Course Cheatsheet
Section 05: Differential Calculus
Limits
Limit Notation
Definition: \(\lim_{x \to a} f(x) = L\) means \(f(x)\) approaches \(L\) as \(x\) approaches \(a\)
Key Properties:
- \(\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)\)
- \(\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)\)
- \(\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)\)
- \(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}\) (if denominator \(\neq 0\))
One-Sided Limits
| Notation | Meaning |
|---|---|
| \(\lim_{x \to a^-} f(x)\) | Limit from the left (values less than \(a\)) |
| \(\lim_{x \to a^+} f(x)\) | Limit from the right (values greater than \(a\)) |
Limit exists if and only if: \(\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)\)
Limits at Infinity
For rational functions \(\frac{P(x)}{Q(x)}\):
| Degree Comparison | Result |
|---|---|
| deg(\(P\)) < deg(\(Q\)) | \(\lim_{x \to \infty} = 0\) |
| deg(\(P\)) = deg(\(Q\)) | \(\lim_{x \to \infty} = \frac{a_n}{b_n}\) (ratio of leading coefficients) |
| deg(\(P\)) > deg(\(Q\)) | \(\lim_{x \to \infty} = \pm\infty\) |
Indeterminate Forms and Evaluation Strategies
Forms that require further analysis:
- \(\frac{0}{0}\) – Factor and cancel, rationalize, or use L’Hopital’s rule
- \(\frac{\infty}{\infty}\) – Divide by highest power, or use L’Hopital’s rule
Not indeterminate:
- \(\frac{k}{0}\) where \(k \neq 0\) results in \(\pm\infty\) (vertical asymptote)
- \(\frac{0}{k}\) where \(k \neq 0\) results in \(0\)
If \(\lim_{x \to a} \frac{f(x)}{g(x)}\) gives the indeterminate form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then:
\[\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\]
provided the limit on the right exists. You may apply L’Hopital’s rule repeatedly if the result is still indeterminate.
Continuity
Three Conditions for Continuity at \(x = a\)
- \(f(a)\) exists (function is defined at \(a\))
- \(\lim_{x \to a} f(x)\) exists (limit exists)
- \(\lim_{x \to a} f(x) = f(a)\) (limit equals function value)
Types of Discontinuities
| Type | Description | Example |
|---|---|---|
| Removable | Limit exists but \(\neq f(a)\) or \(f(a)\) undefined | Hole in graph |
| Jump | Left and right limits exist but are different | Step function |
| Infinite | Function approaches \(\pm\infty\) | Vertical asymptote |
If \(f\) is differentiable at \(a\), then \(f\) is continuous at \(a\). The converse is false: continuous functions may not be differentiable (e.g., \(f(x) = |x|\) at \(x = 0\) has a sharp corner).
Derivatives
Definition
Formal Definition: \[f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\]
Alternative Form: \[f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}\]
Notation
| Notation | Read as |
|---|---|
| \(f'(x)\) | “f prime of x” |
| \(\frac{dy}{dx}\) | “dy dx” (Leibniz notation) |
| \(\frac{d}{dx}[f(x)]\) | “derivative of f with respect to x” |
| \(\left.\frac{df}{dx}\right\|_{x=a}\) | Derivative evaluated at \(x = a\) |
Geometric Interpretation
- \(f'(a)\) = slope of tangent line to \(f(x)\) at \(x = a\)
- Tangent line equation: \(y - f(a) = f'(a)(x - a)\)
Economic Interpretation
| Function | Derivative | Interpretation |
|---|---|---|
| \(C(x)\) = Total Cost | \(C'(x)\) = Marginal Cost | Cost of producing one more unit |
| \(R(x)\) = Revenue | \(R'(x)\) = Marginal Revenue | Revenue from selling one more unit |
| \(P(x)\) = Profit | \(P'(x)\) = Marginal Profit | Profit from one more unit |
Differentiation Rules
Basic Rules
| Rule | Formula |
|---|---|
| Constant | \(\frac{d}{dx}[c] = 0\) |
| Power | \(\frac{d}{dx}[x^n] = nx^{n-1}\) |
| Constant Multiple | \(\frac{d}{dx}[cf(x)] = c \cdot f'(x)\) |
| Sum/Difference | \(\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\) |
Product and Quotient Rules
Product Rule: \[\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)\]
Quotient Rule: \[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}\]
“Low d-high minus high d-low, over the square of what’s below”
Chain Rule
For composite functions \(f(g(x))\): \[\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\]
Leibniz notation: If \(y = f(u)\) and \(u = g(x)\): \[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\]
Common Patterns:
| Function | Derivative |
|---|---|
| \((ax + b)^n\) | \(n(ax + b)^{n-1} \cdot a\) |
| \(\sqrt{f(x)}\) | \(\frac{f'(x)}{2\sqrt{f(x)}}\) |
| \(\frac{1}{f(x)}\) | \(\frac{-f'(x)}{[f(x)]^2}\) |
Forgetting to multiply by the inner derivative when applying the chain rule! Always ask: “What’s inside?”
