Session 09-02 - Differential Calculus Review
Section 09: Exam Preparation
Warm-Up Quiz
Quick Review Questions
Work individually for 5 minutes, then discuss with the class (5 minutes)
Evaluate the limit \(\displaystyle\lim_{x \to 2} \frac{x^2-x-2}{x^2-4}\).
Find \(f'(x)\) for \(f(x) = x^3 \cdot e^{-x}\).
Solve \(3x^2 - 12x + 9 = 0\).
For \(f(x) = x^2 - 6x + 10\), find the vertex and classify it as min or max using the derivative.
Guided Practice A
Practice Problem 1
Limits: Factoring, Rationalizing and Degrees (Section 05)
Work in pairs
Evaluate each limit:
(a) \(\displaystyle\lim_{x \to 5} \frac{x^2-25}{x^2-3x-10}\)
(b) \(\displaystyle\lim_{x \to \infty} \frac{4x^3-x}{2x^3+5x^2}\)
(c) \(\displaystyle\lim_{x \to \infty} \frac{3x^2+1}{x^2-x}\)
(d) \(\displaystyle\lim_{x \to 0} \frac{\sqrt{x+9}-3}{x}\)
Practice Problem 2
Differentiation Rules Drill (Sections 05)
Work in pairs
Find the derivatives:
(a) Product rule: \(f(x) = x^2 \cdot \ln(x)\)
(b) Chain rule: \(g(x) = e^{x^2-3x}\)
(c) Quotient + Chain: \(h(x) = \dfrac{(2x+1)^3}{x-1}\)
Practice Problem 3
Fencing Optimization (Sections 03 & 05)
Work in pairs
A farmer has 200 meters of fencing to enclose a rectangular area along a straight river (no fencing needed on the river side). What dimensions maximize the enclosed area? Let \(x\) be the side perpendicular to the river and express the area as a function of \(x\) alone, then optimize.
Practice Problem 4
Complete Curve Sketch (Sections 05)
Perform a complete curve analysis of:
\[f(x) = (x^2 - 4) \cdot e^{-x}\]
(a) Domain
(b) Zeros (factorization!)
(c) Behavior as \(x \to \pm\infty\) (\(e\)-limits!)
(d) \(f'(x)\) (product rule) and critical points
(e) \(f''(x)\) and inflection points
(f) Sketch
Practice Problem 5
Funktionsscharen (Sections 05)
Work in pairs
For \(f_a(x) = x^3 - 3ax^2 + 4a\) with parameter \(a \in \mathbb{R}\):
(a) Find all extrema as a function of the parameter \(a\).
(b) Find the value of \(a\) for which a local extremum lies on the \(x\)-axis.
(c) Sketch \(f_1(x)\) and \(f_2(x)\) on the same axes.
Guided Practice B
Practice Problem 6
Derivative of Logarithmic Function (Sections 05 & 01)
Work in pairs
For the function \(f(x) = \ln(x^2 + 1)\):
(a) Determine the domain of \(f\).
(b) Find \(f'(x)\) using the chain rule.
(c) Find all critical points and classify them.
(d) Determine \(f''(x)\) and discuss concavity.
(e) Sketch the graph.
Practice Problem 7
Exponential Rate of Change (Sections 05 & 08)
Work in pairs
A bank account grows according to \(A(t) = 5000 \cdot e^{0.04t}\), where \(A\) is in euros and \(t\) is in years.
(a) Find the rate of change \(A'(t)\).
(b) At what rate is the account growing after 10 years?
(c) How long until the account doubles?
(d) Show that the rate of change is proportional to the current balance.
Practice Problem 8
Complete Curve Sketch of Rational Function (Sections 05 & 04)
Perform a complete curve analysis of \(\;f(x) = \dfrac{x}{x^2 + 1}\):
(a) Domain
(b) Symmetry. Hint: \(f(-x) = f(x)\) means symmetry about the \(y\)-axis, \(f(-x) = -f(x)\) means symmetry about the origin.
