Haef Ib - Math Hl s1

HAEF IB - MATH HL TEST 5 DERIVATIVES by Christos Nikolaidis

Name:______Marks:____/100

Date:______Grade: ______

Questions

1. [Maximum mark: 6]

Show from first principles that the derivative of the function f (x)  2x 3  x  2 is

f (x)  6x 2 1

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Find the derivatives of the following functions (do not simplify)

(a) f (x)  e 2x sin x  e3 arctan 2  ln [3 marks]

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(b) g(x)  sin 3 (2x 2 1) [3 marks]

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1 (c) h(x)  x 3 ln(x 4 1)sin 2x [3 marks]

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The tables below show the images of two functions f and g and their derivatives at some values of x

x 1 2 3 4 x 1 2 3 4 f (x) 2 3 -1 3 g(x) 5 3 1 -3 f ' (x) 0 2 5 4 g' (x) 2 1 4 3

Find the derivatives of the following functions when x = 2.

f (x) (a) (b) f x3  g(2x) (c) g f (x) [3+3+3 g(x) 1 marks]

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The line y  4x  2 is tangent to the curve y  mx 3  nx 2 1, at the point where x=1. (a) Find the values of m and n. [5 marks]

(b) Find the normal line at the same point. [2 marks] 5 (c) Find the normal line at the point where x  [2 marks] 3

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Find, showing all your work the coordinates of the stationary points of the curve x  3 f (x)  x 2  x  2 and determine their nature.

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dy Given that the derivative of the function y  cos x is  sin x , find the derivative dx of the inverse function y  arccos x in terms of x.

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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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Find the gradients of the following curves at the points where x = 1.

(a) y  x 2x1 [4 marks]

(b) x 2  y 2  3xy  11 [6 marks]

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The diagram below shows the graph of y  f (x) which passes through the stationary

points A, B, C.

y  f (x)

(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

y  f (x)

[3 marks]

(c) Write down the number of solutions of the equation f (x)  0 . [1 mark]

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dA dB The quantities A and B increase at rates  3 and  2 respectively. Find dt dt

(a) the rate of change of C at the moment when C  17 , given that

C  2A3 1. [3 marks]

(b) the rate of change of D at the moment when D  e , given that

3 ln D  [3 marks] B

(c) The rate of change of F, at the moment when A  B  1, given

F  2 2 B  2B 3 [3 marks]

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A rectangle is enclosed between the x-axis, the curve y  x , and the vertical line x  9 , so that the lower vertices are on the x-axis, one of the upper vertices is on the curve, while the other vertex is on the vertical line. The area of this rectangle is denoted by A.

(a) Write down an expression for A in terms of x. [2 marks] (b) Find the maximum value of A and justify your answer. [4 marks] (c) Hence find the dimensions of the rectangle of maximum area which is enclosed by the curve y 2  x and the vertical line x  9 . [2 marks] (d) Write down the values of x for which we obtain a rectangle of minimum area A. [2 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 m3 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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