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Mathematics 151, Fall 1999, Final Exam
1. a) Use your calculator to approximate
to four decimal places. Show the function exactly as you input it into your calculator.
b) Sketch the graph of the function f(x) = [3 cos(x-2) - x + 3] on your calculator and approximate the x-intercept of the graph to four decimal places. (Be sure your calculator is set to radians.)
2. Use your knowledge of Calculus to compute each of the following limits. If the limit is infinite or does not exist, explain why. (Do not use your calculator except as a double check.)
(a)
(b)
(c)
3. a) State the definition of "f(x) is continuous at a".
b)
Determine values of A and B so that f is continuous at 2.
c) For the values of A and B determined in (b), is f differentiable at 2? Explain.
4. If a particle, moving along a straight line, travels t4/( t + 4 ) feet in t seconds, how fast is it moving 4 seconds after it starts?
5. Evaluate the first and second derivatives of the function
6. Find the volume of the largest right circular cylinder that can be inscribed
in a sphere of radius 3.
7. Find the rate of change of y with respect to x ( i.e. dy/dx ) when x = 0 :
8. a) State the extreme value theorem. Be sure to include all hypotheses and conclusions.
b) Find the absolute maximum and minimum of f(x) = x4 - 8 x2 + 20 on [ -3, 1 ].
9. The dimensions of a rectangle are changing. At a certain instant its width is 10 inches and its length is 40 inches. If, at this instant, its width is increasing at the rate of 2 inches per second, how fast must its length shrink in order for the area of the rectangle to remain constant?
10. Find a function f, such that
11. Use the Fundamental Theorem of Calculus (not your calculator) to evaluate each of the following:
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