Calculus I: Midterm: 09:30-11:00, April 21, 2022
Answer the following questions. Show your work.
1.(30pt) Determine whether the following statements are true (T) or false (F). (True statements should
be true ‘ALWAYS’ without exception.) The right answer gives 3 points, no answer gives 1 point, and a
wrong answer gives 0. Do not justify your answer.
(a) If limx→0 f (x) exits, then the limit should be f (0).
(b) If f (x) is integrable on [0, 1], then f (x) is continuous on [0, 1].
∫1
(c) 0 ex cosx dx ≤ e − 1.
(d) If f ′ (x) ≤ g ′ (x) for (0, 1), then f (x) ≤ g(x) for (0, 1).
(e) If f (x) is differentiable at x = 0, limx→0 f (x) = f (0).
(f) If f (1) = −2 and f ′ (x) ≤ 2 for 1 ≤ x ≤ 4, then f (x) ≤ 3 for 1 ≤ x ≤ 4.
(g) 2sin−1 x = cos−1 (1 − x2 ) for x ≥ 0.
(h) If f (x) is integrable on [0, 1], then
∫1
(i) −2 x−2 dx = −x−1 ]1−2 = − 32 .
∫1
sinx
(j) −1 (x5 − 7x9 + (1+x
2 )4 ) dx = 0.
∫1
0 f (x)dx ≤
∫1
0 |f (x)|dx.
2.(30pt) Find the derivative of the function.
(a) f (x) = x tan−1 (4x)
(b) x2 cosy + sin(xy) = 2y
3.(25pt) Evaluate the followings.
√
∫
3
(a) (tanx + 1+√x x )dx
(b) limx→0+ ( x1 − tan1−1 x )
(c) f (x) =
(c)
∫ e3
e
∫ sin x
x2
cos (t2 )dt
√1 dx
x ln x
4.(15pt) Find the area of the largest rectangle that can be inscribed in the ellipse x2 /a2 + y 2 /b2 = 1.
5.(25pt) Consider f (x) = x2 /ex for [−1, 3],
(a) Find the absolute maximum and minimum values on [−1, 3].
(b) Find the intervals of concavity and the inflection point(s) on [−1, 3].
6.(25pt) Find the value a to make each statement to be true.

(sinx) ln|x|, x ̸= 0
(a) The function f (x) =
is continuous at 0.
a,
x=0
x+a x
(b) limx→∞ ( x−a
) = e.
1