Engineering Mathematics Revision Notes
A function is a rule which operates on an input and produces a single output.
If the rule produces multiple outputs, then it is referred to as a mapping.
Input of a function is called the argument.
Functions can be one-to-one or many-to-one.
Set of x values used as the input is called the domain.
Set of y values produced as a result of the domain is called the range of the function.
Examples
y= mx + c (one-to-one mapping)
domain: -∞<x<∞ range: -∞<y<∞
y= kx2 (many-to-one mapping)
domain: -∞<x<∞ range: 0<y<∞
y= sin x (many-to-one mapping)
domain: -∞<x<∞ range: -1<y<1
Even & Odd Functions
A function is even if the following equation is satisfied. The graph is
symmetrical about the y axis.
A function is odd if it follows this equation. Graph will possess
rotational symmetry about the origin.
Periodic Functions
This is any function that has a definite pattern repeated at regular intervals.
Each complete pattern is known as a cycle. The interval over which the repetition takes
place is the period. E.g. sin wave.
Inverse Functions f-1(x)
This is a function that reverses the original defined rule. The input of y will produce the
output x.
*The inverse function is a reflection of the function in the line y = x.
Only one-to-one functions have an inverse.
,Conic sections are equations of planetary or mechanical orbits (circle, ellipse, hyperbola).
An asymptote is a straight line which is tangent to a curve at infinity. The graph never
touches this line.
Oblique asymptote is where you have a diagonal asymptote and they cross.
Cartesian geometry
1.Straight Line
- A large number of engineering relationships can be
described by a linear relationship/straight line.
- (x,y)
- c is the constant/y-intercept.
- m is the gradient of the graph.
2.Circle
- Not a function as it is many-to-many.
- Equation is represented by this if the
centre is at the origin.
- Most of the time the centre is shifted
out of the origin, so the equation takes
this form.
3.Ellipse
- Has intercepts at x = + a and y = + b.
- The general equation takes the form:
4.Hyperbola
- No y intercept as y = + -b2 so there’s no real solutions.
- This is the equation for a horizontal hyperbola with vertex
at the origin.
- x intercepts are (-a,0) and (a,0).
*Note how the general equations for a circle, ellipse, and hyperbola are all linked.
Example
What type of graph is 4x2 – 16x + 9y2 + 18y – 11 = 0
Complete the square:
2 2
4 [(x - 4x + 4) – 4] + 9 [(y + 2y + 1) – 1] – 11 = 0
2 2
4(x-2) – 16 + 9(y+1) – 9 – 11 = 0
2 2
4(x-2) + 9(y-1) = 36
2 2
4(x-2) + 9(y-1) = 1
36 36
2 2
(x-2) + (y-1) = 1
9 4
2 2
(x-2) + (y-1) = 1 -> ELLIPSE
2 2
3 2
, Asymptotes
Not all curves have asymptotes but some of the simplest ones belong to rational functions.
Sketching rational functions
E.g. f(x) = P(x) = (5-x2)
Q(x) (x+3)
Asymptotes
Vertical:
Q(x) = 0 -> x + 3 = 0 x = -3
Horizontal:
You will have a horizontal asymptote if the degree of the numerator = degree of the
denominator.
Degree of num is less than degree of the denominator. y = 0
Degrees are the same, take the coefficient in front of x2 and divide top by the bottom. E.g.
¾.
Degree of numerator is greater than the bottom -> no horizontal asymptote.
Oblique
These occur when degree of numerator is 1 larger than the denominator.
The oblique asymptote is the result of algebraic long division (not remainder).
->y = -x + 3
The remainder of the long division can be ignored because this will tend to 0 as x becomes
large.
Polar Coordinates
Converting from cartesian to polar: Converting from polar to cartesian:
R = x2 + y2 x = r cos
= tan-1(b/a) y = r sin
Example
Convert r2 = 1/sincos from polar to cartesian form.
Eliminate sin and cos using the above equations:
2 2
r = 1 = r
y/r x x/r xy
1 = 1/yx
y=1
x
A function is a rule which operates on an input and produces a single output.