Special Derivatives
Exponential Functions
| Function | Derivative |
|---|---|
| \(e^x\) | \(e^x\) |
| \(e^{g(x)}\) | \(e^{g(x)} \cdot g'(x)\) |
| \(a^x\) (where \(a > 0\)) | \(a^x \cdot \ln(a)\) |
| \(a^{g(x)}\) | \(a^{g(x)} \cdot \ln(a) \cdot g'(x)\) |
Logarithmic Functions
| Function | Derivative |
|---|---|
| \(\ln(x)\) | \(\frac{1}{x}\) |
| \(\ln(g(x))\) | \(\frac{g'(x)}{g(x)}\) |
| \(\log_a(x)\) | \(\frac{1}{x \cdot \ln(a)}\) |
| \(\log_a(g(x))\) | \(\frac{g'(x)}{g(x) \cdot \ln(a)}\) |
The derivative of \(\ln(\text{something})\) is always \(\frac{\text{derivative of something}}{\text{something}}\).
Trigonometric Functions
| Function | Derivative |
|---|---|
| \(\sin(x)\) | \(\cos(x)\) |
| \(\cos(x)\) | \(-\sin(x)\) |
| \(\tan(x)\) | \(\frac{1}{\cos^2(x)}\) |
| \(\sin(g(x))\) | \(\cos(g(x)) \cdot g'(x)\) |
| \(\cos(g(x))\) | \(-\sin(g(x)) \cdot g'(x)\) |
Linear Approximation
Tangent Line as Approximation
For \(x\) near \(a\), the tangent line provides a good estimate: \[f(x) \approx f(a) + f'(a)(x - a)\]
The change in \(f\) when \(x\) changes by a small amount \(\Delta x\) from \(a\): \[\Delta f \approx f'(a) \cdot \Delta x\]
This is the basis of sensitivity analysis in economics. For example, if \(P'(100) = 5\), then increasing production by 1 unit from 100 increases profit by approximately 5 euros.
Implicit Differentiation
When to Use
- Variables are intertwined (cannot easily solve for \(y\))
- Equations like \(x^2 + y^2 = 25\), \(xy = k\), or \(L^{0.6}K^{0.4} = 100\)
Technique
- Differentiate both sides with respect to \(x\)
- Apply chain rule to terms with \(y\): \(\frac{d}{dx}[y^n] = ny^{n-1} \cdot \frac{dy}{dx}\)
- Collect all \(\frac{dy}{dx}\) terms on one side
- Solve for \(\frac{dy}{dx}\)
Example
For \(xy = 5000\): \[\frac{d}{dx}[xy] = \frac{d}{dx}[5000]\] \[y + x\frac{dy}{dx} = 0\] \[\frac{dy}{dx} = -\frac{y}{x}\]
Graphical Calculus
From \(f(x)\) to \(f'(x)\)
| Feature of \(f(x)\) | Feature of \(f'(x)\) |
|---|---|
| Increasing | \(f'(x) > 0\) (positive) |
| Decreasing | \(f'(x) < 0\) (negative) |
| Local maximum | \(f'(x) = 0\) (crosses from + to -) |
| Local minimum | \(f'(x) = 0\) (crosses from - to +) |
| Steep slope | Large \(|f'(x)|\) |
| Horizontal tangent | \(f'(x) = 0\) |
From \(f'(x)\) to \(f(x)\)
| Feature of \(f'(x)\) | Feature of \(f(x)\) |
|---|---|
| \(f'(x) > 0\) | \(f\) is increasing |
| \(f'(x) < 0\) | \(f\) is decreasing |
| \(f'(x) = 0\) | \(f\) has horizontal tangent (possible extremum) |
| \(f'\) increasing | \(f\) is concave up |
| \(f'\) decreasing | \(f\) is concave down |
Critical Points
Definition: \(x = c\) is a critical point if \(f'(c) = 0\) or \(f'(c)\) is undefined
First Derivative Test:
| Sign change of \(f'(x)\) | Type of critical point |
|---|---|
| + to - | Local maximum |
| - to + | Local minimum |
| No sign change | Neither (inflection point) |
Second Derivative
Concavity:
- \(f''(x) > 0\): Concave up (smile shape)
- \(f''(x) < 0\): Concave down (frown shape)
Inflection Point: Where \(f''(x) = 0\) and concavity actually changes
Second Derivative Test:
- If \(f'(c) = 0\) and \(f''(c) > 0\): Local minimum
- If \(f'(c) = 0\) and \(f''(c) < 0\): Local maximum
- If \(f'(c) = 0\) and \(f''(c) = 0\): Test is inconclusive – use the first derivative test
- First derivative test: Always works, requires creating a sign chart
- Second derivative test: Faster when \(f''\) is easy to compute, but can be inconclusive when \(f''(c) = 0\)
Global Extrema on Closed Intervals
Extreme Value Theorem
If \(f\) is continuous on a closed interval \([a, b]\), then \(f\) attains both an absolute maximum and an absolute minimum on \([a, b]\).
