University of Missouri Columbia Calculus Derivative & Identities Questions

1. (5 pts) Prove that the function f (x) = tan x ? 1 x is strictly2

increasing on (? ? , ? ). Compute the derivative of the inverse function

22

?

2. (5 pts) Let f be a positive function on (0, ?2 ) (f does not have tobe differentiable). Which of the following statements imply that f is1-1? Prove your answer.

(A) f (x) cos x is a strictly increasing function on (0, ?2 );
(B) f (x) sin x is a strictly increasing function on (0, ?2 );
(C) f (x) ln(cos x) is a strictly increasing function on (0, ?2 );(D) f (x)(1 ? sin x) is a strictly increasing function on (0, ?2 ).

3. (5 pts) Let f(x) = xm, m > 0 on the interval [1,2]. Compute thelimit of the lower Riemann sums

lim L(f,Pn)n??

nnnn
corresponding to the partitions P = {1, 2 1 , 2 2 , …, 2 n?1 , 2}. Compare

the result with the integral ?? 2 xmdx.1

Hint: use the formula for the sum of a geometric progression andl’Hospital’s Rule.

atthepointy0 =1? .8

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