- State whether each statement is true or false. Explain your reasoning.
(a) [2] The following use of L’Hopital’s rule is correct: limx→0
sin x
x2 = limx→0
cos x
2x
=limx→0
− sin x
2 = 0
(b) [2] r
0
r2 − x2 + 3
dx = 6r +
πr2
4 , where r > 0.
(c) [2] 2
1
ln x dx < L4, where L4 is the left Riemann sum with four rectangles used to
estimate the area under y = ln x on [1, 2].
2
MATH 1LS3 * Test 3 * 31 March 2020 Name:
Student No.: - Multiple choice questions: circle ONE answer. No justification is needed.
(a) [2] Which of the following differential equations are autonomous?
(I) dy/dx = y2 − x (II) dy/dx = x2 − x (III) dy/dx = y2 − y
(A) none (B) I only (C) II only (D) III only
(E) I and II (F) I and III (G) II and III (H) all three
(b) [2] Consider the differential equation dP/dt = t + P, where P(0) = 120. Using Euler’s
Method with step size h = 5, the approximate value of P(10) is
(A) 1230 (B) 167 (C) 2019 (D) 320
(E) 482 (F) 2386 (G) 4345 (H) none of these
3
MATH 1LS3 * Test 3 * 31 March 2020 Name:
Student No.:
(c) [2] Which of the following formulas is/are correct?
(I)
sec2 x dx = tan x + C (II) 1
x
dx = ln |x| + C (III) 1
1 + x2 dx = arctan x + C
(A) none (B) I only (C) II only (D) III only
(E) I and II (F) I and III (G) II and III (H) all three
(d) [2] Suppose that f(x) is a continuous function with antiderivative F(x). If F(4) = 12
and F(0) = −1, then 4
0 (2f(x) − 5) dx =
(A) 6 (B) 5 (C) −1 (D) 3
(E) −7 (F) 19 (G) 0 (H) none of these
(e) [2] Using T2(x)=1 − x2 as an approximation of f(x) = e−x2
near 0, we
estimate that 1
0
e−x2
dx ≈
(A) −1 (B) 0.333 (C) 1 (D) 0.667
(E) 0.664 (F) 0.822 (G) 0.749 (H) 0.743
4
MATH 1LS3 * Test 3 * 31 March 2020 Name:
Student No.: - [3] Evaluate limx→0
sin2 x
3×2 . - A pie, initially at the temperature of 20oC, is put into an 300oC oven. Let T(t) be the
temperature of the pie at time t. The temperature of the pie changes proportionally to the
difference between the temperature of the oven and the temperature of the pie. Assume
that the proportionality constant is 0.02.
(a) [2] Describe this event as an initial value problem (i.e., write down a differential equation
and an initial condition).
(b) [2] Show that T(t) = 300 − 280e−0.02t is the solution to your initial value problem in
part (a).
5
MATH 1LS3 * Test 3 * 31 March 2020 Name:
Student No.: - The density of monkeys in Kruger National Park in South Africa is given by the function
f(x)=0.004x(250−x) monkeys per kilometre, where x is the distance in km from the main
entrance into the park.
(a) [3] Approximate the area of the region below the graph of f(x)=0.004x(250 − x) and
over the interval [0, 100], using R4 (i.e., right sum with four rectangles). Sketch the function
and the four rectangles involved.
(b) [3] Evaluate 100
0
0.004x(250 − x) dx algebraically. What does this number represent?
6
MATH 1LS3 * Test 3 * 31 March 2020 Name:
Student No.: - Most human papillomavirus (HPV) infections in young women are temporary and have
very little long-term effects. Assume that P(t) is the proportion (or percent) of women
initially infected with the HPV who no longer have the virus at time t (t is measured in
years). The rate of change of P(t) is modelled by the function
p(t)=0.5 − 0.25t
1.5 + 0.5e−1.2t
where 0 ≤ t ≤ 2.
(a) [3] Find 1
0
0.5 − 0.25t
1.5 + 0.5e−1.2t
dt. Round off to two decimal places.
(b) [1] What does the number you obtained in (a) represent in the context of HPV infections?
7
MATH 1LS3 * Test 3 * 31 March 2020 Name:
Student No.: - Evaluate the following integrals algebraically.
(a) [3] 1
0
4xe−0.2×2
dx
(b) [4]
ln(x + 1) dx
Sample Solution
