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UNIT 6 : Differentiation of polynomial, rational and irrational functions
My goals
Introduction
Calculus is concerned with things that do not change at a constant rate. The values of the function called the derivative will be that varying rate of change.
1. It is used economics a lot, calculus is also a base of economics.
2. It is used in history, for predicting the life of a stone.
3. It is used in geography, which is used to study the gases present in the atmosphere.
4. It is mainly used in daily life by pilots to measure the pressure in the air.
Differentiation can help us solve many types of real-world problems. We use the derivative to determine the maximum and minimum values of particular functions (e.g. Cost, strength, amount of material used in a building, profit, loss, etc.). Derivatives are met in many engineering and science problems, especially when modeling the behavior of moving objects.
Required outcomes
After completing this unit, the learners should be able to:
» Use properties of derivatives to differentiate polynomial, rational ad irrational functions
» Use first principles to determine the gradient of the tangent line to a curve at a point.
» Apply the concepts of and techniques of differentiation to model, analyze and solve rates or optimization problems in different situations.
» Use the derivative to find the equation of a line tangent or normal to a curve at a given point.
1. Concepts of derivative of a function
Activity 1
Consider the following figure
provided that the limit exists.
Exercise 1
Find the derivative of
Right-hand and left-hand derivatives
Activity 2
1. Consider the function
Example 3
Exercise 2
Notation
Example 4
Geometric interpretation of derivative
2. Rules of differentiation
a) Constant function and Powers
Activity 3
Derivative of a constant function
From activity 1
Derivative of a power
If n is any real number, then
Particular case
Exercise 3
Find the derivative of the following functions
b) Multiplication by a scalar and product of two functions
Activity 4
Multiplication by a scalar
From activity 2
If f is a differentiable function of x, and c is a constant, then
Derivative of a product
From activity 2:
Exercise 4
Find the derivative of the following functions
c) Sum (difference)of functions
Activity 5
Example 14
Solution
Exercise 5
Find the derivative of the following functions
d) Reciprocal function and quotient
Activity 6
Derivative of the reciprocal function
From activity 4,
From activity 4,
Exercise 6
Find the derivative of the following functions
e) Composite function
Activity 7
Derivative of a composite function: Chain rule
Exercise 7
Successive derivatives
Activity 8
The successive derivatives of a function f are higher order derivatives of the same function.
We denote higher order derivatives of the same function as follows:
Exercise 8
3. Applications of differentiation
Equation of tangent line and normal line
Activity 9
Tangent line
Remark
Normal line
Example 24
a) Find the point where the tangent line is parallel to the bisector of the first quadrant.
b) Find the tangent line to the curve of this function at point
Solution
The tangent line is
Normal line
Example 25
Rates of change
The purpose here is to remind ourselves one of the more important applications of derivatives. That is the fact that
Example 26
Example 27
Critical points
Example 28
Let us determine all the critical points for the function
Example 29
Let us determine all the critical points for the function
Example 30
Let us determine all the critical points of the function
Now, we have two issues to deal with. First the derivative will not exist if there is division by zero in the denominator.
So, we can see from this that the derivative will not exist at t = 3 and t =−2. However, these are not critical points since the function will also not exist at these points. Recall that in order for a point to be a critical point the function must actually exist at that point.
At this point, we have to be careful. The numerator does not factor, but that does not mean that there are not any critical points where the derivative is zero. We can use the quadratic formula on the numerator to determine if the fraction as a whole is ever zero.
Finding absolute extrema
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