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Introductory Calculus

Differentiation

General Concepts The tangent of a curve at point is the line that just touches, by does not intersect, the curve at

point . The slope of this tangent line is equal to the instantaneous rate of change of the curve at

point .

Similarly, the gradient function of a function shows the rate of change of that function across its

entire domain. Differentiation is the process of calculating the gradient function of a given function.

Limits

The limit of a function is the value that the function approaches as approaches a given value.

Limits cannot be found for points on a function that are discontinuous or non-smooth

Gradient of a Line

Suppose we draw a line between points P and Q on a graph of the function

Clearly, the gradient of the line PQ will be:

As we move point Q closer and closer to point P, the value of gets smaller and smaller. Similarly,

the gradient of the line PQ gets closer and closer to the gradient of the tangent of point P.

Thus, we can find the gradient of the tangent of point P by evaluating the following limit:

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Example 1

Find the rate of change of the function when

First Principles

Instead of substituting a particular value of , we can use the same method to calculate the formula

for the rate of change of the function for any value of . This is called the derivative of the function.

Example 2

Find the rate of change of the function

Basic Derivatives

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Basic Rules

The Constant Rule

The Sum Rule

The Product Rule

The product rule is used to differentiate functions which are the multiple of two simpler functions

Example 1

Differentiate the function

Example 2

Differentiate the function

The Quotient Rule

The quotient rule is used to differentiate functions which are one function divided by another

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Example 3

Differentiate the function

Let and

The Chain Rule

Sometimes a function is too complex to differentiate as it is. To make the job easier, we can break a

complex function down into simpler components, each of which can then more easily be

differentiated independently.

The chain rule tells us how to recombine these simpler functions to find the rate of change of the

entire original function.

Thus, if we have a function which can be rewritten in terms of a sub-function

, then

Example 4

Differentiate the function

Example 5

Differentiate the function

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Example 6

Differentiate the function

Further Techniques

Implicit Differentiation

This is used when we want to find the rate of change of with respect to , but where obtaining a

specific expression for with respect to would be difficult or impossible.

For example, we cannot write a specific function for , as two different values of will

satisfy the relation for any given point.

Instead, we differentiate both sides of the equation with respect to .

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Note that we cannot differentiate with respect to directly. We can, however, differentiate is

with respect to , and then multiply the result by

, which will still give us

:

Once we do the above for all terms in the relation, we can make

the subject, and thus solve for

the derivative.

Example 7

Differentiate the relation

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Stationary Points

A stationary point is a point where a function temporarily stops rising or falling; this can occur at a

local minimum, local maximum, minima, maxima, or point of inflection. In other words:

To calculate the type of stationary point, it is useful to complete a ‘gradient table’, whereby the

gradient is calculated either side of a stationary point, and hence the nature of the point becomes

known.

Example 8

Find and state the nature of all the stationary points of the function

slope / \ _ / \

Curvature

The curvature of a function is the rate of change in the gradient of the function – the ‘slope of the

slope’. This can be found by taking the derivative of the derivative of the function.

Just because the slope of a function is zero, does not mean that its curvature is zero. Thus:

At a local maximum, the slope of the gradient of the function must be negative, as the function is

‘preparing’ to descend again. Similarly, the slope of the gradient must be positive at a minimum.

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Inverse Trigonometric Functions

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Integration

General Concepts While differentiation finds the gradient of a particular function, antidifferentiation (or integration)

finds the function for a given gradient

Fundamental Theorem

Suppose that we have a function , and an area under that function of . Suppose further

that we want to calculate the area between point and point .

Clearly the area will be approximately equal to height times width, or . As moves

closer to , approaches zero, and becomes – the infinitesimal.

Thus, the area between two points A and C is given by the sum of the area of all the infinitesimals in

between, which is represented by the notation:

Relation to Antidifferentiation

It turns out that this equation also gives the anti-derivative for the function . In order words:

This occurs because the change in area is roughly equal to the height times the change in width:

Or as approaches zero:

Rearranging gives us:

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Thus we see that the antiderivative of the function turns about to be the area below the function.

Basic Integrals

Basic Rules

Note that indefinite integrals evaluate as functions, and are expressed as

Definite integrals evaluate as numbers (or areas), and are expressed as

Of the rules below, those with definite integral end-points are only relevant to definite integrals.

The Equality Rule

The Sectional Rule

The Constant Rule

The Sum Rule

The Reversal Rule

Integration by Recognition

If a function is too difficult to integrate, we can find the derivative of a related function, and use that

to help us.

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Example 1

Find the derivative of , and hence antidifferentiate

Example 2

Find the derivative of , and hence antidifferentiate

Example 3

Find the derivative of

, and hence antidifferentiate

Calculating Areas

Basic Areas

The area under a function can be found by integrating that function between the two end-points:

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The constants and are known as the terminals of the integral, while , the function to be

integrated, is called the integrand. As the areas below the axis will come out as negative, we must

eliminate this negative sign by taking the negative value of the area (as two negatives make a

positive).

Above and Below Axis

When we have an area that extends both above and below the x-axis, we must first calculate the x-

intercepts so as to enable us to subdivide the area into subareas, and then calculate all these

subareas separately.

Between Functions

To find the area between functions and which do not intersect over the desired

interval , and if , then the area enclosed by the two functions and the lines

and if found by:

Between Intersecting Functions

When the functions do intersect, it is necessary to find the points of intersection and calculate the

area of each region separately, always ensuring that the uppermost function is subtracted fro the

lower function.

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Derivative Proofs

Derivative of

Suppose we have a population of bacteria (N), which doubles every time period. Note that this

assumption of doubling is for convenience only – it could triple or quintuple every time period.

The formula for this population will be , where is the number of elapsed time periods

Thus, the formula for the population after one complete time period is:

Where is the initial population and is the increase per time period (in this case )

Suppose that instead of the whole population increasing by 100% every time period, now the

population increases by 50% each half time period. The formula for this is:

Note that

owing to the lesser rate of increase each time, but now we have two bracket terms

to account for the two iterations each time period.

Thus, if the population increased by 25% four times per time period, we would have:

The increase per time period that would occur with continuous iterations would occur when

,

which can be found be evaluating:

The result is called , the natural rate of growth and decay, and the rate at which a continuously-

compounding value, increasing at a rate such that it would double every interval if it compounded

only once per interval, increases per each time interval.

Derivative of

Here follows proof that

:

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Derivative of

Here follows proof that

: