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The final examination questions for a university-level mathematics course, math125, from december 2001. The questions cover various topics in calculus, including limits, derivatives, integrals, continuity, differentiability, and optimization. Students are required to show their work and use calculators only for numerical computations.
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Math125 Final Examination December 14, 2001
Show all your work/reasoning/computations. You may use results as discussed in class as long as they are quoted correctly Calculators may be used only for numerical computations, that is, no graphing and no programming functions are allowed
1.(15 pts) Find values of the following limits a)
lim x→ 3
x^2 − 9 x^2 − 4 x + 3
b)
lim x→+∞
4 x^2 − 5 x 2 x^2 + 1
2.(15 pts) Differentiate the following functions. a) f (x) = (2x + 1)^5 (3x − 2)^2 (4x + 1)^3
b)
f (x) = (1 +
x
c)
f (x) =
∫ (^) x
0
sin t^2 dt.
4.(10 pts) a) Show that the function f (x) = |x| is continuous at x = 0 using the defintion of conti- nuity of a function.
b) Show that f (x) = |x| is NOT differentiable at x = 0 using the definition of derivative of a function.
5.(10 pts) a) Find the linear approximation of the function f (x) =
x + 1 at x = 3
b) Estimate the value of
3 .95 using the result from the part a.
7.(10 pts) If V is the volume of a cube with edge length x and the cube expands at the constant rate of 300cm^3 /sec.. Find the rate at which the edge length x increases when x = 10cm.
9.(10 pts) A box with a square base and open top must have a volume of 32cm^3. Find the dimension of the box (i.e. the height and the side length of the square) that minimizes the amount/area of the material.
