, is f concave up?
d.) Which values of x, if any, correspond to inflection points on the graph of f?
14.) (5 pts.) Use the graph of f ′^ in question (13.) above to do the following.
a.) Sketch a graph of f ′′.
b.) Assume that f (0) = 0. Sketch a graph of f. (Your graph need only have the right general shape. You do not need to put units on the vertical axis.)
