Calculus I (Math161)
Exam 1 (Spring, 2016)
Department: Id number: Name:
1. (12 pts.) Let f (x) = x3+√x. Then find a δ > 0 such
that
|f (x) − 2| ≤ 1
1000 if |x − 1| ≤ δ.
2. (12 pts.) Suppose that f (x) is differentiable everywhere such that
f (0) = 2, f (π) = 4 and f0(x) ≥ sin x for all x. Then prove that f (x) = 3 − cos x for all x ∈ [0, π].
Department: Id number: Name:
3. (12 pts.) Prove that the identity
sinh−1(tan x) + tanh−1(sin x) = 0 holds for all x ∈ π2,3π2.
4. (12 pts.) Find the linearization of
f (x) =√3
1 + 3x
at x = 0. Also use it to give an approximate value for√3
1.03.
Department: Id number: Name:
5. (14 pts.) Evaluate the limits if it exists. If not, give reasons. (1) (6 pts.) lim x→0+ x1/6sin x (ex− 1)8/7 (2) (8 pts.) lim x→0+ 1 2x 2− 1 + sin(cos−1(tanh x)) x3
6. (12 pts.) Evaluate the limit
lim
x→π 2+
(tan x)(sec x−tan x)
if it exists. If not, give reasons.
Department: Id number: Name: 7. (14 pts.) (1) (6 pts.) If f is continuous on [0, π], prove that Z π 0 xf (sin x)dx = π 2 Z π 0 f (sin x)dx.
(Hint : You may use the substitution u = π − x.) (2) (8 pts.) Evaluate the integral
Z π
0
x sin x 1 + cos2xdx.
8. (12 pts.) Consider the function f (x) = 2x + cos x. It is a strictly increasing function in the sense that f (a) < f (b) if a < b. Find the integralR1ππ f−1(x)2dx.
