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
Forms that require further analysis:
- \(\frac{0}{0}\) — Factor and cancel, or use L’Hôpital’s rule
- \(\frac{\infty}{\infty}\) — Divide by highest power, or use L’Hôpital’s rule
Not indeterminate:
- \(\frac{k}{0}\) where \(k \neq 0\) → \(\pm\infty\) (vertical asymptote)
- \(\frac{0}{k}\) where \(k \neq 0\) → \(0\)
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 you can draw the graph without lifting your pen, the function is continuous at that point.
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! Always ask: “What’s inside?”
Implicit Differentiation
When to Use
- Variables are intertwined (can’t 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) ∪
- \(f''(x) < 0\): Concave down (frown) ∩
Inflection Point: Where \(f''(x) = 0\) and concavity 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
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 endpoints
- 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) | \(0\) |
| \(x^n\) | \(nx^{n-1}\) |
| \(\sqrt{x} = x^{1/2}\) | \(\frac{1}{2\sqrt{x}}\) |
| \(\frac{1}{x} = x^{-1}\) | \(-\frac{1}{x^2}\) |
| \(\frac{1}{x^n} = x^{-n}\) | \(-\frac{n}{x^{n+1}}\) |
| \(e^x\) | \(e^x\) |
| \(\ln(x)\) | \(\frac{1}{x}\) |
Problem-Solving Strategies
Finding Derivatives
- Simplify first if possible (rewrite radicals as powers)
- Identify which rule(s) apply (power, product, quotient, chain)
- Work systematically — don’t 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
Common Mistakes to Avoid
- Forgetting the chain rule inner derivative
- Using power rule on products: \((fg)' \neq f' \cdot g'\)
- Forgetting that \(\frac{dy}{dx}\) appears when differentiating \(y\) terms implicitly
- Not checking if critical points are maxima or minima
- Forgetting units in applied problems