1.1 Introduction to Functions

The concept of a function is fundamental to calculus and to mathematics generally. Calculus is concerned largely with quantities that vary and with the relationships between such quantities. A function provides a precise mathematical way of expressing one quantity in terms of another.

For example, the area of a square depends on the length of its side. If the side length is represented by x, then the area A is given by

Thus, every permitted value of x determines exactly one value of A. In this situation, the area is said to be a function of the side length.

Similarly, if an object moves at a constant speed of 20 metres per second, the distance travelled after t seconds is

The distance d depends on time t. Consequently, d is a function of t.

Functions occur naturally in science, engineering, economics, statistics, computing and everyday quantitative situations. In calculus, they are particularly important because limits, continuity, differentiation and integration are all developed in relation to functions.

1.2 Meaning and Definition of a Function

If a function f maps elements of a set A into a set B, this may be written as

If x∈A and the function assigns the element y∈B to x, then

The element x is called the input or independent variable, while f(x) is the corresponding output or value of the function.

1.3 Function Notation and Evaluation

The notation f(x) is read as “f of x.” It does not mean f multiplied by x. Rather, it denotes the value produced by the function f when the input is x.

Suppose

To determine f(2), every occurrence of x is replaced by 2:

Therefore,

Evaluation at Algebraic Inputs

The input of a function need not be a numerical value. It may itself be an algebraic expression.

Difference Quotient

An expression that will become particularly important when differentiation is introduced is

It measures an average rate of change over an interval of length h.

1.4 Domain, Codomain and Range

A function is not completely described merely by writing a formula. The values that may be used as inputs and the values that may occur as outputs are also important.

The range is therefore contained in the codomain, but the two need not be equal.

Finding the Domain of an Algebraic Function

When no domain is explicitly stated, the natural real domain is usually taken to be all real numbers for which the formula has a real, defined value.

For polynomial functions there is no restriction on real x. Rational functions exclude values that make a denominator zero. Functions involving an even root require the quantity inside the root to be non-negative.

Determining a Range

1.5 Functions of a Real Variable

Such a function may be represented in the form

where D⊆ℝ.

For example,

defines a real-valued function for every real number x. Its natural domain is ℝ.

In contrast,

is real-valued only when x-2>0. The inequality is strict because the expression inside the square root must be positive and the resulting denominator must not be zero. Hence,

Functions of a real variable form the main objects studied in elementary calculus.

1.6 Types of Real Functions

Real functions can be classified according to the form of the rule that defines them. Familiarity with these classes is important because later techniques of limits, differentiation and integration depend partly on the type of function involved.

Constant Functions

A constant function has the form

where c is a fixed real number. Its value does not change when x changes.

For example,

Identity Function

The identity function is defined by

Every input is returned unchanged.

Linear Functions

A linear function in elementary coordinate form is commonly written as

where m and c are constants. Its graph is a straight line, and m represents the slope.

Polynomial Functions

A polynomial function has the general form

where n is a non-negative integer and the coefficients are constants.

Examples include

and

Rational Functions

A rational function is a quotient of two polynomial functions:

where Q(x)≠0.

Radical Functions

A radical function contains a variable within a root. For example,

For a real-valued square-root function,

so the domain is [-4,∞).

Absolute-Value Functions

The absolute-value function is defined by

Thus |x| represents the non-negative distance of x from zero on the real number line.

Exponential Functions

An exponential function has the form

where a>0 and a≠1.

Examples include 2^x, 10^x and e^x.

Logarithmic Functions

The logarithmic function

is the inverse of the exponential function a^x, where a>0 and a≠1. Its real domain is

Trigonometric Functions

The basic trigonometric functions include

They are especially important in calculus because they describe periodic behaviour and arise naturally in geometry, waves, oscillations and rotational motion.

1.7 Equality of Functions

For example, consider

and

Since expansion gives

and both functions have domain ℝ, the two functions are equal.

However, formulas that simplify to the same algebraic expression need not define equal functions if their domains differ.

1.8 Operations on Functions

Functions can be combined using the ordinary algebraic operations of addition, subtraction, multiplication and division.

If f and g are functions, then

and

provided g(x)≠0.

1.9 Composite Functions

A function can be applied to the output of another function. This process is called composition of functions.

The function g acts first, and its output becomes the input of f.

This example shows that function composition is generally not commutative:

Domain of a Composite Function

For f(g(x)) to be defined, x must first belong to the domain of g, and the resulting value g(x) must belong to the domain of f.

1.10 One-to-One and Many-to-One Functions

One-to-One Functions

Equivalently,

Many-to-One Functions

A function is many-to-one when two or more distinct inputs can produce the same output.

For example,

gives

and

Thus, f(x)=x² is not one-to-one on the whole real line.

1.11 Inverse Functions

An inverse function reverses the action of the original function. If f takes an input x to an output y, then its inverse takes y back to x.

Finding an Inverse Function

A convenient algebraic procedure is:

• write y=f(x);
• interchange x and y;
• solve the resulting equation for y;
• write the result as f^-1(x).

Restricting a Domain to Obtain an Inverse

A function that is not one-to-one on its original domain may sometimes become one-to-one when its domain is restricted.

For example, f(x)=x² is not one-to-one on ℝ. If its domain is restricted to x≥0, however, it becomes one-to-one and its inverse is

1.12 Applications of Functions

Functions provide mathematical models for relationships between changing quantities. A mathematical model identifies the relevant variables and expresses one variable in terms of another.

