St Benedict’s College: Calculus Notes
TOPIC - CALCULUS
SKILLS NEEDED BEFORE ATTEMPTING THIS SECTION
1.
2.
3.
Factorising
 Common factor
 Trinomials
 Difference of 2 squares
 Sum/difference of two cubes
 Cubic polynomials (factor theorem and division)
Functions
 Function Notation
 Finding x and y intercepts
 Finding average gradient
 Understanding of maximum and minimum
 Understanding of increasing and decreasing of functions
 Finding equations of straight lines and quadratic (parabolas) functions given certain
information
Exponents and surds
 Use of all exponential rules
 Use of all surd rules
GLOSSARY OF TERMS
Calculus
Limit
The branch of mathematics dealing with limits and derivatives.
Common problems from calculus include finding the slope of a curve, the
maxima and minima of a function and finding the instantaneous rate of
change of a function.
The value that a function or expression approaches as
the domain variable(s) approach a specific value. Limits are written in the
form
Function
Slope of a curve
Average gradient
Derivative
First Derivative
First principles
Power Rule
Tangent
. For example, the limit of
as x approaches 3 is . This is
written
.
A relation for which each element of the domain corresponds to exactly one
element of the range. For every x value there is only one y value.
A number which is used to indicate the steepness of a curve at a particular point.
The slope of a curve at a point is defined to be the slope of the tangent line.
Thus the slope of a curve at a point is found using the derivative.
For a function, this is the change in the y-value divided by the change in the xvalue for two distinct points on the graph.
A function which gives the slope of a curve; that is, the slope of the line
tangent to a function. The derivative of a function f at a point x is commonly
written f'(x).
Same as the derivative. We say first derivative instead of
just derivative whenever there may be confusion between the first derivative and
the second derivative
An algebraic method used to find the derivative of a function (the slope of the
curve at any given point). The formula is always given in assessment situations.
The formula for finding the derivative of a power of a variable (the ‘shortcut’ –
and not using first principles)
A straight line that just touches a point on a curve.
Page 1 of 5
St Benedict’s College: Calculus Notes
Exponent Rules
(useful in calculus)
Second derivative
The derivative of a derivative. Usually written f"(x) or y".
Point of inflection
Concave
A point at which a curve changes from concave up to concave down, or viceversa.
A shape which has an indentation (like a cave – goes in)
Convex
A shape with no indentations (goes out)
y-intercept
The value of the y-coordinate when a graph crosses the y-axis
(where x = 0)
The value of the x-coordinate when a graph crosses the x-axis
(where y = 0)
The x-intercepts.
A solution to an equation of the form f(x) = 0. Roots may be real or complex.
Note: The roots of f(x) = 0 are the same as the zeros of the function f(x).
Sometimes in casual usage the words root and zero are used interchangeably.
Turning points and points of inflection
x-intercept
Root
Stationary points
Turning point
Point of inflection
Maximum of a
Function:
Minimum of a
Function
Optimisation
2-dimensional
3-dimensional
Instantaneous
Rate of Change
The point at which the graph changes from increasing to decreasing or
decreasing to increasing.
The point at which the graph changes from being concave to convex or from
convex to concave
Either a relative (local) maximum or an absolute (global) maximum.
Either a relative (local) minimum or an absolute (global) minimum
Finding the best possible option for requirements (eg finding maximum space
that can be used with a limited amount of fencing)
A ‘flat’ shape where only 2 dimensions are involved (eg length and breadth)
A shape where 3 dimensions are involved (eg length, breadth and height) such
as a rectangular prism (box).
The rate of change at a particular moment. Same as the value of
the derivative at a particular point
For a function, the instantaneous rate of change at a point is the same as
the slope of the tangent line. That is, it's the slope of a curve.
Page 2 of 5
St Benedict’s College: Calculus Notes
SUMMARY OF KEY CONCEPTS
LIMITS
The limit of a function, f(x), is the value of k that the function approaches from both LEFT and RIGHT
as x tends to a given value n.
( )
[The limit of the function as x tends to n is equal to k]
Eg
(
)
DERIVATIVES
The derivative is the rate of change at a point. It is therefore the gradient at a point (or the
instantaneous speed when discussing distance and time)
The derivative of f(x) is:
(
)
( )
( )
[The gradient at a point on f(x)]
NOTATION
The derivative of f(x) is denoted f’(x)
The derivative of y in terms
is denoted
( )
( ) is also sometimes used.
and
All 4 of the above denote the derivative.
SUMMARY OF CALCULUS CONCEPTS
 f’(x)
the derivative of a function f at the point
 To find the derivative from first principles we use:
(
)
( )
[First find f(x+h) and f(x) THEN use the formula]
 Shortcut (power rule):
(
)
(
)
EG
Steps to follow in order to use shortcut (power rule):
- Separate terms
Change surds to variables with exponents
- Ensure variable is in numerator
- Differentiate each term
- Convert negative exponents to positive
- Leave answer in form given
 The derivative of any constant is zero
Page 3 of 5
St Benedict’s College: Calculus Notes
 f’(x) gives you the value of the slope (gradient) of the tangent line to the graph
y = f(x) at x.
REMEMBER:
is the average gradient (approximate slope) of f(x) from one point to another
is the actual slope of f(x) at a point
If
then
= 2x
This is the ‘formula’ for the slope of the curve
.
You can now find the slope at any value of x.
 If f’(x) = 0 then the tangent line is horizontal (it has a gradient of zero) at the point where
The tangent line will be at a turning point in this case.
GRAPHS
Finding the derivative is useful as it tells you if the graph is increasing (f’(x) > 0) or decreasing
(f’(x) < 0).
If f’(x) = 0 there is a stationary point at x (and the tangent is horizontal)
[Not increasing or decreasing]
There are 3 main possibilities in this case:
The sign of f’(x) changes from
negative to positive
OR
The sign of f’(x) changes from positive
to negative
OR
The sign of f’(x) does not change
Local minimum at x
Was decreasing, now increasing
Turning Point
Local maximum at x
Was increasing now decreasing
Turning Point
Was increasing, still increasing
Was decreasing, still decreasing
Point of inflection
Page 4 of 5
.
St Benedict’s College: Calculus Notes
DRAWING THE GRAPH:
 Find the x and y intercepts (by making x = 0 for y-int and x = 0 for x-intercept)
 Use the derivative to find all stationary points
[Solve f(x) = 0 then find corresponding y-values by substituting back in to original function)]
 Decide what type of stationary point it is
 Consider the x-intercepts:
If there is only one, the graph may look like this:
or
If two of the x-intercepts are the same, one of them (the repeated factor) will be the turning point and
may look like this:
OPTIMISATION
If you have a formula for what you are trying to maximise or minimise you can use calculus to help you
find where your formula has a maximum or minimum.
Steps to follow:
 Decide what you are trying to optimise and find a formula for it.
(Decide which variables you are going to use and assign letters to them)
 Get rid of all the variables except one, using the information given
(same constraints/equations linking the variables). You can’t differentiate a formula with more
than one variable.
 Differentiate your formula with respect to the one variable and find the stationary points (make
the derivative equal to zero and solve)
 Check which one gives you the required optimisation
Page 5 of 5