Strategy to Find Global Extrema on \([a, b]\)
- Find all critical points \(c\) in \((a, b)\) where \(f'(c) = 0\) or \(f'(c)\) is undefined
- Evaluate \(f\) at all critical points
- Evaluate \(f\) at both endpoints: \(f(a)\) and \(f(b)\)
- Largest value = absolute maximum
- Smallest value = absolute minimum
Finding critical points but forgetting to check endpoints! The global maximum or minimum may occur at the boundary of the interval. Always create a table comparing all candidates.
Curve Sketching Algorithm
Follow these steps systematically for complete, accurate function graphs:
- Domain and Intercepts
- Find where \(f\) is defined
- \(y\)-intercept: compute \(f(0)\)
- \(x\)-intercepts: solve \(f(x) = 0\)
- Critical Points (\(f'(x) = 0\) or DNE)
- Find all critical points
- Classify using first or second derivative test
- Inflection Points (\(f''(x) = 0\) or DNE)
- Find where concavity changes
- Sign Charts
- Sign chart for \(f'(x)\): increasing/decreasing intervals
- Sign chart for \(f''(x)\): concave up/down intervals
- Asymptotic Behavior
- Vertical asymptotes: where denominator = 0
- Horizontal asymptotes: \(\lim_{x \to \pm\infty} f(x)\)
- End behavior for polynomials
- Complete Sketch
- Plot all key points (intercepts, extrema, inflection points)
- Connect using information from sign charts
Example: Sketching \(g(x) = x^3 - 3x^2\)
| Step | Calculation | Result |
|---|---|---|
| Domain | All reals | \((-\infty, \infty)\) |
| \(y\)-intercept | \(g(0) = 0\) | \((0, 0)\) |
| \(x\)-intercepts | \(x^2(x - 3) = 0\) | \(x = 0, x = 3\) |
| \(g'(x) = 0\) | \(3x^2 - 6x = 3x(x-2) = 0\) | \(x = 0, x = 2\) |
| \(g''(x) = 0\) | \(6x - 6 = 0\) | \(x = 1\) (inflection) |
| \(g''(0) = -6\) | Concave down | Local max at \((0, 0)\) |
| \(g''(2) = 6\) | Concave up | Local min at \((2, -4)\) |
| End behavior | \(x^3\) dominates | \(-\infty\) as \(x \to -\infty\), \(+\infty\) as \(x \to +\infty\) |
Function Determination from Conditions
General Approach
- Choose the function form (e.g., \(f(x) = ax^3 + bx^2 + cx + d\))
- Count unknowns (number of coefficients)
- Identify conditions – you need as many conditions as unknowns
- Set up equations from conditions
- Solve the system
- Verify the solution
Types of Conditions and Their Equations
| Condition Type | What it gives | Number of equations |
|---|---|---|
| Point \((a, b)\) | \(f(a) = b\) | 1 |
| Slope at a point | \(f'(a) = m\) | 1 |
| Extremum at \((a, b)\) | \(f(a) = b\) and \(f'(a) = 0\) | 2 |
| Inflection at \((a, b)\) | \(f(a) = b\) and \(f''(a) = 0\) | 2 |
| Horizontal tangent at \(x = a\) | \(f'(a) = 0\) | 1 |
An extremum at \((a, b)\) provides both a point condition (\(f(a) = b\)) and a derivative condition (\(f'(a) = 0\)). Do not forget the derivative condition!
