THE INTERNATIONAL UNIVERSITY (IU) - VIETNAM NATIONAL UNIVERSITY - HCMC
MIDTERM EXAMINATION
Semester 1, 2016-17 • 8th Nov 2016 • 90 minutes
Head of Dept. of Mathematics:
CALCULUS I
Lecturers:
Assoc. Prof. Nguyen Dinh
TT Duong, JC Harris, HB Minh, NM Quan, MD Thanh
INSTRUCTIONS: Each student is allowed a scientific calculator and a maximum of two
double-sided sheets of reference material (size A4 or similar), stapled together and marked
with their name and ID. All other documents and electronic devices are forbidden.
Question 1. (10 marks)
Find a formula for the inverse of the function
f (x) = x2 + 2x,
x ≥ −1.
Question 2.
a) (10 marks) Evaluate the limit
√
6−x−2
.
lim √
x→2
3−x−1
b) (10 marks) By using the Squeeze Theorem, or otherwise, evaluate the limit
lim (x − π) sin
x→π
π
.
x−π
Question 3.
Let

 cos x,
0,
g(x) =

1 − x2 ,
if x < 0
if x = 0
if x > 0
a) (5 marks) Explain why g(x) is discontinuous at x = 0.
b) (5 marks) Sketch the graph of g(x).
Question 4.
a) (5 marks)
b) (5 marks)
c) (5 marks)
starting from
Let C be a circle with radius 2 centred at the point (2, 0).
Write an equation for the circle C.
Is curve C the graph of a function of x? Explain your answer.
Write parametric equations to traverse C once, in a clockwise direction,
the origin.
—– Continued on next page —–
Question 5.
a) (5 marks) Differentiate the function
2
h(x) = e−x sin(2x).
b) (10 marks) Use logarithmic differentiation to differentiate the function y =
1 ln x
x
.
Question 6. (15 marks)
d2 y
dy
and 2 at the point (0, −2) on the curve 4x2 + y 2 = 4.
Evaluate
dx
dx
Question 7. A particle is moving along an x-axis. Its x-coordinate (in meters) at time
t seconds is given by
x(t) = t3 − 3t + 1, t ≥ 0.
a) (5 marks) Find the velocity of the particle at time t.
b) (5 marks) When is the particle at rest?
c) (5 marks) Find the total distance traveled by the particle during the first 3 seconds.
—– END —–
INTERNATIONAL UNIVERSITY (IU) – VIETNAM NATIONAL UNIVERSITY – HCMC
CALCULUS 1 – MIDTERM EXAMINATION
Semester 1, 2020-21 • Mon 28 Dec 2020 • Duration: 90 minutes
Chair of Dept. of Mathematics:
Lecturers:
Prof. Pham Anh Huu Ngoc
Prof. Nguyen Dinh, Dr. Janet C. Harris, Dr. Nguyen Anh Tu
INSTRUCTIONS:
• Each student is allowed one double-sided A4 sheet of reference material marked with their name
and ID. Calculators are forbidden. All other documents and electronic devices are also forbidden.
• Attempt all questions. Each question carries 10 points. (Total 100 points.)
• Write out your answers in detail, presenting and explaining each step.
1. The piecewise function f comprises two straight line segments and a semi-circle, as shown in the
figure below. Write a formula for the function f .
2. The graph below shows a function g with domain [0, 2] and range [0, 1]. Suppose h(x) = 2g(x−1).
Sketch the graph of h and state its domain and range.
3. Let f be a function defined by
(
f (x) =
x2 −x
3x
k
if x 6= 0
if x = 0
where k is a constant.
a) Find the value of k such that f is continuous at x = 0.
b) With the value of k found in part a), is function f differentiable at x = 0? Explain your answer
clearly. If f 0 (0) exists, find its value.
4. Prove that
lim x2 esin(π/x) = 0.
x→0
PLEASE TURN OVER
5. Find the point on the curve y = 1 + 2ex − 3x at which the tangent line is parallel to the line
x + y = −5.
6. Suppose f (x) and g(x) are differentiable functions, and h = f −1 ◦ g. Assume f (2) = g(1) = 2,
f 0 (2) = 3 and g0 (1) = 6. Find h0 (1).
7. Find the derivative of y = (x2 + 1)sin x .
8. Find the tangent line to the curve y sin x + x cos y = π/2 at the point (π/2, π).
9. A runner exercises on a straight road. His displacement (in kilometers) relative to an origin O is
s(t) = 2t 3 − 9t 2 + 12t, 0 ≤ t ≤ 3, where time t is measured in hours.
a) When is the velocity of the runner zero?
b) Find the total distance traveled by the runner between t = 0 and t = 2.
10. Consider the equation:
2x − 1 = sin x.
a) Show that equation (1) has at least one solution on the interval (0, 2).
b) Show that equation (1) has no solution on [2, ∞) and no solution on (−∞, 0].
Optional Bonus Question (+5 points):
10. c) Show that equation (1) has only one solution in R.
Hint: To answer this you may use anything you have learned in Calculus 1 or in high school.
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