Cost Function

Area as a Function

Temperature Conversion as a Function

The Celsius temperature C corresponding to a Fahrenheit temperature F is given by

Functions and the Development of Calculus

The ideas introduced in this chapter provide the language needed for the remainder of calculus. When a function is represented graphically, its behaviour becomes visible. Limits will allow the behaviour of a function near a point to be studied precisely. Continuity will describe whether the function behaves without a break at a point. Differentiation will measure how rapidly the function changes, while integration will provide methods for accumulation and for determining quantities such as areas and volumes.

For this reason, a sound understanding of functions, domains, ranges, compositions and inverses is not merely preliminary algebra. These ideas form part of the conceptual foundation on which elementary calculus is built.

Chapter Summary

2.1 Introduction to Graphs of Functions

A graph provides a visual representation of the relationship between the input and output values of a function. While an algebraic formula tells us how a function is defined, its graph often reveals features that are less obvious from the formula alone, such as where the function increases or decreases, where it crosses the coordinate axes, whether it is symmetric, and how rapidly it changes.

If a function is given by

then its graph is the set of all points

in the Cartesian plane for values of x belonging to the domain of the function.

Graphs are especially important in calculus because the ideas of limits, continuity, derivatives, maxima, minima and area under a curve all have clear graphical interpretations.

2.2 The Cartesian Coordinate Plane

The Cartesian coordinate plane is formed by two perpendicular number lines:

• the horizontal axis, called the x-axis;
• the vertical axis, called the y-axis.

The point at which the axes intersect is called the origin and is denoted by

Every point in the plane is represented by an ordered pair

The first coordinate specifies horizontal position and the second coordinate specifies vertical position.

The Four Quadrants

2.3 Plotting Points and Reading Coordinates

To plot a point (x,y), begin at the origin. Move horizontally according to the value of x, then move vertically according to the value of y.

For example, to plot

move four units to the right and then three units upward.

To plot

move two units to the left and then five units downward.

To graph a function numerically, one may first construct a table of selected input values and their corresponding outputs.

2.4 Graphs of Linear Functions

A linear function in coordinate form is commonly written as

where m is the slope or gradient and c is the y-intercept.

A positive slope means the line rises from left to right, while a negative slope means it falls.

Equation of a Line Through a Point

If a line has slope m and passes through (x₁,y₁), then

Parallel and Perpendicular Lines

Two non-vertical lines are parallel if they have equal slopes:

Two non-vertical lines are perpendicular if

2.5 Graphs of Quadratic Functions

A quadratic function has the form

Its graph is called a parabola.

If a>0, the parabola opens upward. If a<0, it opens downward.

The Vertex and Axis of Symmetry

For

the x-coordinate of the vertex is

The corresponding y-coordinate is obtained by substituting this value into the function.

The vertical line

is the axis of symmetry.

Intercepts of a Quadratic Graph

The y-intercept is obtained by setting x=0. The x-intercepts are obtained by solving

Completing the Square and Graph Form

A quadratic function can often be written in the form

The vertex is then immediately visible as

2.6 Graphs of Polynomial Functions

A polynomial function may have degree 0, 1, 2, 3 or higher. The degree strongly influences the general shape of its graph.

A polynomial of degree n can have at most n real zeros and at most n-1 turning points.

For example, the cubic function

passes through the origin, is negative for negative x, and positive for positive x.

By contrast,

is non-negative for every real value of x and is symmetric about the y-axis.

Zeros of a Polynomial and Graph Intersections

A zero of a polynomial function is a value of x for which

Graphically, a real zero corresponds to an x-intercept.

2.7 Graphs of Rational Functions

A rational function is of the form

where P and Q are polynomials and Q(x)≠0.

Rational functions may have values of x at which the function is undefined. These excluded values are often associated with vertical asymptotes or removable holes.

Vertical Asymptotes

For the function

the denominator is zero at x=2. The function becomes arbitrarily large in magnitude as x approaches 2 from either side. The line

is a vertical asymptote.

Horizontal Asymptotes

For

the function approaches zero as |x| becomes large. Thus,

is a horizontal asymptote.

2.8 Graphs of Absolute-Value Functions

The basic absolute-value function is

It may be written piecewise as

Its graph is V-shaped, with vertex at the origin.

A more general function

has vertex

2.9 Graphs of Exponential and Logarithmic Functions

Exponential Functions

An exponential function has the form

where a>0 and a≠1.

If a>1, the function increases as x increases. If 0<a<1, the function decreases as x increases.

Every exponential graph of this form passes through

because

Its range is (0,∞), and y=0 is a horizontal asymptote.

Logarithmic Functions

The logarithmic function

is the inverse of

Its domain is (0,∞) and its range is ℝ.

The graph passes through

because

The line x=0 is a vertical asymptote.

2.10 Graphs of Trigonometric Functions

The basic trigonometric functions are periodic. Their graphs therefore repeat after fixed intervals.

The Sine Function

The function

has range

and period

Its zeros occur at

where n is an integer.

The Cosine Function

The function

also has range [-1,1] and period 2π.

The Tangent Function

The function

has period

It is undefined when

that is, at

where n is an integer.

These values correspond to vertical asymptotes of the tangent graph.

2.11 Intercepts and Symmetry

x-Intercepts

An x-intercept occurs where the graph meets the x-axis. At such a point,

Therefore, the x-intercepts of y=f(x) are obtained by solving

y-Intercepts

A y-intercept occurs where the graph meets the y-axis. At such a point,

Therefore, if 0 belongs to the domain, the y-intercept is

Symmetry About the y-Axis

A function is even if

The graph of an even function is symmetric about the y-axis.

Symmetry About the Origin

A function is odd if

The graph of an odd function is symmetric about the origin.

