1. (5 pts) Prove that the function f (x) = tan x − 1 x is strictly
2
increasing on (− π , π ). Compute the derivative of the inverse function
22
π
2. (5 pts) Let f be a positive function on (0, π2 ) (f does not have to
be differentiable). Which of the following statements imply that f is
1-1? Prove your answer.
(A) f (x) cos x is a strictly increasing function on (0, π2 );
(B) f (x) sin x is a strictly increasing function on (0, π2 );
(C) f (x) ln(cos x) is a strictly increasing function on (0, π2 );
(D) f (x)(1 − sin x) is a strictly increasing function on (0, π2 ).
3. (5 pts) Let f(x) = xm, m > 0 on the interval [1,2]. Compute the
limit 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 and
l’Hospital’s Rule.
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