1
IB Mathematics HL 2013
Applications of differential calculus test
Section A: Technology free
Name: ______________________________________
Instructions:

Answers should be stated as exact values

When a question asks you to ‘simplify’ an expression, it is expected that the expression is
presented in its most simplified form
Total marks for Section A=27
Question 1
Consider the function,
with equation
a. For the curve
, show that there are no axes intercepts.
.
1 mark
b. Express
in the form
, where
Hence state the equations of any asymptotes.
are constants.
2
2 marks
c. Find the first and second derivatives.
2 marks
d. Find and classify any stationary points.
4 marks
e. Show that there are no points of inflection.
3
1 mark
f. State over which intervals
i.
Increasing.
ii.
Decreasing.
g. State over which intervals
i.
Concave up.
ii.
Concave down.
is
is
1+1=2 marks
1+1=2 marks
4
h. On the set of axes below, sketch the graph of
.
2 marks
Total 16 marks
5
Question 2
The graph of
of
is provided below. On the accompanying axes draw what could be the graph
.
3 marks
6
Question 3
Consider the function,
.
a. Find the average rate of change between
and
.
2 marks
b. Find the instantaneous rate of change at
.
3 marks
Total 5 marks
7
Question 4
Given the expression
a. Find
.
.
3 marks
b. Hence find the equation of the tangent to the curve which passes through the point
.
3 marks
Total 6 marks
8
END OF SECTION A
Section B: Technology active
Name: _____________________________________
Instructions:
Unless otherwise stated, answers should be stated as either exact values or to an accuracy of
three significant figures
Total marks for Section B=18
Question 1
Given the curve
a. Find
.
.
4 marks
b. Hence find any points of inflection.
9
2 marks
Total 6 marks
Question 2
A particle moves along a straight line so that after
the equation
.
Show that the velocities of the particle when
seconds its displacement
second are
, in metres, satisfies
.
4 marks
Total 4 marks
10
Question 3
A triangle is formed by three lines
and
, where
.
a. Explain why the triangle is a right-angled triangle.
1 mark
b. Hence find the value of
for which the area of the triangle is a minimum.
11
6 marks
Total 8 marks
END OF SECTION B