Mock Exam 05: Foundations through Differential Calculus
Dr. Nikolai Heinrichs & Dr. Tobias Vlćek
Name: _______________________
Duration: 180 minutes
Total Points: 100
Permitted Aids:
- Calculator (non-programmable without graphing capabilities)
- Drawing instruments
- Monolingual dictionary
- No books, notes, or formula sheets
Instructions:
- Work through problems systematically, showing all steps
- Time yourself to practice exam conditions
- Check solutions afterward to identify areas needing review
Problem 1: Service Industry Cost Analysis [36 pts. total]
A consulting firm analyzes its cost structure for providing business advisory services. Financial analysis reveals the following information:
The fixed costs are 180 currency units (abbreviated as “CU” hereafter). At 2 quantity units of service (abbreviated as “Un” hereafter) the marginal costs are 12 CU. At 3 Un the curvature of the cost function changes sign. At 10 Un the total costs amount to 980 CU.
Market research shows that the demand for their services follows a linear function assuming a maximum price of 256 CU and a market saturation at 8 Un.
Part A: Function Development
Determine the cost function \(K(x)\), assuming it is a polynomial of third degree. [13 pts.]
Determine the linear demand function \(p(x)\). [3 pts.]
For verification purposes only:
\(K(x) = 2x^3 - 18x^2 + 60x + 180\)
\(p(x) = -32x + 256\)
Determine the revenue function \(E(x)\) and show that it can be written as \(E(x) = -32x^2 + 256x\). [2 pts.]
Show that the profit function is given by \(G(x) = -2x^3 - 14x^2 + 196x - 180\). [2 pts.]
Part B: Optimization and Business Strategy
Prove that the break-even point is at 1 Un of produce. Explain the significance of this quantity for the company. [3 pts.]
Compute the maximum profit and prove it really is a maximum. [4 pts.]
Decide which price the company has to ask for in order to gain the maximum profit. [3 pts.]
Determine the marginal profit at the profit-maximizing production level. Interpret what this value tells us about the firm’s pricing strategy. [2 pts.]
Compute the minimum variable cost per unit and the short-term lower limit price. [4 pts.]
Problem 2: Exponential Function Analysis [39 pts. total]
Let the function \(f\) be given by \(f(x) = x \cdot e^{-x/2}, \quad x \in \mathbb{R}\)
Part I: Basic Properties and Behavior
Determine the domain of the function \(f\). [1 pt.]
Investigate the asymptotic behavior of \(f\), and determine the \(x\)- and \(y\)-intercepts. [8 pts.]
Explain in complete sentences what the asymptotic behavior tells us about the long-term behavior of this function. [2 pts.]
Part II: Critical Analysis
Compute the first derivative \(f'(x)\) using the product rule. [4 pts.]
Determine the nature (classify as local maximum, local minimum, or saddle point) and the coordinates of all stationary points (critical points where \(f'(x) = 0\)). [5 pts.]
Find the equation of the tangent line at the point \(\left(2, \frac{2}{e}\right)\). [4 pts.]
Compute the angle of intersection \(\alpha\) between the tangent and the \(x\)-axis. [2 pts.]
Find the second derivative \(f''(x)\) and determine where the function changes concavity (inflection points). [4 pts.]
For verification purposes only:
\(f'(x) = e^{-x/2}(1 - x/2)\)
Maximum at \(x = 2\)
Argue whether the following statements are true or false: [3 pts.]
\(f''(2) = 0\)
In the interval \(0 < x < 2\), the function \(f\) is concave down.
\(f'(x)\) possesses a maximum at \(x = 2\).
The tangential gradient of \(f\) at \(x = 0\) is bigger than the slope of the secant in the interval \([0, 2]\).
- Sketch the graph \(G_f\) in the interval \([-2; 10]\). Label stationary points, inflection points, and intercepts. [6 pts.]
Problem 3: Function Determination and Parameter Analysis [25 pts. total]
Part A: Revenue Function Determination
An e-commerce company’s revenue function is modeled by a cubic polynomial \(R(x) = ax^3 + bx^2 + cx + d\), where \(x\) represents thousands of customers served per month.
The following conditions are established from historical data:
- When serving zero customers, revenue is zero: \(R(0) = 0\)
- At 4 thousand customers, the revenue function has an inflection point with a value of 256 CU
- Revenue reaches a local maximum at 8 thousand customers
Translate each condition into a mathematical equation. Explain why the inflection point condition at \((4, 256)\) provides two equations. [4 pts.]
Set up and solve the complete system of equations to find \(a\), \(b\), \(c\), and \(d\). [8 pts.]
Verify that your function satisfies the conditions \(R(4) = 256\) and \(R''(4) = 0\). [2 pts.]
For verification purposes only:
\(R(x) = -2x^3 + 24x^2\)
Part B: Function Family Investigation
Consider the family of functions defined by: \[f_a(x) = (x - a)^2 \cdot e^{-x}, \quad x \in \mathbb{R}, \quad a \in \mathbb{R}\]
Show that for all values of the parameter \(a\), each function \(f_a\) has exactly one zero. State the coordinates of this zero in terms of \(a\). [3 pts.]
Finding the local maximum:
Show that the first derivative can be written as: \[f_a'(x) = e^{-x}(x - a)(a + 2 - x)\] [2 pts.]
Using the factored form above, find all critical points of \(f_a\). Explain why \(x = a\) is not a local extremum. [2 pts.]
Verify that the local maximum occurs at \(x = a + 2\) and calculate \(f_a(a+2)\). [2 pts.]
The maximum value of \(f_a\) is \(y_{max} = 4e^{-(a+2)}\).
Calculate the maximum height when \(a = 0\). [1 pt.]
For which value of \(a\) does the maximum height equal \(4e^{-3}\)? [1 pt.]
Appendix: Practice Materials (Not Part of Examination)
Grading Scale Reference
| Grade | Points Required | Percentage | Self-Assessment |
|---|---|---|---|
| 1 (Excellent) | 91-100 | 91-100% | Outstanding |
| 2 (Very Good) | 77-90 | 77-90% | Strong understanding |
| 3 (Good) | 63-76 | 63-76% | Solid competence |
| 4- (Pass) | 45-62 | 45-62% | Meets requirements |
| 5-6 (Fail) | 0-44 | 0-44% | Needs review |
Focus Areas by Score Range:
If you scored 0-44 points:
- Review fundamental concepts in Sections 01-04
- Practice basic derivative rules and function analysis
- Focus on problem setup and equation-solving techniques
- Consider additional practice with simpler problems first
If you scored 45-62 points:
- Strong foundation, but work on connecting concepts
- Practice optimization problems with economic interpretation
- Improve curve sketching and graphical analysis skills
- Review related rates systematic approach
If you scored 63-76 points:
- Good understanding overall
- Focus on excellence-level problems (parts f-h)
- Practice comprehensive function analysis
- Work on explaining business interpretations clearly
If you scored 77-90 points:
- Very strong preparation
- Fine-tune proof techniques and verification steps
- Practice time management for complex problems
- Review any specific topics where you lost points
If you scored 91-100 points:
- Excellent mastery
- Maintain your preparation level
- Help others understand difficult concepts
- Focus on exam strategy and time optimization
Topic Review Checklist
After completing this exam, identify which topics need review:
Problem 1 Topics:
Problem 2 Topics:
Problem 3 Topics:
Remember: This practice exam is a learning tool. Use your performance to guide your study, not to judge your abilities!