5. DIFFERENTIAL CALCULUS
A PREVIEW OF CALCULUS
Calculus uses not only the ideas and methods from arithmetic, geometry, algebra, coordinate
geometry, trigonometry, and so on but also the notion of limit, which is a new idea that lies at
the foundation of calculus. Calculus provides a system of rules for calculating changing
quantities which cannot be calculated otherwise. It may be mentioned here that the concept
of limit is equally important and applicable also in Integral calculus, and this will be clear when
we study the concept of the definite integral in Chapter 6 (Section 6.5).
There is a fundamental difference between precalculus mathematics and calculus.
Precalculus mathematics is more static, whereas calculus is more dynamic. For instance:
An object traveling at a constant velocity can be analyzed with precalculus
mathematics. To analyze the velocity of an accelerating object, you need calculus.
The slope of a line can be analyzed with precalculus mathematics. To analyze the
slope of a curve, you need calculus.
The area of a rectangle can be analyzed with precalculus mathematics. To analyze
the area under a general curve, you need calculus.
Differential calculus is a subject which can be applied to anything that moves, or changes or
has a shape. It is useful for the study of machinery of all kinds - for electric lighting and
wireless, optics and thermodynamics. It also helps us answer questions about the greatest
and smallest values a function can take.
COMPILED BY T. PAEPAE
,LEARNING OUTCOMES
On completion of this chapter, you will be able to:
Define the concept of a limit.
Use limit-notation and calculate the limit of functions.
Determine the rate of change.
Find the gradient of the secant and tangent lines.
Find the instantaneous rate of change and the derivative derived from first principles.
Use special rules to find the derivative and determine the derivative of different algebraic
functions.
Use special rules to determine the derivative of transcendental functions such as
exponential, logarithmic, and trigonometric functions.
Define, denote and determine higher derivatives.
Explain and use applications of differentiation to determine the maximum and minimum
turning points and the point of inflection for graphs.
Use the maximum and minimum turning points in sketch graphs.
Apply differentiation in rectilinear and circular motion problems.
Apply differentiation to optimization (maxima and minima) problems.
1
,5.1 THE LIMIT OF A FUNCTION
Why it is important to understand: The Limit of a Function
In the operation of computing the antiderivative, the concept of limit is involved indirectly. On
the other hand, in defining the definite integral of a function, the concept of limit enters the
process directly. Thus, the concept of limit is involved in both, differential and integral calculus.
In fact, we might define calculus as the study of limits. It is therefore important that we have a
deep understanding of this concept.
SPECIFIC OUTCOMES
On completion of this study unit, you will be able to:
Explain the meaning of a limit and be familiar with the notation used for limits.
Explain some properties of limits.
Calculate the limit of algebraic functions.
0
Determine the special indeterminate limit .
0
Determine the special indeterminate limit .
2
, 5.1.1 Limits – An Informal Approach
The two broad areas of calculus known as differential and integral calculus are built on the
foundation concept of a limit. In this section, the approach to this important concept will be
intuitive, concentrating on understanding what a limit is using numerical examples. In the next
section, the approach will be analytical, that is, we will use algebraic methods to compute the
value of a limit of a function.
Limit of a Function – Informal Approach
Suppose you are asked to sketch the graph of the function f given by
x3 1
f ( x) , x 1. (1)
x 1
For all values other than x 1, you can use standard curve-sketching techniques. However,
at x 1, it is not clear what to expect. To get an idea of the behaviour of the graph of f near
x 1, you can use two sets of x -values – one set that approaches 1 from the left and one set
that approaches 1 from the right, as shown in the table.
x approaches 1 from the left. x approaches 1 from the right.
x 0,75 0,9 0,99 0,999 1 1,001 1,01 1,1 1,25
f (x) 2,313 2,710 2,970 2,997 ? 3,003 3,030 3,310 3,813
f (x) approaches 3. f (x) approaches 3.
