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University University of Winnipeg
Course MATH-1104-003 Introduction to Calculus II
Year 2026
Author Christine
Subject Mathematics
File Type PDF

19 views 12 pages

Assignment Instructions

MATH-1104 — Introduction to Calculus II University of Winnipeg | Winter 2026 | Dr. Ross Stokke
Test 1 — Questions
Saturday, February 7, 2026 | 90 minutes | Total: 80 marks
Special Instructions:
• All questions must be answered in the space provided. Show all work and justify answers.
• Calculators, mobile phones, smart watches, notes, etc. are not allowed.
• All electronic devices must be turned OFF and put away.
Question 1 [4 marks]
You are given that f is a function satisfying
2
f′′(x) = cos x + 12x and f(0) = 0,
What is f(x)?
Question 2 [4 marks]
Find the derivative of the function
Question 3 [8 marks]
f(π) = 2.
2 x2
g(x) = ∫ sin√(t + 1) dt x
x
Let g(x) = ∫ f(t) dt, where f is the continuous function with the graph shown.
0
The graph shows y = f(x) as a piecewise linear function: rising from (0, 0) to a peak of y = 4 around x = 4, remaining at y = 4 until x = 12 (forming a rectangle of area 4 × 4 = 16 from x = 4 to x = 8, plus triangular areas). The total area above the x-axis from 0 to 12 equals 64. Then f decreases linearly below zero from x = 12 to x = 24, forming a triangle with base 12 and height 8.
(a) [2 marks] Evaluate g(12).
(b) [2 marks] Evaluate g(24).
(c) [2 marks] Determine the intervals on which g is increasing and decreasing.
(d) [1 mark] Where does g have a maximum value?
(e) [1 mark] State the name of the theorem that tells us that g is differentiable on (0, 24).
Question 4 [6 marks]
(a) [3 marks] Express the definite integral as a limit of Riemann sums. (Use the right endpoints
of each subinterval as your sample points.) Do not evaluate the limit.
∫
52 ln(1 + x ) dx
2

(b) [3 marks] Express the limit
n
lim Σ [1 / √(1 + 2i/n)] · (2/n)
b
as a definite integral ∫ f(x) dx. Do not evaluate the integral.
a
n→∞ i=1

Question 5 [33 marks]
Find each of the following integrals.
(a) [5 marks]
232
∫ (3/(4x ) + sec x tan x − x√(x ) + 7/√(1 − x ) + cos(3x)) dx
(b) [4 marks]
23
(x + x − 1) dx
0
(c) [4 marks]
∫ (x − √x) / ■x dx (d) [4 marks]
34
∫ x / √(5 + x ) dx
(e) [4 marks]
52
∫ x √(x + 4) dx
(f) [4 marks]
2
∫ sin(1/x) / x dx
(g) [4 marks]
∫
ln 5 x x
e / (5 + e ) dx
0
(h) [4 marks]
2 ∫ sec x tan x / (1 + sec
(i) [4 marks]
∫
2026 4
x sin(x) dx
32
∫ |x − 4| dx
0
Question 6 [7 marks]
∫
−2026
(j) [4 marks]
x) dx
55
(a) [4 marks] If ∫ f(x) dx = 2 and ∫ g(x) dx = 3, use properties of the definite integral to find
22
5 2 22
∫ (4f(x) − g(x)) dx + ∫ f(x) dx − ∫ f(x)g(x) dx.
55
(b) [3 marks] If ∫ f(x) dx = 6 and ∫ f(x) dx = −2, use properties of the definite integral to find
13
3 1
Question 7 [5 marks]
Prove the Mean Value Theorem for Integrals:
If f is continuous on [a, b] with a < b, then
b
f(c) = [1/(b − a)] ∫ f(t) dt for some c ∈ [a, b].
∫
f(x) dx.
2 5 −22
a

Total: 4 + 4 + 8 + 6 + 33 + 7 + 5 = 67 marks
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