Mini-Mock Exam 01: - Business Foundations

BFP Mathematics Course
Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Overview

This assessment is a tool for us to understand your current mathematical strengths and to identify areas where we can focus our efforts. Please read these instructions carefully:

  • Total Time: Approximately 140 minutes
  • No Grades: This test will not be graded. Its purpose is purely diagnostic. Be honest about what you know and don’t know.
  • Solve or Explain: Your primary goal is to solve each task. If you get stuck or cannot solve a task, use the space provided to write a short, clear sentence explaining why. Examples: “I have forgotten the quotient rule for derivatives,” “I don’t know how to set up a transition matrix,” or “I am unsure what ‘asymptote’ means.” This information is extremely valuable.
  • Confidence Scale: For each task, please circle a number from 1 (not confident at all) to 7 (very confident) that reflects how you feel about your ability to tackle that specific task.
  • Allowed Materials: You may use a scientific calculator.

Section 1: Mathematical Foundations & Algebra

Suggested time: approximately 15 minutes.

Task 1.A: Manipulating Terms

Confidence (circle one): 1 2 3 4 5 6 7

Simplify the expression completely and show your steps: \[ \left(\frac{x^{4}\,y}{x^{-2}\,y^{3}}\right)^{2}\cdot y \]

Task 1.B: Business Application with Percentages

Confidence (circle one): 1 2 3 4 5 6 7

A company buys a product for a net price of €150. The company adds a 40% markup to determine the selling price. The final selling price must also include a 19% Value Added Tax (VAT). What is the final gross selling price?

Task 1.C: Logarithms & Scientific Notation

Confidence (circle one): 1 2 3 4 5 6 7

  1. Convert \(3.2 \times 10^{-4}\) to decimal form.
  2. Solve for \(x\): \(\log_2(x) + \log_2(4) = 5\).

Task 1.D: Factorization & Absolute Value

Confidence (circle one): 1 2 3 4 5 6 7

  1. Factor completely: \(x^2 - 5x + 6\).
  2. Solve: \(|2x - 3| = 7\).

Task 1.E: Number Systems

Confidence (circle one): 1 2 3 4 5 6 7

Classify each as \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), or \(\mathbb{R}\) (choose all that apply):

  1. \(\sqrt{2}\)
  2. \(-\frac{5}{2}\)
  3. \(0.\overline{3}\)

Section 2: Equations & Basic Functions

Suggested time: approximately 25 minutes.

Task 2.A: System of Linear Equations

Confidence (circle one): 1 2 3 4 5 6 7

A small company produces two types of desks: Standard and Premium. A Standard desk requires 2 hours of assembly and 1 hour of finishing. A Premium desk requires 3 hours of assembly and 2 hours of finishing. The company has 180 assembly hours and 100 finishing hours available each week. Let \(x\) be the number of Standard desks and \(y\) be the number of Premium desks. Set up a system of two linear equations and solve for \(x\) and \(y\) so that all available hours are used exactly.

Task 2.B: Quadratic Functions in Business

Confidence (circle one): 1 2 3 4 5 6 7

The profit \(P\) (in thousands of €) for selling \(x\) hundred units is \[ P(x) = -2x^{2} + 20x - 32. \]

  1. Find the break-even points (where profit is zero).
  2. Determine the number of units that maximizes the profit.

Task 2.C: Linear Inequality

Confidence (circle one): 1 2 3 4 5 6 7

Solve the inequality \(3x - 5 \leq 2x + 7\) and represent the solution set in interval notation.

Task 2.D: Exponential Equation

Confidence (circle one): 1 2 3 4 5 6 7

Solve for \(x\): \[ 5 \cdot 2^{\,x-1} = 40. \]

Section 3: Advanced Functions & Analysis

Suggested time: approximately 20 minutes.

Task 3.A: Analysis of a Rational Function

Confidence (circle one): 1 2 3 4 5 6 7

Given \[ f(x) = \frac{2x - 4}{x + 1}. \]

  1. State the domain of \(f\).
  2. Find any \(x\)- and \(y\)-intercepts.
  3. Find the equations of the vertical and horizontal asymptotes.

Task 3.B: Trigonometric Functions in Sales

Confidence (circle one): 1 2 3 4 5 6 7

The quarterly sales of ice cream (in thousands of units) can be modeled by \[ S(t) = 15 \sin\left(\frac{\pi}{2} t - \frac{\pi}{2}\right) + 25 \] where \(t\) is time in quarters (\(t = 1\) is the end of the first quarter).

  1. What are the sales at the end of the second quarter (\(t = 2\))?
  2. What is the maximum sales volume predicted by this model?

Task 3.C: Exponential/Logarithmic Model

Confidence (circle one): 1 2 3 4 5 6 7

For \(f(x) = A \cdot e^{kx}\), suppose \(f(0) = 3\) and \(f(2) = 12\).

  1. Find \(A\) and \(k\).
  2. State whether the function is growing or decaying and give the continuous growth rate.

Task 3.D: Composition, Domains, and Inverse

Confidence (circle one): 1 2 3 4 5 6 7

Let \(g(x) = \sqrt{x - 1}\) and \(h(x) = \ln(x)\).

  1. Find the domain of \(g \circ h\) and of \(h \circ g\).
  2. Find the inverse of \(f(x) = \frac{2x - 1}{x + 3}\) and state its domain.