2.12 Transformations of Graphs

Many graphs can be obtained from a familiar basic graph by shifting, reflecting or stretching it. Understanding transformations reduces the need to construct every graph from a large table of values.

Vertical Translation

If

is shifted upward by k>0, the new graph is

A downward shift by k is represented by

Horizontal Translation

The graph of

is obtained by shifting y=f(x) to the right by h units.

The graph of

is shifted to the left by h units.

Reflection in the x-Axis

The graph of

is the reflection of y=f(x) in the x-axis.

Reflection in the y-Axis

The graph of

is the reflection of y=f(x) in the y-axis.

Vertical Stretch and Compression

For

all y-values are multiplied by a.

If |a|>1, the graph is stretched vertically. If 0<|a|<1, it is compressed vertically.

Horizontal Stretch and Compression

For

the horizontal scale is altered by the factor 1/|b|.

If |b|>1, the graph is compressed horizontally. If 0<|b|<1, it is stretched horizontally.

2.13 Reading and Interpreting Graphs

A graph should be interpreted as mathematical information rather than merely as a picture. Important features include the domain, range, intercepts, intervals of increase or decrease, turning points, maximum and minimum values, symmetry and asymptotic behaviour.

Increasing and Decreasing Behaviour

A function is increasing on an interval if its output values rise as the input values increase. It is decreasing on an interval if its output values fall as the input values increase.

For example,

decreases as x moves from negative values toward zero, reaches a minimum at x=0, and then increases for positive x.

Maximum and Minimum Values

A graph may possess a highest or lowest value over part or all of its domain.

Intersections of Two Graphs

If two functions

and

intersect, their y-values are equal at the point of intersection. Hence, the x-coordinates of intersections satisfy

Graphs as Preparation for Calculus

The graph of a function provides an intuitive introduction to several central ideas in calculus. A graph can suggest whether a function approaches a particular value near a point, which leads to the concept of a limit. A graph without a break at a point suggests continuity. The steepness of a graph introduces the idea of a derivative, while the region between a curve and an axis leads naturally to integration.

For this reason, graphical understanding complements algebraic manipulation. A reliable study of calculus requires both: algebra provides exact symbolic methods, while graphs reveal the behaviour and structure of functions.

Chapter Summary

3.1 Introduction to Limits

The concept of a limit is one of the central ideas of calculus. It provides a precise way of describing the value that a function approaches when the independent variable approaches a particular point. Limits make it possible to study the behaviour of functions even at points where direct substitution is impossible or where the function itself is not defined.

The ideas of continuity, differentiation and integration all depend fundamentally on limits. In differentiation, a derivative is defined as a limit of an average rate of change. In integration, definite integrals can be constructed from limits of sums.

Consider the function

As x approaches 3, the value of f(x) approaches 5. Symbolically,

This statement means that by choosing x sufficiently close to 3, the value of x+2 can be made as close to 5 as desired.

3.2 Intuitive Meaning of a Limit

The notation is

A limit describes what happens to the function near a point. It does not necessarily depend on the actual value of the function at that point.

3.3 Limit Notation

The expression

is read as “the limit of f(x) as x approaches a is L.”

The symbol x→a means that x is allowed to take values increasingly close to a, but it need not actually equal a.

3.4 Limits from Tables and Graphs

A limit can often be estimated numerically by constructing a table of function values near the point of interest.

Graphical Interpretation

On a graph, the statement

means that the points on the graph approach a height of L as their x-coordinates approach a from both sides.

The actual point (a,f(a)) may lie at (a,L), may lie somewhere else, or may not exist at all. The limit depends on the nearby behaviour.

3.5 Left-Hand and Right-Hand Limits

A two-sided limit examines the behaviour of a function as x approaches a point from both directions.

Left-Hand Limit

Right-Hand Limit

A finite two-sided limit exists precisely when both one-sided limits exist and are equal:

if and only if

3.6 Laws of Limits

Limit laws allow complicated limits to be evaluated from simpler ones. Suppose

and

Then, whenever the resulting expressions are defined, the following laws apply.

3.7 Direct Substitution

For many elementary functions, the first method to try when evaluating a finite limit is direct substitution.

If substituting x=a produces an ordinary real number, then that number is often the required limit.

3.8 Indeterminate Forms and Algebraic Simplification

Sometimes direct substitution produces the expression

This is called an indeterminate form. It does not mean that the limit is zero, nor does it mean that the limit is automatically undefined. Instead, it indicates that further simplification is required before the limit can be determined.

Common techniques for resolving elementary 0/0 forms include factorisation, cancellation and rationalisation.

3.9 Limits Involving Factorisation

3.10 Limits Involving Rationalisation

When square roots occur and direct substitution gives 0/0, multiplication by a conjugate often removes the indeterminate form.

3.11 Limits Involving Trigonometric Functions

Two important limits used frequently in elementary calculus are

and

where the angle x is measured in radians.

3.12 Infinite Limits and Vertical Asymptotes

A function may increase or decrease without bound as x approaches a particular value. Such behaviour is described using an infinite limit.

For example,

becomes arbitrarily large and positive as x approaches zero from either side. Hence,

This notation indicates unbounded growth; it does not mean that infinity is an ordinary real number.

3.13 Limits at Infinity

Limits at infinity describe the behaviour of a function when x becomes arbitrarily large in the positive or negative direction.

For example,

As the denominator grows without bound, the fraction approaches zero.

Rational Functions at Infinity

For a rational function, the relative degrees of the numerator and denominator often determine the limit as x→∞ or x→-∞.

Horizontal Asymptotes

If

or

then the line

is a horizontal asymptote in the corresponding direction.

