Summary Mathematics for Pre-Master Mid-Term TISEM Tilburg University
14 views 0 purchase
Module
350897-B-6 (350897B6)
Institution
Tilburg University (UVT)
This is an extensive summary of the video lectures of each week (class notes) screens of important example questions discussed during the Q&A, and more examples. Studying this and practicing the old exams will guarantee that you pass the course! Good luck.
Content Exam October:
From book Mathematics for Business Economics by Hamers, Kaper and Kleppe:
- Chapter 1
- Chapter 2, except 2.3.3
- Chapter 3
- Chapter 4.1 and 4.2
- Chapter 5.1, 5.3 and 5.7
- All examples related to minimum function are dismissed.
Lecture Week 1, Chapter 1
Function
= With a function you have one or more input variables (independent variable), and an output
variable (dependent variable).
Example: input variable = x, and then the function is x2 (whatever you put in, you must square it), and
the output is y.
So the function is: y(x) = x2
D(p) = 10-p
z(xy) = √x + xy2
Domain (D) of a function = set of values that you can use as an input.
Depends on what the input variable is, example: price (p) cannot be negative so p ≥ 0. Also depends
on what the output variable is, example: demand (D) cannot be negative.
So, for D(p) = 10-p → 0 ≤ p ≤ 10 OR p∈[0,10]
if boundaries are included: .. ∈[.. , ..]
if boundaries are not included: .. ∈(.. , ..)
x∈(0,∞)
Another example: z(xy) = √x + xy2 → x cannot be negative, as it is put in a square root in the function.
So x ≥ 0.
Range = set of all possible outputs.
Example:
1
,Input: litres of petrol sold (q)
Output: revenue (R)
Function: R(q) = 1.65q
Domain: 0 ≤ q ≤ 5000
Range: 0 – 8250
Graph:
A zero of a function
= solution of the equation
y(x) = 0
y is in this case 0, and the points that come out are all the points of the graph crossing the x-as.
In this case, only mention the outcomes of x! so x = …
Thus, y is always zero so no need to write it down.
Intersection point = the points where two graphs intersect
Elementary functions
1. Constant function: y(x) = c
- Always same answer
- Example: y(x) = 3 or y(x) = 5 1/3
- Graph: horizontal line crossing the y-as at one point
2. Linear function: y(x) = ax + b
- a≠0
- a = slope
- b crosses the y-as.
- Graph:
- Zero of a linear function:
solve y(x) = 0
-
- a = y 2 – y 1 / x 2 – x1
2
,- Example linear functions: y(x) = 5x + 3 and z(x) = -4x
a. Determine the zeros of y(x) and z(x).
So: y(x) = 0
5x + 3 = 0
5x = -3
x = -3/5
z(x) = 0
-4x = 0
x=0
b. Draw the graphs of y(x) and z(x).
Find two points and connect them.
c. Determine the intersection point of the graphs of y(x) and z(x).
y(x) = z(x)
5x + 3 = -4x
5x + 4x = -3
9x = -3
x = -3/9 = -1/3
y(-1/3) = 5 * -1/3 + 3 = 4/3
3. Quadratic functions: y(x) = ax2 + bx + c
- a≠0
- Graph:
a > 0, ∪
a < 0, ∩
- Example: y(x) = x2 – 5x + 2
a. Zeros of a quadratic function:
→ IMPORTANT!!!
x = 5 + √ or x = 5 - √
b. Graph:
a = 1, so ∪.
Already have the points on x-as, now find the point where x=0, so the y-as. This is c.
- Discriminant (D): b2 – 4ac
D > 0: 2 zeros → 2 points on x-as
D = 0: 1 zero → 1 point on the x-as
D < 0: no zero → no point on x-as
3
, - Example: y(x) = 2x2 + px + 4 ½. Determine the values of p such that:
a. y(x) has no zeros:
First: solve D = 0
D = b2 – 4ac
D = p2 – 4 * 2 * 4 ½
D = p2 – 36 = 0
p2 = 36
p = √36 = 6 or -6
So:
It must be: -6 < p < 6
b. y(x) has one zero:
D = p2 – 36
So p2 must be 36, so p = √36 = 6 or -6.
c. y(x) has two zeros:
D = p2 – 36
p = √36 = 6 or -6.
to get 2 zeros: p < -6 or p > 6
Solving inequalities
1. Set the inequality to zero. (Example: h(x) ≥ 0, h(x) > 0, h(x) ≤ 0, h(x) < 0).
2. Determine the zeros of h.
- Solve h(x) = 0
3. Determine the sign chart of h.
4. Find the solution set from the sign chart.
Example: Find all x such that 2x2 + 3x + 2 ≤ 4x + 3.
So:
1.
2x2 + 3x – 4x + 2 – 3 ≤ 0
h(x) = 2x2 – x – 1 ≤ 0
2.
h(x) = 0
4
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller kellymonka. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £6.31. You're not tied to anything after your purchase.
