110-Calculus-Physics
Midterm I
Instructor: Ray Lai
October 27th, Fall 2021
Name:
Student ID ]:
Instructions:
1. This exam consists of 8 Problems with total of 110 points.
2. The maximum of the midterm is 100 points.
3. Put away books, notes, calculators, cell phones, and other elec-
tronic devices. No discussion allowed during the exam.
4. It might be a good idea to finish the simpler questions first.
Good luck!
Honor Pledge
I will not give or receive aid on this examination in any
format, such as browse through internet or mobile phone.
(In particular, this exam is NOT an open-book exam.) This
includes discussing the exam with students who have not
yet taken it. I understand that if I am aware of cheating
on this examination, I have an obligation to inform Pro-
fessor Lai. I also understand that Professor Lai will follow
NCKU’s Academic Integrity Policy if he detects acts of
academic dishonesty.
Signature:
1
2
x
1. (20 points) Consider the function f (x) = x+1 with the
domain Dom(f ) = (−∞, −1) ∪ (−1, ∞).
(a) (5 points) Prove that f (x) is one to one. (Do not use
the graph.)
(b) (5 points) Find the inverse function y = f −1 (x) and
the domain Dom(f −1 ).
(c) (5 points) If f (x) has the same tangent line slope at
two different points A = (a, f (a)) and B = (b, f (b)),
then what is the relation of a and b?
(d) (5 points) Following (c), find two different points A
and B on the graph of f (x) such that the line AB is
the normal line to both tangent lines of f (x) at A and
B.
2. (24 points) Determine if the following limits exists or
not. Provide your full reasoning in detail.
sin(θ − 1)
(a) (8 points) lim 2
θ→1 θ + θ − 2
!
1 1
(b) (8 points) lim p −
t→0 t 1 + |t| t
p
(c) (8 points) lim 2
x +x+x
x→−∞
3. (24 points) Compute the derivatives.
(a) (6 points) Use the definition of the derivative to deter-
d
mine dx ln |x|. (Consider x > 0 and x < 0. You don’t
need chain rule here.)
d x2 − 4
(b) (8 points) Find .
dx x2 + 4
x=2
3
d2
(c) (10 points) Find 2 (cot(x) + sec(x)).
dx
d10
(d) (8 points) Find 10 (x8 log7 x), where log7 x := ln x
ln 7 .
dx
4. (16 points) Suppose that f (x) is a differentiable func-
tion defined on (−1, 1) and
f (x)
lim 2 = L.
x→0 x
Consider the function
(
f (x) sin( x1 ) if 0 < |x| < 1
g(x) = .
A if x = 0
(a) (10 points) Find f (0) and f 0 (0).
(b) (6 points) For which A ∈ R the function g(x) is con-
tinuous at x = 0? Why? (Hint: Find out lim |g(x)|
x→0
first.)
5. (18 points) This is about inverse trigonometry.
(a) (6 points) Define tan−1 (x) and specify its domain and
range. √
(b) (6 points) Find tan−1 ( 3).
(c) (6 points) Simplify (sec(tan−1 (x)))2 into a rational
polynomial in x.
6. (8 points) Suppose that f (x) is continuous on the
closed interval [0, 1] and that 0 ≤ f (x) ≤ 1 for every x ∈
[0, 1]. Show that there exists a number c ∈ [0, 1] such that
f (c) = c.
