WiSe 2025/2026 - Mini-Mock Exam 03
BFP Mathematics Course

Grading

Problem 1 Problem 2 Total Grade
___ /28 ___ /22 ___ /50 ___

Exam Information

Reading Time 10 minutes
Working Time 90 minutes
Total Points 50

Student Information

Name ________________________________________________

Guidelines

  • The exam duration is 90 minutes (50 points). All 2 problems must be completed.
  • Please write your name and student ID number on the cover sheet and each worksheet.
  • Only use the provided paper. Using other papers will invalidate the exam.
  • The task sheets are part of the examination and must be submitted.
  • Pencils and red pens are not allowed.
  • During the exam, conversations (except quietly with the supervisor), copying from others, and holding up work are considered attempts at cheating.
  • Only writing materials, a non-programmable, non-graphing calculator, and drawing instruments may be used.
  • No formula sheets, notes, or books are permitted.
  • Carrying smartphones, mobile phones, tablets, smartwatches, and similar devices, even when turned off, is prohibited and considered an attempt at cheating.

I wish you much success!

Problem 1: E-Commerce Platform Optimization [28 pts. total]

An online marketplace analyzes its pricing and demand relationships for a new product category. Market research reveals strategic information about customer behavior and cost structures.

Part A: Demand and Revenue Analysis

The demand function is linear with a maximum willingness to pay of 150 currency units (CU) when no units are sold. At a price of 30 CU, customers would purchase 40 units (Un).

  1. Determine the linear demand function \(p(x)\) expressing price as a function of quantity. Show all steps. [4 pts.]
  1. Show that the revenue function is given by \(R(x) = 150x - 3x^2\). Start from your demand function. [3 pts.]

Part B: Cost Structure

The company has fixed costs of 800 CU per month and variable costs that follow the function \(V(x) = 2x + 0.5x^2\) where \(x\) represents the quantity produced.

  1. Express the total cost function \(C(x)\) and compute the cost of producing 25 units. [4 pts.]

For verification purposes only:

  • \(p(x) = 150 - 3x\)
  • \(C(x) = 800 + 2x + 0.5x^2\)

Part C: Profit Optimization

  1. Determine the profit function \(P(x)\) and find the quantity that maximizes profit. Use the vertex formula and verify that this is indeed a maximum. [7 pts.]
  1. Calculate the break-even points by solving \(P(x) = 0\). Explain their significance for the business using complete sentences. [5 pts.]

Part D: Practical Constraints

  1. Due to warehouse limitations, the company can only stock a maximum of 20 units at any time. Determine:
    • The profit at this constraint level
    • The price that should be charged at this quantity
    • Whether the constraint is binding (affecting the optimal solution)
    Provide business reasoning for your conclusions. [5 pts.]

Problem 2: Function Analysis and Business Application [22 pts. total]

Consider the function \(f(x) = -0.25x^2 + 4x + 5\) which models the daily profit (in hundreds of CU) of a restaurant based on the number of staff members \(x\).

Part A: Function Properties [12 pts.]

  1. Determine the domain that makes sense in this business context. Explain your reasoning using complete sentences. [2 pts.]
  1. Find the vertex of the function using the vertex formula \(x = -\frac{b}{2a}\). Show your calculation and interpret its meaning for the restaurant. [4 pts.]
  1. Determine where the profit equals zero (x-intercepts). Use the quadratic formula and explain what these points represent for the business. [3 pts.]
  1. The restaurant currently employs 12 staff members. Calculate the current profit and determine how many additional staff would optimize profit. [3 pts.]

Part B: Transformations and Composition [10 pts.]

The restaurant plans to expand to a tourist location where:

  • All costs increase by 20% (affecting the entire profit function)
  • An additional fixed cost of 300 CU per day is incurred
  1. Write the transformed profit function \(g(x)\) for the tourist location incorporating the cost increase and additional fixed costs. Show the transformation steps. [3 pts.]
  1. If the minimum acceptable daily profit is 500 CU (5 hundreds), determine the range of staff numbers that achieve this for:
    • The original location
    • The tourist location
    Show your work algebraically. [4 pts.]

  1. The company uses a staffing agency that provides workers according to the function \(w(d) = 2d + 6\), where \(d\) is the number of days in advance the request is made.

    Express the profit as a composite function \((f \circ w)(d)\) for the original location and evaluate the profit when ordering 3 days in advance. [3 pts.]

NoteGrading Reference
Grade Percentage
1 (Excellent) ≥ 90%
2 (Very Good) ≥ 77%
3 (Good) ≥ 63%
4- (Pass) ≥ 45%
5-6 (Fail) < 45%

Note: Passing grade requires at least 45% of total points.

TipAppendix A – Terms used in phrasing of problems
Verb Task
name, state, give A reasoning does not have to be given unless explicitly demanded.
decide A reasoning does not have to be given unless explicitly demanded.
assess The judgment provided needs to be explained.
describe, characterize A description requires suitable wording and usage of technical terminology. A reasoning does not have to be provided.
explain, illustrate The explanation provides information which allows to comprehend a graphical depiction or a mathematical procedure.
interpret, construe An interpretation establishes a relation between e.g. a graphical depiction, a term or the result of a calculation and the provided context.
substantiate, reason, prove, show Statements and issues are to be confirmed by logical induction. The method can be freely chosen unless stated otherwise. The chosen method needs to be explained.
evaluate, calculate, compute, verify The computation needs to be illustrated starting from an ansatz.
determine, identify The method can be freely chosen unless stated otherwise. The chosen method needs to be explained.
investigate The method can be freely chosen unless stated otherwise. The chosen method needs to be explained.
graph, plot All diagrams and plots have to be drawn accurately with care.
sketch The sketch needs to contain all essential pieces of information.