WIHN0001 Name:
Summer 2022 Matriculation ID:
Final Exam
5/8/2022
Time Limit: 90 Minutes
Exam: Mathematics in Natural and Economics Science I
Examiner: Prof. Dr. Jingui Xie
This exam contains 12 pages (including this cover page) and 5 questions. Total of points is
100.
Instructions:
• A paper(A4 size) of cheat sheet and the pocket calculator are allowed.
• For calculations and answers, please use blank sheets. Write name and matriculation
number on every page.
• State all formulas used and show all calculations! Results (numbers) without calcula-
tions score zero points!
• Do not use a pen with red ink.
• Record your answers legibly (readably) and simplify the expression as much as possible.
• With respect to rounding, 3 places after the decimal point should be kept.
• You are requested to answer all of the questions. Points for each of the questions are
provided with each question.
Grade Table (for examiner use only)
Question: 1 2 3 4 5 Total
Points: 20 20 20 20 20 100
Score:
WIHN0001 Final Exam - Page 2 of 12 5/8/2022
1 0 −1
1. (20 points) Let A = 0 1 0 .
1 2 1
(a) (5 points) Compute the determinant of the matrix A.
(b) (5 points) Compute the inverse of the matrix A.
(c) (5 points) Compute det(A−1 · A−1 ).
WIHN0001 Final Exam - Page 3 of 12 5/8/2022
(d) (5 points) Consider the linear system
1+k 0 −1 x1 1
(A + kI) · x = 0 1+k 0 x2 = 2
1 2 1+k x3 3
Find for what value(s) of k this system has (i) a unique solution? (ii) no solution?
(iii) infinitely many solutions?
WIHN0001 Final Exam - Page 4 of 12 5/8/2022
2. (20 points) .
(a) (6 points) Determine the following limit if it exists
R x − 1 t2
e 2 dt
lim 0 3
x→0 x + x5
(b) (7 points) Given e4 > 17, prove that f (x) = ex − 12 x2 − x − 5 has a unique zero
in the interval (0,4). Find an approximate value for this zero by using Newton’s
method once (one iteration), with x0 = 1.
WIHN0001 Final Exam - Page 5 of 12 5/8/2022
(c) (7 points) Prove that
5n − 3n ≥ 2n
for any positive integers n.
WIHN0001 Final Exam - Page 6 of 12 5/8/2022
3. (20 points) Define the following function.
Z x+1
f (x) = −t2 + t + 1dt
x
(a) (6 points) Calculate the first derivative. Is f (x) monotonic?
(b) (5 points) Find the region where f (x) is convex and the one where f (x) is concave.
(c) (5 points) Write Taylor’s formula for f (x) around the point x = 0 with n = 2.
WIHN0001 Final Exam - Page 7 of 12 5/8/2022
(d) (5 points) Find the maximal value of f (x) and limx→+∞ f (x).
WIHN0001 Final Exam - Page 8 of 12 5/8/2022
4. (20 points) (a) (6 points) Define
(
2x + 1, x < 0
f (x) =
e−x − x, x ≥ 0
Compute the following integral.
Z 1
f (x)dx
−1
(b) (7 points) Compute the following integral.
Z 3p √
1+ 1+x
√ dx
0 2 1+x
WIHN0001 Final Exam - Page 9 of 12 5/8/2022
(c) (7 points) Compute the following integral.
Z e2
(x3 + x) · ln(x)dx
1
WIHN0001 Final Exam - Page 10 of 12 5/8/2022
5. (20 points) A firm produces two different kinds, A and B, of a commodity. The profit
of producing x units of A and y units of B is f (x, y) = −x2 − y 2 + 4x + y + 10, where
x > 0 and y > 0.
(a) (5 points) Find critical point(s) of f (x, y).
(b) (5 points) Use the second derivative test to classify critical point(s).
WIHN0001 Final Exam - Page 11 of 12 5/8/2022
(c) (10 points) The firm must produce at least b > 0 units of the commodity in total,
i.e. x2 + y 2 ≤ b. To find the best producing level, we must solve the following
optimization problem:
f ∗ (b) = max − x2 − y 2 + 4x + y + 10
x,y
s.t. x2 + y 2 ≤ b
x, y ≥ 0
Find the optimal solution.
Hint: the optimal solution depends on b.
WIHN0001 Final Exam - Page 12 of 12 5/8/2022
Additional Paper
