I. INTRODUCTORY PROBLEMS

1. Find *y* the following:

(i) (ii)

(iii) (iv)

(v) (vi)

(vii) (viii)

(ix)

2. In each of the following, find F(x) given that F'(x) = f(x):

(a) (b)

(c) (d)

(e) (f)

(g)

3. Find f(x) given that f'(x) = 2x-2 and f(1) = 4. [ans: ]

- Find f(x) given that f'(x) = and f'(1) = 3.
- At all points on a certain curve . The point (2, 4) belongs to the curve. Find the equation of the curve. [ans:

6. Find the equation of the curve given that the gradient at any point P(x, y) is and that the point (3,3) belongs to the curve.

7. For the function find the rule that defines the function F, where F'(x) = f(x) and F(1) = 4.

8. Find the equation of a curve given that at any point P and that when x=3, and y = -3.

9. Find the rule that defines a function f, given that f'(x) = for all x, and f'(0) = 1 and f(0) = 2.

10. A curve contains the point (0,4) and its gradient is (x-1)(x+2) at any point on the curve. Find the equation of the curve.

11. At all points on a certain curve . The point (3,6) belongs to the curve and its tangent at this point is inclined at 45o to the X-axis. Find the equation of the curve.

12. Find the primitives of

(a) (b) (c) (d) .

13. In each of the following, find f(x) given that:

(a) (b)

(c) (d)

14. Given that , find *s* in terms of *t* given that s=4 when t=1.

15. Given that , t__>__0 and and s=0 when t=0, find

(a) t when (b) s when .

[ans: (a) 12 (b) 432]

16. Given for all t __>__ 0 and and x = 4 when t = 0, find t when

(a) (b) *x* = 0.

[ans: (a) 5/2 (b) 1 or 4]