lOMoARcPSD|3207746
Exam 2015, questions - Math exam sample for midterm
Differential & Integral Calculus II (Concordia University)
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lOMoARcPSD|3207746
CONCORDIA UNIVERSITY
Department of Mathematics & Statistics
Number
205
Date
8 March, 2015
Only approved calculators are allowed
Show all your work for full marks
Course
Mathematics
Examination
Midterm
Special
Instructions:
1. (10 marks): (a) Graph the function f (x) =
to calculate
R1
(
−x
√− 1
− 1 − x2
Sections
All
Duration
1 h 30 min
if x ≤ 0
, and use it
if x > 0
f (x) dx in terms of area.
−3
(b) Let for continuous functions f and g we know
R0
f (x) dx = 2,
−3
R5
g(x) dx = 4 . Calculate
0
R5
0
R5
f (x) dx = 6,
−3
[3 − 2f (x) + 5g(x)]dx, or explain why it is impossible.
2. (6 marks): Use the Fundamental Theorem of Calculus to find the derivative of the function
F (x) =
Zx2
esin t dt
x
.
3. (10 marks): Calculate the following indefinite integrals
Z
x
dx
(b)
cos x cos3 (sin x) dx
x2 + 3x + 2
√
4. (6 marks): Let f (x) = x. Find a value c on the interval [1,4] such that f (c) is equal to
the average value of f (x) on that interval [1,4].
(a)
Z
5. (12 marks): Evaluate the following definite integrals (do not approximate):
(a)
Z2
0
√
x5
dx
1 + 2x2
(b)
Zπ
x2 sin(2x) dx
0
6. (6 marks): Sketch the graphs of functions y = |x| − 2 and y = 4 − x2 , and find the area
of the region bounded by these graphs.
Bonus. (3 marks): Prove that for any function f (x) integrable on the interval [a, b] the right
and left Riemann sums Rn and Ln are connected by the relation
f (b) − f (a)
R n − Ln =
(b − a)
n
Downloaded by Bob Lungu (bob@freiman.cc)