3.14 Applications and Significance of Limits

The importance of limits extends beyond the direct evaluation of expressions. Limits provide the mathematical foundation for describing instantaneous change, continuity and accumulation.

Average Rate of Change

For a function f(x), the average rate of change between x=a and x=a+h is

When h is allowed to approach zero, the limiting value, when it exists, gives the instantaneous rate of change. This will lead directly to the derivative.

Limits and Continuity

A function behaves continuously at a point when its limiting value agrees with its actual function value there. In symbols, continuity at x=a requires

This relationship will be developed fully in the next chapter.

Limits and Differentiation

The derivative of f at a point x is defined through the limit

provided the limit exists. Thus, the theory of differentiation rests directly on the concept of a limit.

Limits and Integration

A definite integral can also be interpreted through a limiting process in which a region is divided into many narrow subregions and the corresponding sums approach a fixed value. In this way, limits also provide the theoretical basis for the calculation of areas and volumes.

Chapter Summary

4.1 Introduction to Continuity

Continuity is one of the fundamental ideas used to describe the behaviour of functions. Informally, a function is continuous over a region when its values change without sudden breaks, jumps or interruptions. Although this intuitive description is useful, calculus requires a more precise definition based on limits.

Consider the polynomial function

At x=2,

Also,

Thus, the value approached by the function as x approaches 2 is exactly the value of the function at 2. This agreement between the limiting value and the actual function value is the essential idea of continuity at a point.

Continuity is important because many of the standard procedures of calculus rely on functions behaving predictably. It also provides the bridge between the theory of limits developed in the preceding chapter and the study of derivatives that follows.

4.2 Continuity at a Point

This definition states that the value approached by the function near a must coincide with the actual value assigned to the function at a.

4.3 The Three Conditions for Continuity

The definition of continuity at x=a can be expressed through three conditions. All three conditions must be satisfied.

If any one of these conditions fails, the function is not continuous at the point.

Repairing a Removable Discontinuity

In the preceding example, continuity can be restored by defining

Then the function value and limiting value agree:

This illustrates an important feature of a removable discontinuity: the discontinuity can be eliminated by assigning the function the appropriate value at the missing point.

4.4 Continuity on an Interval

Continuity may be considered not only at a single point but also over an interval.

For a closed interval [a,b], the endpoints require one-sided conditions.

4.5 One-Sided Continuity

At endpoints of intervals, and especially for piecewise-defined functions, one-sided continuity is useful.

A function is right-continuous at x=a if

Similarly, it is left-continuous at x=a if

4.6 Continuity of Elementary Functions

Many functions commonly encountered in elementary calculus are continuous throughout their natural domains. Recognising this fact allows many continuity and limit problems to be handled efficiently.

Polynomial Functions

Every polynomial function is continuous for every real number.

For example,

is continuous on

Rational Functions

A rational function

is continuous at every point where

Radical Functions

A square-root function is continuous wherever its radicand is non-negative.

Exponential and Logarithmic Functions

Exponential functions of the form

are continuous for every real number.

Logarithmic functions of the form

are continuous for

Trigonometric Functions

The sine and cosine functions are continuous for all real values of their arguments:

The tangent function

is continuous wherever cos x≠0. Thus, it is discontinuous at

4.7 Algebra of Continuous Functions

Continuity is preserved under the standard algebraic operations. Suppose f and g are continuous at x=a. Then the following functions are also continuous at a whenever they are defined:

and

provided g(a)≠0.

4.8 Continuity of Composite Functions

Continuity is also preserved under suitable composition.

In limit form,

4.9 Discontinuity and Its Types

Several forms of discontinuity occur frequently in elementary calculus.

Removable Discontinuity

A removable discontinuity occurs when the limit exists but the function is either undefined at the point or has a value different from the limiting value.

For example,

is undefined at x=1. For x≠1,

Therefore,

The discontinuity can be removed by defining f(1)=2.

Jump Discontinuity

A jump discontinuity occurs when the left-hand and right-hand limits both exist as finite numbers but are unequal.

Infinite Discontinuity

An infinite discontinuity occurs when the function becomes unbounded as the variable approaches a point.

For example,

has an infinite discontinuity at x=2, where the graph has the vertical asymptote

Oscillatory Discontinuity

A function may also fail to approach a single value because it oscillates indefinitely near a point. A standard example is

As x→0, the quantity 1/x grows in magnitude and the sine function oscillates between -1 and 1 increasingly rapidly. Consequently,

does not exist.

4.10 Continuity of Piecewise-Defined Functions

A piecewise-defined function uses different formulas on different parts of its domain. The main points requiring examination are usually the boundary points where the defining formula changes.

4.11 Determining Unknown Constants for Continuity

A common application of the definition of continuity is to determine an unknown constant in a piecewise-defined function. The value of the constant is chosen so that the left-hand limit, right-hand limit and function value agree at the joining point.

4.12 The Intermediate Value Property

One of the important consequences of continuity is that a continuous function cannot move from one value to another without passing through all intermediate values.

A particularly useful case occurs when f(a) and f(b) have opposite signs. In that situation, zero lies between the two function values, so the continuous function must have at least one zero between a and b.

4.13 Continuity and Differentiability

Continuity and differentiability are closely related, but they are not identical concepts. A derivative measures instantaneous rate of change, while continuity describes whether the function value agrees with its limiting behaviour.

An important theorem states:

Thus,

The converse is not generally true. A function can be continuous at a point without being differentiable there.

A Continuous Function That Is Not Differentiable

Consider

The function is continuous at x=0 because

However, the graph has a sharp corner at the origin. The slope from the left is -1, while the slope from the right is 1. Therefore, there is no single derivative at x=0.

