Section 09: Exam Preparation
Work individually for 5 minutes, then discuss with the class (5 minutes)
Evaluate the limit \(\displaystyle\lim_{x \to 0} \frac{e^{2x}-1}{x}\).
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.
\(e\)-Limits and Special Limits (Sections 05)
Work in pairs
Evaluate each limit:
(a) \(\displaystyle\lim_{x \to 0} \frac{e^{3x}-1}{x}\)
(b) \(\displaystyle\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^n\)
(c) \(\displaystyle\lim_{x \to \infty} \frac{3x^2+1}{x^2-x}\)
(d) \(\displaystyle\lim_{x \to 0} \frac{\ln(1+x)}{x}\)
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: \(h(x) = \dfrac{(2x+1)^3}{x-1}\)
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.
Complete Curve Sketch (Sections 05)
Work in pairs
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
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.
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.
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.
Complete Curve Sketch of Rational Function (Sections 05 & 04)
Perform a complete curve analysis of:
\[f(x) = \frac{x}{x^2 + 1}\]
(a) Domain
(b) Symmetry
(c) Zeros and \(y\)-intercept
(d) Asymptotes (vertical? horizontal?)
(e) \(f'(x)\) and critical points (extrema)
(f) \(f''(x)\) and inflection points
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.
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.
Tangent Line Challenge (Sections 02 & 05)
Think individually (2 min), then work in groups of 3-4
Find the equation of all tangent lines to \(f(x) = x^3 - 3x\) that pass through the point \((0, 2)\). A tangent line at \(x = a\) has the form \(y = f'(a)(x - a) + f(a)\). Require that this line passes through \((0, 2)\) and solve for \(a\).
Newton’s Cooling Law (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 \(T'(t) = -0.1(T(t) - 20)\) — the rate of cooling is proportional to the temperature difference.
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 \(e^{2x}-1 = (e^x-1)(e^x+1)\))
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?
Session 09-02 - Differential Calculus Review | Dr. Nikolai Heinrichs & Dr. Tobias Vlćek | Home