1. MODULE 4: CALCULUS Differentiation Integration
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

2. DIFFERENTIATION
Differentiation = process of finding the derivative or the gradient function or the differential coefficient of a given function (monomial or polynomial).
Differentiation can be done by ‘working from the first principles’ also known as ‘differentiating by the limiting process’ or ‘by rates of change’.
Differentiation can also be done simply by multiplying the term by its index and then decrease the index by 1 in case of a monomial; and adding individual derivatives in case of a polynomial.
dy/dx means the derivative of y with respect to (wrt) x
More complicated rules are used to differentiate more complex functions
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

3. DIFFERENTIATION ctd Rules of differentiation
1.If y = xn ; dy/dx = nxn – 1
2. If y = axn; dy/dx = anxn – 1
3. If y = constant; dy/dx = 0
4. If y = ex; dy/dx = ex
5. If x is in radians
5.1. If y = sin (x); dy/dx = cos (x)
5.2. If y = cos (x) ; dy/dx = - sin (x)
5.3. If y = tan (x) ; dy/dx = sec2 (x)
6. If y = ln(x), dy/dx = 1/x.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

4. DIFFERENTIATION ctd Rules of differentiation ctd
7. Composite function or function of a function rule: if y = f(u) and u = f(x), dy/dx = dy/du.du/dx. Aka ‘Chain rule’.
8. Product rule: if y = u.v, with u = f(x) and v = g(x); dy/dx = v.du/dx + u.dv/dx
9. Quotient rule: if y = u/v, with u = f(x) and v = g(x); dy/dx = v.du/dx – u.dv/dx v2
10. Higher derivatives, e.g. 2nd , 3rd, ... Derivatives are obtained by re-differentiating the previous level derivative, i.e you differentiate the 1st derivative to get the second derivative.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

5. DIFFERENTIATION ctd Work out the following
Using the limiting process, find dy/dx if (a) y =x2 + 3, (b) y = 2x3 + x2 – 5
Find dy/dx if (a) y = 1/x; (b) y = x2 + x
Differentiate y = (2x – 3)5 wrt x
Some oil is spilt onto a level surface and spreads out in the shape of a circle. The radius r cm of the circle is increasing at a rate of 0.5cm/s. At what rate is the area of the circle increasing when the radius is 5cm?
The displacement, s, of a particle is given by s = 75t – t3 ( t  0 ). (i) find the velocity, v, where v = ds/dt; (ii) at what value of t is v = 0?
Differentiate cos (a)/[1 + sin (a)] wrt a
If y = (3x2 – x + 4)(2x +x3) find dy/dx
Given that y = [2x4 +3x2 ]/ [4x5 – 6x3], find dy/dx
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

6. DIFFERENTIATION ctd Applications: Increasing and decreasing functions
As x increases, y = f(x) is increasing if dy/dx > 0 and y = f(x) is decreasing if dy/dx < 0.
Tangent and Normal
The gradient for the tangent to y = f(x) is dy/dx, whilst the gradient for the normal to y = f(x) is –(dy/dx)– 1 [this is because the product of gradients of  lines is always – 1 and the tangent and normal to the same curve at the same point are ).
Stationary points: minimum and maximum
A function y = f(x) has a stationary point where dy/dx = 0. There are three stationary points: maximum, minimum and point of inflexion. The stationary is a maximum if d2y/dx2 = < 0, and it is a minimum if d2y/dx2 > 0. If d2y/dx2 = 0, then the sign test must be used to determine the type of stationary point (i.e. Check the sign of dy/dx around the stationary point: +0- gives max; -0+ gives min and -0- or +0+ give point of inflexion).
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

7. DIFFERENTIATION ctd Work out the following:
Find the range of values for which the function y = x3 – 6x2 – 15x + 3 is increasing
Find the equations of the tangent and the normal to the curve y = x2 – 2x – 3 at the point where it meets the positive x-axis.
Find the nature of the stationary points on the curve y = 4x3 – 3x2 – 6x + 2
A cylindrical can (with lid of radius r) is made from 300cm2 of thin metal sheet. (a) show that its height h cm is given by h = (150 - r2)/r; (b) find r and h so that the can will contain the maximum possible volume.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

8. INTEGRATION
Integration is the reverse process of differentiation, hence it is sometimes called “anti-differentiation”: to Differentiate, multiply by the index and Decrease it by 1; to Integrate, Increase the index by 1 and then divide by the new index.
If y = x2 + 3; dy/dx = 2x [ differentiation].
If dy/dx = 2x; y = x2 + C [Integration].
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

9. INTEGRATION ctd Rules of integration (some): indefinite integral
xndx = xn+1/(n+1) [n  -1],
dx/x = 1/x dx = ln/x/,
axdx = ax/ln(a),
ln(x)dx = x[ln(x) – 1],
log(x)dx = x[log(x) – log(e)],
sin(x)dx = - cos(x), [with x in radians]
cos(x)dx = sin(x), [with x in radians]
tan(x)dx = ln/sec(x)/. [with x in radians];
For definite integrals, see page in msword (UNILUS- REFRESHER MATHS-INTEGRATION)
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

10. INTEGRATION ctd Work out the following: Integrate wrt x:
2x3 - 3x + 1,
(b) 2x – 3)2,
(c)(x4 - 3x + 1)/2x3,
Find (a) cos(t) d(t), (b) 10xdx
The gradient of the tangent at a point on a curve is given by X2 + x – 2. Find the equation of the curve if it passes through (2,1).
Integrate x – 1/x2 wrt x and within the boundaries x = 1 and x = 2.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

11. INTEGRATION ctd Work out the following ctd
5.The curve y = ax2 + bx + c passes through the points (0, - 2) and (1, - 3) and its gradient where x = 2 is 5. Find (a) the value of a, of b and of c;(b) The area under the curve between the lines x = 2 and x = 3.
6. The diagram shows part of the curve y = x2 and the line y = 4. The line AB is drawn through A(0,2) with gradient - 1 to meet the curve at B. Find (a)the coordinates of B; (b) the ratio of the shaded areas P to Q. y
y =4
A Q y = x2
P B x 6.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

12. INTEGRATION ctd Work out the following ctd
7. If y = x2 for 0  x  2 is completely rotated round the x – axis, find the volume of the solid of revolution produced.
8. If y = x2 for 0  y  4 is completely rotated round the y – axis, find the volume of the solid of revolution produced.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR