TOPIC: INTRODRODUCTION TO CALCULUS
Calculus is a branch of mathematics which was developed by Newton
(1642-1727) and Leibnitz (1646-1716) to deal with changing quantities.
EXPECTED OUTCOMES:
A: Differentiation.
1. Differentiate functions from first principles.
2. Differentiate functions using the formula
3. Calculate equations of tangents and normals
B: Integration
Find indefinite integrals
Evaluate simple definite integrals
Find the area under the curve
A: DIFFERENTIATION.
1. DIFFERENTIATING FUNCTIONS FROM FIRST PRINCIPLES.
EXAMPLES:
1. If , from first principle.
SOLUTION
DATA:
[Plug in these functions in the formula above]
2. Find from first principle for the function .
SOLUTION:
DATA.
[plug in these in the formula above]
EXERCISE:
1. Find for each of the following functions by first principle.
Expected Answers:
2. DIIFFERENTIATING FUNCTIONS USING THE FORMULA:
A. The Derivate of
Given a function then it follows that .
Examples
1. Given that , find .
SOLUTION:
NOTE THAT: The derivate of any constant is Zero (0)
2. Given that , find .
SOLUTION:
3. Find the derived function of
SOLUTION:
B. The Derivate of
The derivate of the function is given by the formula
EXAMPLE:
1. If .
SOLUTION:
C. The Derivate of a product. (Product Rule)
If we can let . From it follows
that, the derivative of a product is given by the formula
Example:
1. Given that , find .
SOLUTION:
We let ---
From the above it follows that =
C: THE DERIVATE OF A QUOTIENT
If is a ratio of functions where are also functions of
, the derivative of the function with respect to is given by the
formula
Example:
1. Differentiate .
SOLUTION:
We let and
EXERCISE:
1. Differentiate
2. Differentiate
3. Differentiate
EXPECTED ANSWERS:
1.
2.
3.
TANGENTS AND NORMALS
If is a curve, we can find the gradient at any point on the curve. This gradient is
equal to the gradient of the tangent to the curve at that point.
If the gradient of the tangent is and that of the normal line is it
follows that
The tangent and the normal are perpendicular to each other at
the point of contact. Example (FINDING THE GRADIENT OF THE
TANGENT AND THENORMAL)
1. Find the gradient of the tangent and the normal to the curve
at the point where .
SOLUTION
is the equation of the curve.
The gradient of the tangent is =
and since the value of
To find the gradient of the normal, recall that
The gradient of the tangent, and the gradient of the
normal,
Example (FINDING THE EQUATION OF THE TANGENT AND THE
NORMAL)
1. Find the equation of the tangent and the normal to the curve
at the point where .
SOLUTION
We have the value for x. So let’s find the corresponding value for
y.
The point is
The gradient of the tangent is and that of the normal is at
the point
Equation of the tangent
Equation of the normal
EXERCISE
1. Find the equation of the tangent and the normal to the curve at
the point
Expected Answers: for the tangent and for the normal.
INTEGRATION
The inverse of differentiation or the reverse of differentiation is called
integration.
Since integration is the reverse of differentiation the following steps must
be taken:-
1. Increase the power of the variable by 1.
2. Divide the term (variable term) by the new power.
3. Then finally add the arbitrary term C
Example (INDEFINITE INTEGRALS)
An indefinite integral must contain an arbitrary constant (C). An integral of
the form is called an indefinite integral.
1. Integrate the following gradient functions
(a) (b)
SOLUTION:
(a) (b)
EXERCISE
1. Integrate the following gradient functions
(a) (b)
EXPECTED ANSWERS:
1(a) (b) +c
DEFINITE INTEGRALS
A definite integral is an integral performed between the limits. Thus
is an integral performed between the limiting values a and b
for x.
NOTE that
Example
1. Evaluate the definite integral of between
Integrals can be used to compute the area under a given curve.
Example
1. Find the area of the region bounded by the curve , the ordinates
x=1 and x=2 and the x- axis.
EXERCISE:
1. Find the area under the curve between x=1 and x=3 Expected
Answer
2. Find the area enclosed by the x – axis, the curve and the
straight lines x=3 and x= 5. Expected Answer. 102 squared units.
