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Solutions to the final exam of math 105, sections a and d, held on december 13, 2001. The exam covers various topics in calculus, including limits, derivatives, antiderivatives, and parametric curves.
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Name Final Exam Math 105, sections A and D December 13, 2001
Total
(a) lim x→ 1 x
(^5) − x (^4) − 2 x (^3) + 2x (^2) + x − 1 x^5 − 2 x^4 + 2x^2 − x
(b) lim x→ 0 e
x (^) − cos x sin x
f ′(x) = (x^ −^ 3)
(^2) (x + 3) (x + 1)^3
and f ′′(x) = 6 (x^ −^ 3) (x^ + 5) (x + 1)^4
For what values of x is f (x) increasing, and for what values is it decreasing? For what values of x is f (x) concave up, and for what values is it concave down? What are the local maxima and minima (peaks and valleys) of f (x)? What are its inflection points? Explain. For extra credit, find a possible f (x).
(a) If a(x) = ln (sin x), find a′(x). (b) If b(x) = ln (cos x), find b′(x).
(c) Evaluate either
cot x dx or
csc x dx (your choice)
(d) Evaluate either
tan x dx or
sec x dx (your choice)
(i)
12 x^5 − 8 x + 5 + 6x−^4 + 3 sin x − 4 cos x
dx
(ii)
∫ (^) (x + 6)(x + 1) x^2 dx