(c) Zeros and \(y\)-intercept
(d) Asymptotes (vertical? horizontal?)
(e) \(f'(x)\) and critical points (extrema)
(f) \(f''(x)\) and inflection points
Coffee Break
Team Problems A
Practice Problem 9
Business Profit Optimization (Sections 05 & 03)
Think individually (2 min), then work in groups of 3-4
A company’s revenue and cost functions are:
\[R(x) = 1000x \cdot e^{-0.01x}, \qquad C(x) = 500 + 8x\]
where \(x\) is units produced.
(a) Find the profit function \(P(x)\).
(b) Find the production level that maximizes profit.
(c) Confirm with the second derivative test.
(d) Calculate the maximum profit.
Practice Problem 10
Oblique Asymptote & Full Sketch (Sections 05 & 04)
Think individually (2 min), then work in groups of 3-4
\[f(x) = \frac{x^2 + 3x}{x - 1}\]
(a) Perform polynomial long division.
(b) Identify the oblique asymptote and vertical asymptote.
(c) Find zeros and \(y\)-intercept.
(d) Use derivatives to find extrema.
(e) Complete the sketch showing the oblique asymptote.
Practice Problem 11
Tangent Line Challenge (Section 05)
Think individually (2 min), then work in groups of 3-4
Let \(f(x) = 2x^3 + 3x^2 - 36x + 4\). At which points is the tangent to the graph parallel to the line \(y = 36x - 5\)? Give the equation of each of these tangent lines.
Coffee Break
Team Problems B
Practice Problem 12
Cooling Coffee (Sections 05 & 04)
Think individually (2 min), then work in groups of 3-4
A cup of coffee cools according to \(T(t) = 20 + 60e^{-0.1t}\) (temperature in degrees C, time \(t\) in minutes).
(a) What is the initial temperature of the coffee?
(b) What is the room temperature (asymptote)?
(c) Find \(T'(t)\) and the rate of cooling at \(t = 5\) minutes.
(d) When does the coffee reach 40 degrees C?
(e) Show that \(T'(t) = -0.1(T(t) - 20)\), i.e., the rate of cooling is proportional to the temperature difference.
Practice Problem 13
Limits by Algebraic Manipulation (Sections 05)
Think individually (2 min), then work in groups of 3-4
Evaluate the following limits using algebraic techniques (factoring, expanding, rationalizing):
(a) \(\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x^2 - 5x + 6}\) (factor numerator and denominator)
(b) \(\displaystyle\lim_{x \to \infty} \frac{5x^3 - 2x}{3x^3 + x^2 - 1}\) (divide by highest power)
(c) \(\displaystyle\lim_{x \to 0} \frac{e^{2x} - 1}{e^{x} - 1}\) (factor using \(e^{2x}-1 = (e^x-1)(e^x+1)\))
Practice Problem 14
Funktionsschar: Common Points and Tangent Lines (Sections 05)
Think individually (2 min), then work in groups of 3-4
For the family of functions \(f_t(x) = x^2 - 2tx + t^2 + t\) with \(t \in \mathbb{R}\):
(a) Show that \(f_t(x) = (x - t)^2 + t\).
(b) Find the vertex of \(f_t\) as a function of \(t\).
(c) Find the value(s) of \(t\) for which the vertex lies on the \(x\)-axis (i.e., \(f_t\) has a double zero).
(d) Find the tangent line to \(f_1(x)\) at \(x = 3\) and to \(f_2(x)\) at \(x = 3\). Do these tangent lines intersect? If so, where?
Wrap-Up & Next Steps
Key Takeaways
- Differentiation rules (product, chain, quotient) are the core toolkit for curve analysis and optimization
- Complete curve sketches require systematic analysis: domain, symmetry, zeros, asymptotes, derivatives, and inflection points
- Exponential and logarithmic functions appear throughout business applications, so their derivatives and limits are essential
- Funktionsscharen (function families) test your ability to work with parameters and generalize results
- Algebraic manipulation (factoring, rationalizing, long division) is key to evaluating limits and simplifying derivatives