MATH HL
TEST
DERIVATIVES
by Christos Nikolaidis
Name:____________________________________ Marks:____/100
Date:________________ Grade: ______
Questions
1. [Maximum mark: 6]
1 + e2x
(a) Show that the function f ( x ) = is increasing. [5 marks]
1 − 2e 2 x
−1
(b) Determine whether f ( x) exists or not. [1 mark]
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Page 1
2. [Maximum mark: 14]
Find the derivatives of the following functions
3
(a) f ( x) = e ln x
+ 2 tan 2 3x (b) g ( x) =
x2 +1
(c) h( x) = 23 x + log 23 x + log 23 23 x (d) h( x) = ln x x + ln π
[4+3+3+4 marks]
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Page 2
3. [Maximum mark: 5]
5
Show from first principles that the derivative of the function f ( x) = 2x 3 + − 1 is
x
5
f ′ ( x) = 6 x 2 − 2
x
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Page 3
4. [Maximum mark: 6]
Consider the differentiable function g (x) with some values given by
x 0 1 2 3 4
g (x) 2 3 4 5 8
g ′(x) 3 5 2 0 1
dy
Find for the following functions at the given points
dx
(a) y = g ( x) × ln x at x = 2 [2 marks]
(b) y = ln( g ( x)) at x = 1 [2 marks]
(c) y = g (ln x) at x = 1 [2 marks]
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Page 4
5. [Maximum mark: 9]
The line y = −4 x + m is tangent to the curve y = 2 x 3 + nx 2 + 1 at x = 1.
(a) Find the values of m and n. [4 marks]
(b) Find the normal line to the curve at x = 1. [3 marks]
5
(c) Find the normal line to the curve at x = [2 marks]
3
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Page 5
6. [Maximum mark: 9]
ln x
Consider the function f ( x) = . Find
x3
(a) the first and the second derivative of f (x) ; simplify your answers. [4 marks]
(b) the x-coordinate of the stationary point; determine its nature. [3 marks]
(c) the x-coordinate of the point of inflection; justify your answer. [2 marks]
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Page 6
7. [Maximum mark: 9]
A cubic function has a maximum at (0,5) and a point of inflexion at (1,1). Find
(a) an expression of the cubic function. [6 marks]
(b) the coordinates of the minimum point; justify that it is a minimum. [3 marks]
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Page 7
8. [Maximum mark: 7]
The point A(π,π) lies on the curve
x 2 + sin y − xy = c
(a) Show that the curve passes through the origin. [2 marks]
(b) Find the equation of the tangent line to the curve at the point A. [4 marks]
(c) Find the equation of the normal line to the curve at the origin. [1 mark]
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Page 8
9. [Maximum mark: 8]
The equal sides of an isosceles triangle have a fixed length of 10cm each. The angle θ
contained by the two equal sides is decreasing at a rate of 0.1π rad/sec. Find at the
moment when the triangle is an equilateral
(a) the rate at which the area is changing [4 marks]
(b) the rate at which the length of the third side is changing. [4 marks]
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Page 9
10. [Maximum mark: 7]
The diagram below shows the graph of y = f ′(x) . The tangent lines on this curve at
points A, B, C are parallel to the x-axis.
y
y = f ′(x)
A
C
x
(a) Sketch the graph of the function y = f ′′(x) , by indicating the x-intercepts
y
y = f ′′(x)
[3 marks]
(b) Sketch a graph for the function y = f (x) , given that the function is positive.
Indicate on the graph any local maxima, minima and the points of inflection.
y
y = f (x)
[4 marks]
Page 10
11. [Maximum mark: 8]
1
Show by mathematical induction that the n-th derivative of f ( x) = + e −3 x , is given
x
by
n!
f ( n ) ( x) = (−1) n n +1 + 3 n e −3 x for all n ≥ 1 .
x
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Page 11
12. [Maximum mark: 12]
A rectangle ABCD is enclosed between the x-axis, the curve y = 2 x , and the
vertical line x = 9 , so that the lower vertices are on the x-axis with coordinates A(a,0)
and B(9,0) respectively, the vertex D is on the curve, while the vertex C is on the
vertical line. The area of this rectangle is denoted by S.
(a) Find an expression of S in terms of a. [3 marks]
(b) Find the exact maximum value for the area S ; justify your answer. [5 marks]
(c) Hence find the dimensions of the rectangle of maximum area which
is enclosed by the curve y 2 = 4 x and the vertical line x = 9 . [2 marks]
(d) Write down the values of a for which we obtain a rectangle of
minimum area S. [2 marks]
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Page 12
