Self Test Mathematics
Preparation for the master Statistical Science
at Leiden University, The Netherlands
Part 1: Functions and graphs
Exercise 1
Calculate the following without using a calculator, and simplify the answers.
(a)
39
14
+
8
3
=
(b)
19
8
−
2
3
=
(c)
9
4
32
13
=
(d)
33
15
×
×
43
11
(e) 4 × 5 +
(f)
1
2
=
4
2
×4+3
(g) 5 × (4 − 32 )
Exercise 2
Simplify by removing the brackets.
(a) 2(x + 3)
(b) −3(3 + 2x − 3x2 )
(c) 2(x + 3 + x(4 + x))
(d) (3x − 1)(x2 + 3 − x)
Exercise 3
Sketch these functions and write down their domain and range.
(a) f (x) = x2
1
(b) g(x) = ex
(c) h(x) = (x − 1)2
Exercise 4
Given
f (x) = x2 ,
g(x) = ex ,
h(x) = (x − 1)2 .
Specify the functions:
(a) (g + h)(x)
(b) (f h)(x)
(c) (g − f )(x)
(d) 3h(x)
(e) f ◦ g(x)
Exercise 5
Give the function definition for a parabola that intersects the x-axis at −5 and 2, and
the y-axis at −1. Make a sketch of this parabola.
Exercise 6
Solve the following equations, or show that there is no solution.
(a) x2 − 2x − 15 = 0
(b) 2x2 − 2x − 8 = 4
(c) 3x2 − 5x + 1 = 7x − 2
(d) −8x2 + 12x + 9 = 2x2 + 3x − 7
(e) x(2x + 3) + (x + 3)(x − 1) = 4
Exercise 7
Sketch the curve corresponding to
(x + 2)2 (y − 4)2
+
=1
9
4
Explain how you got your answer.
2
Exercise 8
Simplify
(a)
6
√3
81
(b) 3log9 (4)
(c)
3
√9
729
(d) log6 (216)
(e) log4 (16) + 2 log2 (8)
√
(f) logb (x2 − 3x + 2) − 3 logb ( 3 x − 1)
Exercise 9
Solve for x
(a) 3x+16 = 81
(b) e4x − 3e2x = 0
(c) e2x − 7ex + 12 = 0
(d) 3x+a = 27
(e) 18x+3 + 6 = 62
2 −1
(f) 3x
= 92x
Part 2: Derivatives and Extremes
Exercise 10
Find the derivative f 0 (x) of
(a) f (x) = e3x
2
(b) f (x) = 6 tan(x) + 2 ln(x)
(c) f (x) = cos(x) sin(x)
(d) f (x) =
ex
cos(x)
(e) f (x) = (3 + x)(3 + y)
(f) f (x) = ln(3x)
3
(g) f (x) = sin(3x2 + 2)
(h) f (x) =
2x8 +sin(x)
e3x +cos(2x)
(i) f (x) = e4x (22x − 3 sin(x))
p
(j) f (x) = 3 ln(x)
(k) f (x) = ln(ax + bx2 )
Exercise 11
Use the derivatives of the following functions to find their interval(s) of increase and
interval(s) of decrease:
(a) f (x) = x2 + x3
(b) f (x) = (x3 − 2)2
Exercise 12
x3
Consider the two functions f (x) =
+ 2x + 4 and g(x) = x5 + 6. Show that f (x) and
g(x) intersect at x = 1, and that their tangent lines at this point of intersection are
equal.
Exercise 13
Determine the local and global extrema of the following functions
2
(a) f (x) = xe−2x on the interval [−4, 3].
(b) h(x) =
2x3 −x
x3 −4x
Exercise 14
Find the equation of the tangent line of f (x) = ex (3x − 1)2 at x = 23 .
Exercise 15
Suppose you want to make a wooden cuboid with a square base (like in the figure below)
with a volume of S cm3 . The wood to make the edges costs 2 euro’s per cm, and the
blocks to connect edges cost 3 euro each. Show that, to minimise the costs, you have to
make a cube (i.e. all faces are square).
4
Exercise 16
We know the Product and Quotient rules for differentiation. Find the derivative f 0 (x)
for the function
f (x) = g(x)h(x) ,
with g(x) and h(x) arbitrary, differentiable functions.
5