Introductory Calculus

Objective

A. Differentiation Students should be able to:

1. use the concept of the derivative at a point x = c as the gradient of the tangent to the graph at x = c

2. define the derivative at a point as a limit

3. use the f ' (x) and dy/dx notation for the first derivative of f(x)

4. use (d/dx)xⁿ = nx^{n - 1} where n is any real number

5. use (d/dx) sin x = cos x and

(d/dx) cos x = - sin x

6. use simple rules of derivatives to find derivatives of sums and multiples of functions

7. use Specific Objectives 4, 5 and 6 above to calculate derivatives of polynomials and trigonometric functions

8. apply the chain rule in the differentiation of composite functions

9. differentiate products and quotients of simple polynomials and trigonometric functions

10. use the concept of the derivative as a rate of change

11. use the concept of stationary points

12. determine the nature of stationary points

13. locate stationary points, maxima and minima, by considering sign changes of the derivative

14. calculate the second derivative, f '' (x)

15. interpret the significance of the sign of the second derivative

16. use the sign of the second derivative to determine the nature of stationary points

17. obtain equations of tangents and normals to curves

B. Integration Students should be able to:

1. recognize integration as the reverse process of differentiation

2. use the notation

Study Resources

3. show that the indefinite integral represents a family of functions which differ by constants

4. use simple rules of integration

5. integrate functions of the form (ax + b)ⁿ where a, b, n are real and n is not - 1.

6. find indefinite integrals using formulae and integration theorems

7. integrate simple trigonometric functions

8. compute definite integrals

9. formulate the equation of a curve given its gradient function and a point on the curve

10. apply integration to:

(i) find the area of the region in the first quadrant bounded by a curve and the lines parallel to the y-axis

(ii) find volumes of revolution about the x-axis, for polynomials up to and including degree 2