Hence,

This distinction becomes particularly important when derivatives are studied in the following chapters.

4.14 Applications and Significance of Continuity

Continuity provides a mathematical description of smooth and predictable variation. Many physical quantities are modelled by continuous functions over suitable intervals. Examples include the position of a moving object as a function of time, temperature variation over time, the volume of water in a steadily filled container, and many idealised relationships used in science and engineering.

Using Continuity to Evaluate Limits

If f is known to be continuous at x=a, then

This means that the limit can be evaluated simply by substitution.

Continuity in Mathematical Modelling

Continuity as Preparation for Differentiation

The derivative is defined through a limiting process. Before a function can be differentiable at an ordinary point, it must first be continuous there. A break, jump or infinite discontinuity prevents the derivative from existing at that point.

For this reason, continuity is an essential checkpoint when analysing the behaviour of a function. In later chapters, continuity will support the study of derivatives, tangents, rates of change, maxima and minima, and curve sketching.

Chapter Summary

5.1 Introduction to the Derivative

The derivative is one of the central concepts of differential calculus. It measures how rapidly one quantity changes with respect to another. When a quantity y depends on another quantity x, the derivative expresses the instantaneous rate at which y changes as x changes.

If

then the derivative of f with respect to x may be written as

or

The derivative has both an algebraic and a geometrical interpretation. Algebraically, it is defined by a limit. Geometrically, it gives the slope of the tangent to the graph of a function at a point.

5.2 Average Rate of Change

Before studying instantaneous change, it is useful to consider average change over an interval.

Suppose a function y=f(x) changes from f(x₁) to f(x₂) as x changes from x₁ to x₂. The average rate of change is

Average Rate of Change as Secant Slope

If the points

and

lie on the graph of f, then the average rate of change is also the slope of the secant line joining P and Q.

5.3 Instantaneous Rate of Change

An average rate of change describes change over an interval. In many applications, however, the rate of change at a particular instant or at a particular point is required.

Suppose the interval begins at x and has width h. The average rate of change over the interval is

If h is made smaller and smaller, the second point approaches the first. When the limit exists,

gives the instantaneous rate of change of the function at x.

5.4 Derivative from First Principles

Differentiation from first principles means finding a derivative directly from its limit definition rather than using an already established differentiation rule.

The basic procedure is:

• find f(x+h);
• form f(x+h)-f(x);
• divide by h;
• simplify;
• take the limit as h→0.

5.5 Geometrical Meaning of the Derivative

The derivative has a direct geometrical interpretation. Consider a curve y=f(x) and a point

A second point nearby may be written as

The slope of the secant line through P and Q is

As h→0, the point Q approaches P. If the secant slopes approach a definite value, that value is the slope of the tangent to the curve at P.

Equation of a Tangent

If the tangent to y=f(x) at x=a has slope f'(a), then its equation is

Equation of a Normal

The normal is perpendicular to the tangent. If the tangent slope is m, where m≠0, then the normal slope is

5.6 Derivative Notation

Several notations are commonly used for derivatives. If y=f(x), then the first derivative may be written as

or

The notation dy/dx is particularly useful when applying differentiation rules because it clearly indicates the variable with respect to which differentiation is being performed.

5.7 Differentiability

Differentiability is a stronger property than continuity. If a function is differentiable at a point, then it must be continuous at that point.

However, continuity does not always imply differentiability.

Failure of Differentiability at a Corner

Consider

At x=0, the function is continuous. But the slope to the left is -1 while the slope to the right is 1. Therefore, the derivative does not exist at the origin.

A derivative may fail to exist at:

• a discontinuity;
• a sharp corner;
• a cusp;
• a point with a vertical tangent in the ordinary finite-derivative sense.

5.8 Derivatives of Constant and Power Functions

Derivative of a Constant

If

where c is a constant, then

This is expected geometrically because the graph of a constant function is horizontal.

The Power Rule

5.9 Basic Rules of Differentiation

Constant Multiple Rule

If c is a constant, then

Sum Rule

Difference Rule

5.10 Higher-Order Derivatives

The derivative of a function may itself be differentiated. This produces a second derivative. Repeating the process gives higher-order derivatives.

If

then the second derivative may be written as

or

The third derivative may be written as

or

5.11 Derivative as Velocity and Acceleration

A major application of differentiation occurs in motion. If the position of a particle at time t is represented by s(t), then velocity is the derivative of position:

Acceleration is the derivative of velocity:

Equivalently,

5.12 Significance of the Derivative in Calculus

The derivative provides a precise mathematical description of local change. It determines the slope of a curve, the instantaneous velocity of a moving object, the rate at which one variable changes with respect to another, and the behaviour of a function near a point.

The derivative also provides the foundation for many later applications, including:

• finding tangents and normals;
• determining increasing and decreasing intervals;
• locating maximum and minimum values;
• solving optimization problems;
• analysing motion;
• sketching curves accurately;
• studying rates in physical, economic and biological models.

Chapter Summary

6.1 Introduction to Differentiation Techniques

Differentiation from first principles provides the theoretical foundation of the derivative, but it is not the most efficient method for differentiating complicated functions. Once the fundamental derivative rules have been established, they can be applied systematically to polynomial, rational, trigonometric, exponential, logarithmic and composite functions.

The choice of rule depends on the structure of the function. A product requires the product rule, a quotient usually requires the quotient rule, and a function nested inside another function requires the chain rule.

6.2 Review of the Power, Sum and Difference Rules

For a power function,

If several differentiable terms are added or subtracted, they may be differentiated separately.