To interpret the numerical information in (1) graphically, observe that for every number x 1,
the function f can be simplified by cancellation:
f ( x)
x 3 1 x 1 x 2 x 1 x2 x 1
x 1 x 1
3
A PREVIEW OF CALCULUS
Calculus uses not only the ideas and methods from arithmetic, geometry, algebra, coordinate
geometry, trigonometry, and so on but also the notion of limit, which is a new idea that lies at
the foundation of calculus. Calculus provides a system of rules for calculating changing
quantities which cannot be calculated otherwise. It may be mentioned here that the concept
of limit is equally important and applicable also in Integral calculus, and this will be clear when
we study the concept of the definite integral in Chapter 6 (Section 6.5).
There is a fundamental difference between precalculus mathematics and calculus.
Precalculus mathematics is more static, whereas calculus is more dynamic. For instance:
An object traveling at a constant velocity can be analyzed with precalculus
mathematics. To analyze the velocity of an accelerating object, you need calculus.
The slope of a line can be analyzed with precalculus mathematics. To analyze the
slope of a curve, you need calculus.
The area of a rectangle can be analyzed with precalculus mathematics. To analyze
the area under a general curve, you need calculus.
Differential calculus is a subject which can be applied to anything that moves, or changes or
has a shape. It is useful for the study of machinery of all kinds - for electric lighting and
wireless, optics and thermodynamics. It also helps us answer questions about the greatest
and smallest values a function can take.
COMPILED BY T. PAEPAE
,LEARNING OUTCOMES
On completion of this chapter, you will be able to:
Define the concept of a limit.
Use limit-notation and calculate the limit of functions.
Determine the rate of change.
Find the gradient of the secant and tangent lines.
Find the instantaneous rate of change and the derivative derived from first principles.
Use special rules to find the derivative and determine the derivative of different algebraic
functions.
Use special rules to determine the derivative of transcendental functions such as
exponential, logarithmic, and trigonometric functions.
Define, denote and determine higher derivatives.
Explain and use applications of differentiation to determine the maximum and minimum
turning points and the point of inflection for graphs.
Use the maximum and minimum turning points in sketch graphs.
Apply differentiation in rectilinear and circular motion problems.
Apply differentiation to optimization (maxima and minima) problems.
1
,5.1 THE LIMIT OF A FUNCTION
Why it is important to understand: The Limit of a Function
In the operation of computing the antiderivative, the concept of limit is involved indirectly. On
the other hand, in defining the definite integral of a function, the concept of limit enters the
process directly. Thus, the concept of limit is involved in both, differential and integral calculus.
In fact, we might define calculus as the study of limits. It is therefore important that we have a
deep understanding of this concept.
SPECIFIC OUTCOMES
On completion of this study unit, you will be able to:
Explain the meaning of a limit and be familiar with the notation used for limits.
Explain some properties of limits.
Calculate the limit of algebraic functions.
0
Determine the special indeterminate limit .
0
Determine the special indeterminate limit .
2
, 5.1.1 Limits – An Informal Approach
The two broad areas of calculus known as differential and integral calculus are built on the
foundation concept of a limit. In this section, the approach to this important concept will be
intuitive, concentrating on understanding what a limit is using numerical examples. In the next
section, the approach will be analytical, that is, we will use algebraic methods to compute the
value of a limit of a function.
Limit of a Function – Informal Approach
Suppose you are asked to sketch the graph of the function f given by
x3 1
f ( x) , x 1. (1)
x 1
For all values other than x 1, you can use standard curve-sketching techniques. However,
at x 1, it is not clear what to expect. To get an idea of the behaviour of the graph of f near
x 1, you can use two sets of x -values – one set that approaches 1 from the left and one set
that approaches 1 from the right, as shown in the table.
x approaches 1 from the left. x approaches 1 from the right.
x 0,75 0,9 0,99 0,999 1 1,001 1,01 1,1 1,25
f (x) 2,313 2,710 2,970 2,997 ? 3,003 3,030 3,310 3,813
f (x) approaches 3. f (x) approaches 3.
To interpret the numerical information in (1) graphically, observe that for every number x 1,
the function f can be simplified by cancellation:
f ( x)
x 3 1 x 1 x 2 x 1 x2 x 1
x 1 x 1
3