Funktionsscharen (Function Families)
Definition
A Funktionenschar is a family of functions depending on a parameter (usually \(t\), \(a\), or \(k\)):
- Notation: \(f_t(x)\) or \(f(x, t)\)
- Example: \(f_t(x) = x^2 - tx + 1\)
Common Problem Types
| Question | How to solve |
|---|---|
| For which \(t\) does \(f_t\) have exactly one zero? | Set discriminant \(\Delta = 0\) |
| For which \(t\) does \(f_t\) have two zeros? | Set discriminant \(\Delta > 0\) |
| For which \(t\) does \(f_t\) have an extremum at \(x = a\)? | Solve \(f_t'(a) = 0\) for \(t\) |
| For which \(t\) is \(f_t(a) = c\)? | Substitute and solve for \(t\) |
| For which \(t\) does \(f_t\) have an inflection at \(x = a\)? | Solve \(f_t''(a) = 0\) for \(t\) |
Strategy
- Identify the condition (zeros, extrema, inflection points, function values)
- Set up the equation involving the parameter
- Substitute the given point (if specified)
- Solve for the parameter \(t\)
- Verify the answer makes sense
These are heavily tested! Practice setting up equations from verbal conditions. Remember to differentiate with respect to \(x\) while treating the parameter \(t\) as a constant.
Business Applications
Marginal Analysis
Key Relationships:
- Profit is maximized when \(P'(x) = 0\), equivalently when \(R'(x) = C'(x)\)
- Produce more if \(R'(x) > C'(x)\) (marginal revenue exceeds marginal cost)
- Produce less if \(R'(x) < C'(x)\)
Optimization Strategy
- Find the function to optimize (profit, cost, revenue)
- Take the derivative and set equal to zero
- Solve for critical points
- Test using second derivative or first derivative test
- Check endpoints if domain is restricted
- Verify the answer makes business sense
Average Cost
Average Cost Function: \(AC(x) = \frac{C(x)}{x}\)
Minimum average cost occurs when \(AC'(x) = 0\), which happens when \(AC(x) = C'(x)\) (average cost equals marginal cost).
Quick Reference: Derivative Rules
| Function | Derivative |
|---|---|
| \(c\) (constant) | \(0\) |
| \(x^n\) | \(nx^{n-1}\) |
| \(\sqrt{x} = x^{1/2}\) | \(\frac{1}{2\sqrt{x}}\) |
| \(\frac{1}{x} = x^{-1}\) | \(-\frac{1}{x^2}\) |
| \(\frac{1}{x^n} = x^{-n}\) | \(-\frac{n}{x^{n+1}}\) |
| \(e^x\) | \(e^x\) |
| \(a^x\) | \(a^x \cdot \ln(a)\) |
| \(\ln(x)\) | \(\frac{1}{x}\) |
| \(\log_a(x)\) | \(\frac{1}{x \cdot \ln(a)}\) |
| \(\sin(x)\) | \(\cos(x)\) |
| \(\cos(x)\) | \(-\sin(x)\) |
| \(\tan(x)\) | \(\frac{1}{\cos^2(x)}\) |
Problem-Solving Strategies
Finding Derivatives
- Simplify first if possible (rewrite radicals as powers, expand simple products)
- Identify which rule(s) apply (power, product, quotient, chain)
- Work systematically – do not skip steps
- Simplify the result
Optimization Problems
- Draw a picture if applicable
- Define variables and write the objective function
- Express in terms of one variable using constraints
- Differentiate and find critical points
- Check endpoints if domain is restricted
- Interpret the result in context
Curve Sketching Exam Strategy
- Always start with the domain – know where the function exists
- Compute \(f'\) and \(f''\) before anything else – you will need both
- Create organized sign charts for \(f'\) and \(f''\)
- Label all key points in your sketch with coordinates
- Check your sketch against end behavior – does the overall shape make sense?
- Forgetting the chain rule inner derivative
- Using the power rule on products: \((fg)' \neq f' \cdot g'\)
- Confusing the quotient rule sign: numerator is \(f'g - fg'\), not \(fg' - f'g\)
- Forgetting that \(\frac{dy}{dx}\) appears when differentiating \(y\) terms implicitly
- Not checking if critical points are maxima or minima (always verify!)
- Forgetting to check endpoints when finding global extrema on \([a,b]\)
- Concluding that \(f''(c) = 0\) means inflection point without checking sign change
- Forgetting units in applied problems
- In Funktionsscharen: differentiating with respect to \(t\) instead of \(x\)