6.3 The Product Rule

When two functions are multiplied together, differentiating each separately and multiplying the results is generally incorrect. The product rule must be used.

6.4 The Quotient Rule

When one differentiable function is divided by another, the quotient rule provides the derivative.

6.5 The Chain Rule

The chain rule is used when one function is contained inside another. Such a function is called a composite function.

Suppose

and

Then y=f(g(x)).

6.6 Differentiation of Trigonometric Functions

When angles are measured in radians, the fundamental derivatives of the basic trigonometric functions are

Trigonometric Functions with the Chain Rule

6.7 Differentiation of Exponential Functions

The natural exponential function has the important property

For a general exponential function,

where a>0 and a≠1.

6.8 Differentiation of Logarithmic Functions

The natural logarithm satisfies

More generally,

For logarithms to base a,

6.9 Implicit Differentiation

Not every relationship between x and y is conveniently written in the explicit form y=f(x). Sometimes the variables appear together in one equation. In such cases, implicit differentiation is useful.

The key principle is to differentiate both sides with respect to x, remembering that y is itself a function of x.

6.10 Logarithmic Differentiation

Logarithmic differentiation is useful when a function involves complicated products, quotients or variables occurring in both the base and the exponent.

The method involves taking the natural logarithm of both sides and then differentiating implicitly.

6.11 Differentiation of Composite Algebraic Functions

More complicated functions may require more than one differentiation rule. The most efficient method is to work from the outermost structure inward.

6.12 Higher-Order Derivatives of Standard Functions

Higher-order derivatives are often required in the study of acceleration, curvature, maxima, minima and curve sketching.

6.13 Choosing an Appropriate Differentiation Rule

Efficient differentiation depends on recognising the structure of a function before beginning the calculation.

Chapter Summary

9.1 Introduction to Definite Integration

Indefinite integration produces a family of antiderivatives. A definite integral, however, produces a numerical value associated with a function over a specified interval. It is one of the principal tools for measuring accumulated change.

If a quantity varies continuously with respect to another quantity, integration can be used to determine its total accumulated effect over an interval. Applications include area, displacement, total production, accumulated flow and volume.

A definite integral has two limits of integration and is written in the form

The numbers a and b are called the lower and upper limits of integration respectively.

9.2 Meaning and Notation of a Definite Integral

Unlike an indefinite integral, a definite integral does not require an arbitrary constant C. Once the limits are specified, the result is a definite number.

9.3 Definite Integrals as Limits of Sums

The concept of a definite integral originates from the idea of dividing an interval into many small subintervals and adding the contributions from each part.

Suppose [a,b] is divided into n equal subintervals of width

If x_i^* is a selected point in the ith subinterval, then the sum

approximates the accumulated contribution over the interval.

When the number of subdivisions increases without bound and the widths approach zero, the limiting value is the definite integral:

9.4 Geometrical Interpretation of Definite Integrals

When f(x)≥0 on [a,b], the definite integral

is equal to the area between the graph y=f(x), the x-axis and the vertical lines x=a and x=b.

When the graph lies below the x-axis, the integral contributes negatively. Thus, a definite integral gives signed area or net area, not necessarily the total geometric area.

9.5 Properties of Definite Integrals

Definite integrals satisfy several useful algebraic properties.

Equal Limits

Reversal of Limits

Additivity over Intervals

If c lies between a and b, then

Constant Multiple Rule

Sum and Difference Rules

Order Property

If

for every x∈[a,b], then

9.6 The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus establishes the central connection between differentiation and integration.

The notation

means

This theorem is extremely important because it allows definite integrals to be evaluated using antiderivatives instead of calculating limits of sums directly.

Accumulation Form of the Theorem

If

where f is continuous, then

Thus, differentiation and definite integration are inverse processes in a precise mathematical sense.

9.7 Evaluation of Definite Integrals

9.8 Definite Integrals Involving Trigonometric Functions

9.9 Definite Integrals Involving Exponential Functions

9.10 Substitution in Definite Integrals

Substitution can be applied directly to definite integrals. When the variable is changed, the limits should also be changed to values corresponding to the new variable.

9.11 Symmetry Properties of Definite Integrals

Symmetry can simplify many definite integrals.

Odd Functions

If f is odd, so that

then over a symmetric interval,

Even Functions

If f is even, so that

then

9.12 Total Change from a Rate Function

If F'(x)=f(x), then the Fundamental Theorem of Calculus gives

This principle is often called the net change theorem. It states that the integral of a rate of change gives the net change in the original quantity.

9.13 Average Value of a Function

9.14 Significance of Definite Integration

Definite integration provides a rigorous method for calculating accumulated quantities over intervals. Its importance lies not only in area calculations but in its ability to convert rates into totals.

For example, integrating velocity over time gives displacement, integrating a flow rate gives total volume, and integrating a density over a region can give total mass under appropriate conditions.

The Fundamental Theorem of Calculus makes these calculations practical by connecting accumulated change to antiderivatives.

Chapter Summary

10.1 Introduction to Applications of Definite Integrals

One of the most important applications of definite integration is the measurement of geometric quantities. Areas bounded by curves and volumes of solids generated by rotation can often be calculated exactly using definite integrals.

The central principle is accumulation. A region or solid is imagined as being divided into many extremely small parts. The contribution of each small part is expressed mathematically, and integration adds these contributions over the required interval.

In area problems, the essential quantity is usually a vertical or horizontal strip. In volume problems, the basic element may be a thin disk, washer or cylindrical shell.

10.2 Area Under a Curve

Suppose f(x)≥0 on [a,b]. The area under the curve y=f(x) and above the x-axis is

10.3 Area Between a Curve and the x-Axis

If a function is entirely above the x-axis on the interval, its definite integral gives the geometric area directly.

