Name____________________________________________________ Calculus Revision: No Calculators Allowed 1. Let g(x) = (a) ln x x2 , for x > 0. Use the quotient rule to show that g ʹ′( x) = 1 − 2 ln x x3 . (4) (b) The graph of g has a maximum point at A. Find the x-coordinate of A. (3) (Total 7 marks) 2. Consider the function f with second derivative f′′(x) = 3x – 1. The graph of f has a minimum ⎛ 4 358 ⎞ point at A(2, 4) and a maximum point at B ⎜ − , ⎟ . ⎝ 3 27 ⎠ (a) Use the second derivative to justify that B is a maximum. (3) (b) Given that f′ = 3 2 x – x + p, show that p = –4. 2 (4) (c) Find f(x). (7) (Total 14 marks) 3. Let g(x) = 2x sin x. (a) Find g′(x). (4) (b) Find the gradient of the graph of g at x = π. (3) (Total 7 marks) IB Questionbank Maths SL 1 4. ⎛ 3 ⎞ The graph of the function y = f(x) passes through the point ⎜ , 4 ⎟ . The gradient function of f is ⎝ 2 ⎠ given as f′(x) = sin (2x – 3). Find f(x). (Total 6 marks) 5. Let f(x) = ax 2 x +1 , –8 ≤ x ≤ 8, a ∈ . The graph of f is shown below. The region between x = 3 and x = 7 is shaded. (a) Show that f(–x) = –f(x). (2) IB Questionbank Maths SL 2 (b) Given that f′′(x) = 2ax( x 2 − 3) ( x 2 + 1) 3 , find the coordinates of all points of inflexion. (7) (c) 6. It is given that a ∫ f ( x)dx = 2 ln( x 2 + 1) + C . (i) Find the area of the shaded region, giving your answer in the form p ln q. (ii) Find the value of 8 ∫ 2 f ( x − 1)dx . 4 (7) (Total 16 marks) 1 3 x + 2x2 – 5x. Part of the graph of f is shown below. There is a maximum 3 point at M, and a point of inflexion at N. Consider f (x) = (a) Find f ′(x). (3) (b) Find the x-coordinate of M. (4) (c) Find the x-coordinate of N. (3) (d) The line L is the tangent to the curve of f at (3, 12). Find the equation of L in the form y = ax + b. (4) (Total 14 marks) IB Questionbank Maths SL 3 7. (a) (b) Find Find ∫ 2 1 1 (3 x 2 − 2) dx. ∫ 2e (4) 2x 0 IB Questionbank Maths SL dx . (3) (Total 7 marks) 4 8. On the axes below, sketch a curve y = f (x) which satisfies the following conditions. x f (x) −2 ≤ x < 0 0 –1 0 < x <1 1 1<x≤2 2 f ′ (x) f ′′ (x) negative positive 0 positive positive positive positive 0 positive negative (Total 6 marks) IB Questionbank Maths SL 5 9. 10. The function f is given by f (x) = 2sin (5x – 3). (a) Find f " (x). (b) Write down ∫ f ( x ) dx . (Total 6 marks) If f ʹ′(x) = cos x, and f ⎛⎜ π ⎞⎟ = – 2, find f (x). ⎝ 2 ⎠ Working: Answer: ...................................................................... (Total 4 marks) IB Questionbank Maths SL 6 11. The diagram below shows the shaded region R enclosed by the graph of y = 2x 1 + x 2 , the x-axis, and the vertical line x = k. y y = 2x 1+x 2 R x k (a) Find dy . dx (3) (b) Using the substitution u = 1 + x2 or otherwise, show that 3 ∫ 2 2 x 1 + x dx = (1 + x2) 2 + c. 3 2 (3) (c) Given that the area of R equals 1, find the value of k. (3) (Total 9 marks) 12. x The diagram shows part of the graph of y = e 2 . y x y = e2 P ln2 IB Questionbank Maths SL x 7 (a) Find the coordinates of the point P, where the graph meets the y-axis. (2) The shaded region between the graph and the x-axis, bounded by x = 0 and x = ln 2, is rotated through 360° about the x-axis. (b) Write down an integral which represents the volume of the solid obtained. (4) (c) Show that this volume is π. IB Questionbank Maths SL (5) (Total 11 marks) 8

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