Section 4: Differential Calculus

Suggested time: approximately 25 minutes.

Task 4.A: Derivative Rules & Tangent Line

Confidence (circle one): 1 2 3 4 5 6 7

  1. Differentiate \(y = (x^2 + 1) \cdot e^{3x}\).
  2. Compute \(y'(0)\) for \(y(x) = \frac{\sin(x)}{x^2 + 1}\).
  3. Find the equation of the tangent line to \(y = \sqrt{2x + 1}\) at \(x = 4\).

Task 4.B: Limits, Continuity, and Difference Quotient

Confidence (circle one): 1 2 3 4 5 6 7

  1. Compute \(\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\).
  2. Let \(f(x) = \frac{x^2 - 1}{x - 1}\) for \(x \neq 1\) and \(f(1) = 3\). Is \(f\) continuous at \(x = 1\)? Justify.
  3. For \(f(x) = x^2 - 3x\), compute the difference quotient \(\frac{f(x + h) - f(x)}{h}\) and simplify.

Task 4.C: Extrema and Inflection

Confidence (circle one): 1 2 3 4 5 6 7

Consider \(f(x) = x^3 - 6x^2 + 9x\).

  1. Find all critical points and classify them (local max/min) using derivatives.
  2. Find any inflection points and intervals of concavity.

Task 4.D: Recovering a Function from its Derivative (Micro)

Confidence (circle one): 1 2 3 4 5 6 7

Given \(f'(x) = 3x^2 - 4x + 1\) and \(f(0) = 2\), find \(f(x)\).

Section 5: Integral Calculus & Applications

Suggested time: approximately 20 minutes.

Task 5.A: Total Profit from Marginal Profit

Confidence (circle one): 1 2 3 4 5 6 7

The marginal profit is \[ P'(x) = -3x^{2} + 24x + 27, \] where \(x\) is the number of units sold. Compute the total change in profit from the 4th to the 10th unit using a definite integral.

Task 5.B: Financial Mathematics

Confidence (circle one): 1 2 3 4 5 6 7

An investment of €5,000 earns 4% interest per year, compounded annually. What is the future value after 10 years?

Task 5.C: Integration Techniques & Area

Confidence (circle one): 1 2 3 4 5 6 7

  1. Compute the indefinite integral \(\int \frac{2x}{x^2 + 1} \, dx\).
  2. Evaluate \(\int_0^1 x \cdot e^{x^2} \, dx\) (use substitution).

Section 6: Probability Theory

Suggested time: approximately 20 minutes.

Task 6.A: Conditional Probability

Confidence (circle one): 1 2 3 4 5 6 7

A factory has two machines, A and B. Machine A produces 60% of output with a 3% defect rate; Machine B produces 40% with a 5% defect rate.

  1. Draw a tree diagram to represent this situation.
  2. What is the probability that a randomly selected item is defective?
  3. Given an item is defective, what is the probability it came from Machine A?

Task 6.B: Combinatorics

Confidence (circle one): 1 2 3 4 5 6 7

A department has 10 employees. A project team with three distinct roles is needed: project leader, main developer, and tester. How many different teams can be formed if no one can hold more than one role?

Task 6.C: Binomial Distribution

Confidence (circle one): 1 2 3 4 5 6 7

Let \(X \sim \text{Bin}(n = 20, p = 0.3)\).

  1. Compute \(P(X = 5)\).
  2. Find \(E[X]\) and \(\mathrm{Var}(X)\).
  3. Interpret the expectation in a business context.

Task 6.D: Contingency Table & Predictive Values

Confidence (circle one): 1 2 3 4 5 6 7

A screening test has sensitivity 90% and specificity 95%. In a population with 5% prevalence:

  1. Build a \(2 \times 2\) contingency table per 10,000 people.
  2. Compute Positive Predictive Value and Negative Predictive Value.

Section 7: Statistics & Linear Algebra

Suggested time: approximately 15 minutes.

Task 7.A: Descriptive Statistics

Confidence (circle one): 1 2 3 4 5 6 7

Monthly revenue (in thousands of €) over six months: 12, 15, 14, 18, 20, 23. Compute the mean and the sample standard deviation.

Task 7.B: Transition Matrices

Confidence (circle one): 1 2 3 4 5 6 7

Two streaming services, StreamA and StreamB, compete for a fixed market. Currently: StreamA has 70%, StreamB has 30%. Each month:

  • StreamA retains 80% of its customers, 20% switch to StreamB.
  • StreamB retains 90% of its customers, 10% switch to StreamA.
  1. Write the transition matrix \(T\) for this process.
  2. Calculate the market share after one month.
  3. Calculate the market share after three months

Task 7.C: Descriptive Stats—Median & IQR

Confidence (circle one): 1 2 3 4 5 6 7

For the dataset: 8, 12, 12, 13, 15, 18, 21

  1. Compute the median and the interquartile range (IQR).
  2. Which summary (mean vs median) is more robust to outliers? Briefly explain.

Task 7.D: Matrix Inverse & Solving a System

Confidence (circle one): 1 2 3 4 5 6 7

  1. Find the inverse of \(A = \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}\) if it exists.
  2. Solve \(A \mathbf{x} = \mathbf{b}\) for \(\mathbf{x}\), where \(\mathbf{b} = \begin{bmatrix} 5 \\ 3 \end{bmatrix}\).