If the rule produces multiple outputs, then it is referred to as a mapping.
Input of a function is called the argument.
Functions can be one-to-one or many-to-one.
Set of x values used as the input is called the domain.
Set of y values produced as a result of the domain is called the range of the function.
Examples
y= mx + c (one-to-one mapping)
domain: -∞<x<∞ range: -∞<y<∞
y= kx2 (many-to-one mapping)
domain: -∞<x<∞ range: 0<y<∞
y= sin x (many-to-one mapping)
domain: -∞<x<∞ range: -1<y<1
Even & Odd Functions
A function is even if the following equation is satisfied. The graph is
symmetrical about the y axis.
A function is odd if it follows this equation. Graph will possess
rotational symmetry about the origin.
Periodic Functions
This is any function that has a definite pattern repeated at regular intervals.
Each complete pattern is known as a cycle. The interval over which the repetition takes
place is the period. E.g. sin wave.
Inverse Functions f-1(x)
This is a function that reverses the original defined rule. The input of y will produce the
output x.
*The inverse function is a reflection of the function in the line y = x.
Only one-to-one functions have an inverse.
,Conic sections are equations of planetary or mechanical orbits (circle, ellipse, hyperbola).
An asymptote is a straight line which is tangent to a curve at infinity. The graph never
touches this line.
Oblique asymptote is where you have a diagonal asymptote and they cross.
Cartesian geometry
1.Straight Line
- A large number of engineering relationships can be
described by a linear relationship/straight line.
- (x,y)
- c is the constant/y-intercept.
- m is the gradient of the graph.
2.Circle
- Not a function as it is many-to-many.
- Equation is represented by this if the
centre is at the origin.
- Most of the time the centre is shifted
out of the origin, so the equation takes
this form.
3.Ellipse
- Has intercepts at x = + a and y = + b.
- The general equation takes the form:
4.Hyperbola
- No y intercept as y = + -b2 so there’s no real solutions.
- This is the equation for a horizontal hyperbola with vertex
at the origin.
- x intercepts are (-a,0) and (a,0).
*Note how the general equations for a circle, ellipse, and hyperbola are all linked.
Example
What type of graph is 4x2 – 16x + 9y2 + 18y – 11 = 0
Complete the square:
2 2
4 [(x - 4x + 4) – 4] + 9 [(y + 2y + 1) – 1] – 11 = 0
2 2
4(x-2) – 16 + 9(y+1) – 9 – 11 = 0
2 2
4(x-2) + 9(y-1) = 36
2 2
4(x-2) + 9(y-1) = 1
36 36
2 2
(x-2) + (y-1) = 1
9 4
2 2
(x-2) + (y-1) = 1 -> ELLIPSE
2 2
3 2
, Asymptotes
Not all curves have asymptotes but some of the simplest ones belong to rational functions.
Sketching rational functions
E.g. f(x) = P(x) = (5-x2)
Q(x) (x+3)
Asymptotes
Vertical:
Q(x) = 0 -> x + 3 = 0 x = -3
Horizontal:
You will have a horizontal asymptote if the degree of the numerator = degree of the
denominator.
Degree of num is less than degree of the denominator. y = 0
Degrees are the same, take the coefficient in front of x2 and divide top by the bottom. E.g.
¾.
Degree of numerator is greater than the bottom -> no horizontal asymptote.
Oblique
These occur when degree of numerator is 1 larger than the denominator.
The oblique asymptote is the result of algebraic long division (not remainder).
->y = -x + 3
The remainder of the long division can be ignored because this will tend to 0 as x becomes
large.
Polar Coordinates
Converting from cartesian to polar: Converting from polar to cartesian:
R = x2 + y2 x = r cos
= tan-1(b/a) y = r sin
Example
Convert r2 = 1/sincos from polar to cartesian form.
Eliminate sin and cos using the above equations:
2 2
r = 1 = r
y/r x x/r xy
1 = 1/yx
y=1
x
