MATHEMATICS FOR PREMASTER - detailed summary of all lectures up to midterm
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Course
Mathematics
Institution
Tilburg University (UVT)
Extensive and detailed summary of all lectures, examples/ exercises from week 1 - 7 of the course mathematics for premaster's. All information and material needed for midterm is included
Function is a prescription that turns an input into an output
- ALL FUNCTIONS HAVE JUST 1 OUTPUT
- Functions with 1 variable, have just 1 input
- Functions with more variables, have more than 1 input
Function = is a formula (prescription) which calculates a number, the function value, for any feasible
value of the variable x
The function of the variable x, is the input for the formula Y
The set of feasible values of D of x is called the domain of the function
o Sometimes there are restrictions on what you can plug in -> for instance looking at
exam grades, it is only possible to receive a number between 0-10. So the output is
restricted to 0-10, which is your domain for this specific function. However there are
not always restrictions, and so not a limited domain
The set of all possible function values is called the range of the function
o The range of a function is the complete set of all possible resulting values of the
dependent variable (y, usually), after we have substituted the domain.
o the definition means: The range is the resulting y-values we get after substituting all
the possible x-values
The graph of a function is the visual representation of the function
Squares can never be negative -> if you take a square of a negative number it becomes positive
A zero of a function (nulpunt), y(x) is a solution of the equation, where you put the function value
equal to zero
Y(x) = 0
You do not put x=0 into the function – because this has nothing to do with the zero function
itself
You are going to find all x’s (all values of x) for which it means that if you put such an x into
the function, you will get a zero as result
this is the graph that belongs to finding all the zero points
-> there are 4 points on the X-axis where the line touches the zero point -> at this point you have a
zero of the function
,An intersection point of the graph of a function y(x), with the graph of another function z(x) is a point
(a,b) such that
- A is a solution of the equation y(x) = z(x)
- B = y(a) (=z(a))
Graphs can have multiple intersections as well
Most often the intersection point of 2 graphs = the break-even point
LIST OF ELEMENTARY FUNCTIONS
Constant functions -> y(x) =c
- Everytime you get the same answer – it doesn’t matter wat you put in, the answer is
constant
- It gives the same output, regardless of the input
- A constant function has a horizontal line graph (the height of the line, depends on the value
of the constant)
What are the zeros of a constant function = zero is where the x value equals y value -> but in a
constant function the y line never crosses the x-axis. This means that almost always there is no zero
in a constant function
Only when you have y(x) = 0 – so when it has to equal zero, than all x’s are equal to zero.
Because in this case the line is equal to the x-axis which is the zero line
All other functions are never zero
Linear functions -> y(x) = ax + b where a ≠ 0
- In a linear function a is never 0
- The constants can both be positive and negative -> is always the last number (without the x)
- The value is the first number ax -> where the x represents a value
- The letters (a,b,c) represent numbers
Y(x) = -3x +5
Y(x) = 2x +1
A determines the slope of the line (positive it is sloping upwards - the function value
becomes larger) because a >0,
When the slope is downwards, the function value becomes smaller. And the a <0
, Zero of a function is defined by the X value, where the corresponding function value equals zero
The intersection point is the point where X is equal to y(x) -> it needs to correspond to the correct y
value (both axis need to be mentioned). Intersection point consist of 2 coordinates, both the X which
is calculated and the Y value
Quadratic functions -> y(x) = Ax2 + bx + c, where a ≠ 0
If a would be equal to zero, it would not be a quadratic function
There must always be a ‘to the power’ -> macht teken for a quadratic function
Combination between quadratic (x2), linear and a constant number
Y(x) = ax2 +bx + c y(x) = 2x2 +5x +3
- Ax2 = the quadratic value
- Bx = the linear value
- C = the constant value
Lecture 2
When you have a formula x(2x + 7) = 0
This means that the ‘product’ is equal to zero
You should have at least one of the elements be made 0
This can be either the x=0 or 2x+7 = 0
When you take the last one -> 2x+7 = 2*0+7 = x=-3,5
When Y(x) does not have any zeros -> D ≤ 0 (discriminant is lower or equal to 0)
POWER FUNCTIONS
Positive integer (whole number/ round number -> never a decimal) -> negative numbers are
not considered in this case
X to the power of K = xk = a power function, where k is a integer, larger than 0 -> this K is
called the ‘degree’
X to the power of 1 = X0 = 1 -> for all X (so doing something to the power of 0 = 1)
X is free to be combined with all numbers, does not have to be integer = x is free
Only K is limited to integer numbers
What are the zeros of a positive integer power function? = if you look for zeros of the third degree
power function y(x) = X3 = x *x*x = 0
- In this case you have a product, that consist of 3 things -> however you need it to equal to
zero, which means one element must be 0
- If one x = 0 , all x’s in this case are zero – because they are the same
- However for this x = 0
- So the zeros of a positive integer power function = WATCH minute 38.00
Negative integer power functions
General = y(x) = x-k
Domain = x is not 0
x-1 = 1/x1 = 1/x
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