If the function is entirely below the x-axis, however, the definite integral is negative. Geometric area must be positive, so the magnitude of the integral is taken.

10.4 Total Area When a Curve Crosses the x-Axis

If a curve crosses the x-axis within the interval, a single definite integral gives net signed area rather than total geometric area. The interval must therefore be divided at the zeros of the function.

10.5 Area Between Two Curves

Suppose two functions y=f(x) and y=g(x) bound a region between x=a and x=b, with

Then the area between the curves is

10.6 Determining the Limits of Integration

In many area problems, the limits of integration are not provided directly. They must be obtained by finding where the bounding curves intersect.

If the curves are

and

their intersection points satisfy

10.7 Areas with Respect to y

Some regions are more conveniently described by horizontal strips rather than vertical strips. In that case, the curves are written in the form

and

If the right-hand curve is x=f(y) and the left-hand curve is x=g(y), then

In words: right curve minus left curve.

10.8 Introduction to Volumes of Solids of Revolution

A plane region can generate a three-dimensional solid when rotated about a line. Such a solid is called a solid of revolution.

For example, rotating a region under a curve about the x-axis may produce a solid whose cross-sections perpendicular to the axis are circular disks.

Definite integration allows the volumes of such solids to be obtained by summing the volumes of extremely thin elements.

10.9 Volume by the Disk Method

Suppose a region under the curve

from x=a to x=b is rotated about the x-axis and f(x)≥0.

A typical cross-section is a circular disk of radius

Its area is

Therefore, the volume is

10.10 Volume by the Washer Method

If the rotating region does not extend all the way to the axis of rotation, each cross-section may have a hole. Such a cross-section is called a washer.

If R(x) is the outer radius and r(x) the inner radius, then the cross-sectional area is

Therefore,

10.11 Rotation About the y-Axis

When a region is rotated about the y-axis, one possible method is to write x as a function of y and use horizontal disks or washers.

If

is rotated about the y-axis, the disk formula becomes

10.12 Volume by Cylindrical Shells

Another useful method for volumes of revolution is the cylindrical shell method. It is particularly convenient when vertical strips are rotated about the y-axis.

A vertical strip at position x generates a thin cylindrical shell with:

• radius x;
• height f(x);
• circumference 2πx;
• small thickness dx.

Therefore,

10.13 Applications to Physical and Geometrical Problems

The formulas for areas and volumes are not restricted to purely abstract curves. They provide models for real geometric and physical structures whenever the dimensions can be represented by suitable functions.

Cross-Sectional Interpretation

If a solid has cross-sectional area A(x) at position x, then its volume over the interval [a,b] is

The disk and washer methods are special cases of this general principle.

Recovering a Physical Quantity from a Rate

10.14 Choosing an Appropriate Area or Volume Method

Successful applications of integration depend on translating the geometry of the problem into an appropriate integral. The method should be chosen according to the orientation of the region and the axis of rotation.

Chapter Summary

7.1 Introduction to Systems of Particles

Many objects encountered in mechanics are not isolated particles. A car, a rigid body, a collection of interacting masses and even a planet may be regarded as a system made up of many particles. Instead of describing the motion of every individual particle separately, mechanics often simplifies the analysis by studying the system as a whole.

The particles within a system may exert forces on one another. They may also experience forces from objects outside the system. The distinction between these two kinds of force is fundamental.

7.2 Internal and External Forces

Internal forces occur in equal and opposite pairs according to Newton's third law. Consequently, when the motion of the whole system is considered, internal forces cancel in pairs in the total force balance. The motion of the centre of mass is therefore controlled by the resultant external force.

7.3 Total Mass and Total Momentum of a System

For a system of particles with masses m₁, m₂, ..., m_n, the total mass is

The total linear momentum of the system is the vector sum of the momenta of all its particles:

7.4 Centre of Mass of a System

The centre of mass provides a convenient single point with which to describe the overall translational motion of a system.

For discrete particles,

The centre of mass need not coincide with the location of any actual particle. It may even lie outside the physical material of an object, as can occur for a ring.

7.5 Centre of Mass in One Dimension

For particles positioned along the x-axis,

7.6 Centre of Mass in Two and Three Dimensions

For a system distributed in a plane, the coordinates of the centre of mass are

and

For three-dimensional systems,

7.7 Centre of Mass of Continuous Bodies

For an extended body whose mass is continuously distributed, the summation used for particles becomes an integral:

In one dimension,

If the body is uniform and possesses symmetry, the centre of mass often lies at the geometric centre. For example, the centre of mass of a uniform straight rod lies at its midpoint, while that of a uniform circular disc lies at its geometric centre.

7.8 Motion of the Centre of Mass

The velocity of the centre of mass is obtained by differentiating its position:

For a system of particles,

Therefore, the total momentum is

Differentiating again gives

7.9 Centre of Mass and Momentum Conservation

If the net external force on a system is zero,

Hence,

Therefore the centre of mass either remains at rest or moves with constant velocity. Internal interactions may be violent, as in an explosion, but they cannot alter the motion of the centre of mass of an isolated system.

7.10 Introduction to Relative Motion

Motion is always described with respect to a reference frame. An object may be stationary in one reference frame and moving in another. Relative motion provides a systematic way of relating observations made from different moving frames.

For example, a passenger sitting in a moving bus has zero velocity relative to the bus but a non-zero velocity relative to the road.

