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MA110 Mock Test 2
Name:
Time Allowed: 80 minutesTotal Value: 75 marks
Number of Pages: 9
Instructions:
Non-programmable, non-graphing calculators are permitted. No
other aids al lowed.
Check that your test paper has no missing, blank, or illegible
pages. Note that test questions appear
on both sides of the paper.
Answer in the spaces provided.
Show all your work. Insucient justication will result in a loss
of marks.
1. [6 marks] Find a formula for y = f(x) given that f00 (x) =
4e2x + 5 sin x and f(0) = f0 (0) = 2.2. [10 marks] Evaluate each of
the following limits.
(a) limx!0+1
x 1
sin x
(b) limx!1+ (ln x)(x1)
3. [10 marks]
(a) Suppose g and h are functions such that g (x) = ln (1 + h
(p
x)), h (2) = 1 and h0 (2) = 1.
Find g0 (4).
(b) Use logarithmic dierentiation to evaluate f0 (0) where
f(x) =(cos x)sinx (3x 1)
3px5 + x 8
.
4. [6 marks] Use dierentials to approximate the value of3p
7:95.
5. [6 marks] Determine the equation of the tangent to the
curve
cos(x y) = yex 2
at the point (x; y) =
0;
2
.
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6. [8 marks] COMPLETE PART A OR PART B OF THIS QUESTION
(a) Use dierentiation to prove the following identity:
2sin1 (x) = cos1
1 2x2
, x 0.(b) Use the Intermediate Value Theorem and Rolles Theorem
to prove that the equation
sin(x)
2x = 1
1
p2has exactly one real solution.
7. [10 marks] Consider the function f(x) = x + ln
x2 1
.
(a) Determine the intervals of increase and decrease for the
graph of f. State any relative minimum
and relative maximum points. You may use the fact that f0 (x)
=x2 + 2x 1
x2 1 .(b) Determine the intervals of concavity for the graph of
f. State any points of inection. You may
use the fact that f00 (x) =2
x2 + 1
(x2 1)2
.
(c) Given that y = f(x) only has one intercept, at approximately
(1:2; 0), use your answers to
part (a) and (b) to sketch a possible graph for f.
8. [10 marks] A rectangular box with an open top is to have a
volume of 48 m3. The length of the baseis to be twice the width.
The material for the base costs $8 per square metre. Material for
the sideswill cost $6 per square metre. Find the cost of producing
the cheapest such box.
9. [7 marks] Consider the function f(x) = ex
x2 + 1
.
(a) Determine the absolute minimum value and the absolute
maximum value of f on the interval[2; 0].
(b) Using your answers to part(a) and the properties of denite
integrals, determine a closed interval
which contains the value of0
R2
ex x2 + 1 dx.[You do not need to evaluate the integral
itself.]
0R2
ex
x2 + 1
dx
10. [3 marks] Let f(x) =
(1 x if 1 x 0p1 x2 if 0 < x 1 .
Use the graph of y = f(x) to evaluate1R1
f(x) dx by interpreting it in terms of areas.
11. [8 marks] Consider the area between the curve f(x) = x2 + 4
and the x-axis from x = 1 to
x = 3.
(a) Represent the area using an appropriate denite integral.
(b) Determine the required area by using a Riemann sum and right
endpoints.
You may use the fact thatnPi=1
i2 =n (n + 1)(2n + 1)
6and
nPi=1
i =n (n + 1)
2.
[No marks will be given for using the Fundamental Theorem of
Calculus to evaluate the integral.]
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