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MATH HL
TEST
DERIVATIVES
by Christos Nikolaidis
Name:____________________________________
Date:________________
Questions
1. [Maximum mark: 6] Show from first principles that the derivative of the function f ( x)= 2 x^3 +x+ 2 is f( x)= 6 x^2 + 1 ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... .....................................................................
Marks:____/
Grade: ______
Find the derivatives of the following functions (do not simplify)
(a) f ( x)= e^2 x^ sinx+e^3 arctan 2 +ln π [3 marks]
(b) g ( x)= sin^3 ( 2 x^2 + 1 ) [3 marks]
..................................................................... (c) h ( x) x^3 ln(x^41 )sin 2 x 1 = + [3 marks] ..................................................................... ..................................................................... 3. [Maximum mark: 9] The tables below show the images of two functions f and g and their derivatives at some values of x
Find the derivatives of the following functions when x = 2.
(a) g(fx^ ()x +) 1 (b) f ( x) 3 + g( 2 x) (c) g( f(x)) [3+3+3 marks]
x 1 2 3 4 f (x) 2 3 -1 3 f ' (x) 0 2 5 4
x 1 2 3 4 g(x) 5 3 1 - g' (x) 2 1 4 3
Find, showing all your work the coordinates of the stationary points of the curve 2
x x f x^ x and determine their nature.
..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... 6. [Maximum mark: 5] Given that the derivative of the function y = cosx is (^) dx^ dy^ = −sinx, find the derivative of the inverse function y = arccosx in terms of x. ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... ..................................................................... .....................................................................
A cubic function has a maximum at (0,5) and a point of inflexion at (1,1). (a) Find an expression of the cubic function in terms of x. [7 marks] (b) Find the x-coordinate of the minimum point and justify that it is a minimum. [3 marks]
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The diagram below shows the graph of y = f(x)which passes through the stationary points A, B, C.
(a) Complete the following table of signs (with + or – appropriately) x A B C y =f′(x ) [2 marks] (b) Sketch the graph of the function y = f′(x), by indicating the x-intercepts
[3 marks] (c) Write down the number of solutions of the equation f ′′^ ( x)= 0. [1 mark]
.....................................................................
C
B
A
y =f(x )
y = f′(x )
The quantities A and B increase at rates dA dt^ = 3 and dB dt^ = 2 respectively. Find (a) the rate of change of C at the moment when C = 17 , given that C = 2 A^3 + 1. [3 marks] (b) the rate of change of D at the moment when D = e, given that D (^) B ln = 3 [3 marks] (c) The rate of change of F, at the moment when 2 3 A = B= 1 , given F = 2 Α B+ 2 B [3 marks]
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A tank has the shape of a regular square pyramid with apex down, as shown below.
The side of the square base has length x = 6 metres while the depth of the tank is 10 metres. Water flows into the tank at 10 m^3 per minute. Find (a) the rate of change of the depth of the water at the instant when the water is 5 meters deep. [7 marks] (b) Given that the tank is empty at the beginning, find the rate of change of the depth of the water after 90 seconds. [3 marks]
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