1. Differentiate, Expressing Your Answers with Positive Indices

Further Calculus, Vectors, Revision of Units 1 & 2

1. Differentiate, expressing your answers with positive indices:

4 1 (a) (3x  2) (b) (x 2  4) 2 (c) 5  4x

1  4 2 (d) 2 3 (e) (f) (x  3x) x  2 x 2  7

2. Differentiate, expressing your answers with positive indices: (a) 2sin 3x (b) cos3 x (c) 5sin2 x 1 1 (d) cos (3  2x) (e) (f) sin x cos x 5 3. A curve has equation y  , where x   1 . 4x 1 4 Find the equation of the tangent to this curve at the point where x = 1.

4. A curve has equation y = 2sin ½  , 0    .

2 (a) Find the exact value of y when  = 3  . dy (b) Find and evaluate it for  = 2  . d 3

2 (c) Hence show that the equation of the tangent to the curve at the point where  = 3  2 can be written in the form 2y    (2 3  3  ). 5. Find: (a)  (x 1) 4 dx (b)  (2x 1)5 dx (c)  (3  2x)3 dx (d)  (x 1) 4 dx 1  3 (e)  (3  x) 2 dx (f)  (x  4) 2 dx (g)  (6x  5) 2 dx (h)  x  3dx

6. Evaluate: 1 1 4 2 dx (2x 1)3dx (1 x)3dx 4  x dx (a) (b) (c) (d)  1 2  0  1  0 (x  2)  1   4  5        7. a =  2  , b =   3 and c =  2 . (a) Find the components of the vector a  2b + 3c.        2  1   0  (b) Calculate the lengths of a + b and a – b . 8. Find the coordinates of the points dividing the line joining A(0 , 1 , 5) and B(0 , 4 , 5) internally and externally in the ratio 3 : 2. 9. P(0 , 2 , 1) and Q(0 , 1 , 2). Calculate the size of POQ where O is the origin.

 2   k      10. a =  1 , b =  1  . Find k if a is perpendicular to b.      3 1

11. In this right angled triangle a = 2 and b = 4

(a) Find the exact value of c . b c (b) Show that b.(a + c) = 16  3 a

y 12. The diagram shows the graph of a quadratic function f(x) and a linear function g(x). f(x) g(x) Find, in terms of x: O 1 3  4 x (a) the equation of the parabola f(x). (b) the equation of the line g(x). 4  (c) the values of x for which f(x)  g(x).

13. When an oil tanker is travelling at a speed of x km/h it uses up fuel at a rate of (1 + 00000625x3) tonnes per hour.

(a) Prove that the total amount of fuel used on a 16000 km voyage at a speed of x km/h is 16000 given by the function F(x) =  x 2 . x

The captain of the tanker wants to minimise the amount of fuel used in this voyage.

(b) Find the optimum speed at which the tanker should travel to keep the fuel consumption at a minimum. (c) Find the actual amount of fuel used for the voyage if the captain manages to maintain the optimum speed throughout the voyage.

14. Write 2cos2 x – 4cos x + 5 in the form a(cos x + b)2 + c