7.11 Relative Position and Relative Velocity

Let r_A and r_B be the position vectors of objects A and B relative to a common reference frame. The position of A relative to B is

Differentiating with respect to time,

7.12 Relative Acceleration

Differentiating the relative velocity equation gives the acceleration of A relative to B:

If both objects have the same acceleration, their relative acceleration is zero, and their relative velocity remains constant.

7.13 Relative Motion in One Dimension

When two objects move along the same straight line, relative motion can often be analysed by simple addition or subtraction of signed velocities.

Same Direction

For objects moving in the same direction, their relative speed is the difference between their speeds.

Opposite Directions

For objects moving towards one another, their closing speed is the sum of their speed magnitudes.

7.14 Relative Motion in Two Dimensions

When motion occurs in a plane, relative velocity must be treated as a vector. Components are usually resolved along perpendicular axes and combined.

7.15 Boats, Rivers, Wind and Moving Media

Relative velocity is particularly important when an object moves through a medium that is itself moving. Examples include a boat crossing a flowing river and an aircraft flying through moving air.

For a boat,

For an aircraft,

Chapter Summary

8.1 Introduction to Rotational Motion

Translational motion describes the movement of an object from one position to another. Rotational motion describes motion in which points of a body move around an axis. A rotating wheel, a ceiling fan, the hands of a clock and a spinning disc are familiar examples.

Many ideas from linear mechanics have rotational counterparts. Position corresponds to angular position, velocity to angular velocity, acceleration to angular acceleration, force to torque, mass to rotational inertia and linear momentum to angular momentum.

8.2 Angular Position and Angular Displacement

The orientation of a rotating body can be described by an angular position θ, measured relative to a chosen reference line.

The SI unit of angular displacement is the radian.

Hence,

8.3 Angular Velocity

Average angular velocity is

Instantaneous angular velocity is

Its SI unit is radian per second, rad/s.

8.4 Angular Acceleration

Average angular acceleration is

Instantaneous angular acceleration is

Its SI unit is rad/s².

8.5 Rotational Kinematics

For constant angular acceleration, rotational motion obeys equations directly analogous to those of uniformly accelerated linear motion.

8.6 Relationship Between Linear and Angular Quantities

For a point at distance r from a fixed axis of rotation, the arc length travelled is

provided θ is measured in radians.

Differentiating with respect to time gives

For tangential acceleration,

A point moving in a circle also has radial or centripetal acceleration

8.7 Torque and Moment of a Force

A force can change the translational motion of a body, but its ability to produce rotation depends on where and how the force is applied.

The magnitude of torque is

where r is the distance from the axis to the point of application, F is the force magnitude and θ is the angle between r and F.

Equivalently,

where r_⊥ is the perpendicular moment arm.

8.8 Torque as a Vector Product

Torque is a vector quantity and is defined by the cross product

Its direction is perpendicular to the plane containing r and F, determined by the right-hand rule.

In two-dimensional mechanics, counterclockwise torque is often assigned a positive sign and clockwise torque a negative sign, although the sign convention may be reversed if used consistently.

8.9 Rotational Form of Newton's Second Law

For a rigid body rotating about a fixed axis, the rotational analogue of Newton's second law is

where I is the moment of inertia about the axis and α is the angular acceleration.

The moment of inertia measures resistance to angular acceleration. Its detailed calculation and standard forms will be developed further in the next chapter.

8.10 Angular Momentum of a Particle

Since p=mv,

The magnitude is

If the particle moves perpendicular to the radius vector,

The SI unit of angular momentum is

8.11 Angular Momentum of a Rigid Body

For a rigid body rotating about a fixed principal axis, the angular momentum may be written as

This is analogous to the translational relation

The relation L=Iω is especially useful when the angular momentum is parallel to the rotation axis.

8.12 Conservation of Angular Momentum

Torque is related to the time rate of change of angular momentum:

If the resultant external torque is zero,

and therefore

For rotation about a fixed axis,

8.13 Rotational Work, Energy and Power

A torque acting through angular displacement can perform rotational work. For a constant torque,

provided torque and angular displacement correspond to the same rotational direction.

The rotational kinetic energy of a rigid body rotating about a fixed axis is

The rotational form of the work-energy theorem is

Rotational power is

8.14 Rotation of Coordinate Axes

Sometimes it is useful to describe the same point using a new coordinate system whose axes have been rotated relative to the original axes. This does not change the physical point; it changes only the coordinates used to describe it.

Suppose the new axes x' and y' are obtained by rotating the original axes counterclockwise through an angle φ. The coordinates are related by

and

Conversely,

and

The distance from the origin remains unchanged:

8.15 Polar Coordinates

Cartesian coordinates locate a point using perpendicular distances x and y. For rotational and circular motion, polar coordinates are often more natural.

The relationship between Cartesian and polar coordinates is

and

Conversely,

and, with the quadrant taken into account,

Unit Vectors in Polar Coordinates

In planar polar coordinates, the unit vectors are e_r in the radial direction and e_θ in the direction of increasing angle.

Unlike the fixed Cartesian unit vectors, these unit vectors change direction as the particle moves around the origin. Their time derivatives are

and

Velocity in Polar Coordinates

For position

the velocity is

Thus, motion may contain both a radial component and a transverse component.

Acceleration in Polar Coordinates

The acceleration is

For uniform circular motion, r is constant, ṙ=0, r̈=0 and θ̇=ω is constant. Therefore,

which is the familiar centripetal acceleration directed toward the centre.

8.16 Applications of Rotational Dynamics

Rotational mechanics combines angular kinematics, torque, rotational inertia, angular momentum and energy methods. These concepts allow the analysis of wheels, pulleys, rotating shafts, gears, flywheels and many other mechanical systems.

Chapter Summary