AP Calculus AB
review & study guide

1. A limit describes a trend: what value a function approaches as $x$ approaches a number, not necessarily what its value is at that point.
2. $f(a)$ does not need to be defined for the limit as $x$ approaches $a$ to exist.
3. Creating a table and/or observing the graph can be helpful for estimating limits.
4. A limit exists only if both one-sided limits exist and match.

Growth of exponential and logarithmic functions

• Exponential functions ($e^x$) outgrow any polynomial as $x \to \infty$
• Logarithmic functions ($\ln(x)$) grow slower than any power of $x$ as $x \to \infty$
• For limits at infinity:
  • If denominator grows faster (e.g., $e^x$), limit $\to 0$
  • If numerator grows faster as denominator $\to 0$, limit $\to \pm\infty$
  • Compare dominant terms for large $x$

Squeeze theorem

• Used to evaluate limits by bounding $f(x)$ between two functions with known limits
• If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x\to a}g(x) = \lim_{x\to a}h(x) = L$, then $\lim_{x\to a}f(x) = L$
• Especially useful for oscillating functions and trigonometric limits

Special trigonometric limits

• $\lim_{x\to 0} \frac{\sin(x)}{x} = 1$
• $\lim_{x\to 0} \frac{1 - \cos(x)}{x} = 0$
• For $\lim_{x\to 0} \frac{\sin(kx)}{x} = k$
  • Use substitution or multiply/divide to match standard forms

Example strategies and challenge problems

• Use substitution to rewrite limits in standard forms (e.g., $u = kx$)
• Factor denominators to isolate special limit forms
• For limits like $\frac{\sin(ax)}{\sin(bx)}$ as $x \to 0$, answer is $\frac{a}{b}$ by matching to $\frac{\sin(ax)}{ax} \cdot \frac{bx}{\sin(bx)}$

Continuity at a point

• Three conditions: f(a) defined, lim f(x) exists, and lim f(x) = f(a)
• All three must hold; failure of any one means discontinuous at x = a
• Closed interval continuity also requires matching one-sided limits at endpoints

Types of discontinuities

• Removable (hole): limit exists but ≠ f(a), or f(a) undefined
• Jump: one-sided limits both exist but differ; common in piecewise functions
• Infinite: limit = ±∞, indicating a vertical asymptote

Piecewise continuity problems

• Check continuity only at breakpoints (polynomials/rationals are continuous on their own intervals)
• Set one-sided limits equal to each other and to the function value to solve for unknown constants

Intermediate Value Theorem (IVT)

• If f is continuous on [a, b] and L is between f(a) and f(b), then f(c) = L for some c in [a, b]
• Guarantees no "skipping" of values on a continuous interval
• AP requirement: explicitly state continuity on [a, b] and that f(a) < L < f(b) before invoking IVT

Chain rule basics

• Used for differentiating composite functions: $f(g(x))$
• Formula: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$
• Differentiate outer function, multiply by derivative of inner function

Chain rule with common functions

• Power functions: $(ax+b)^n \to n(ax+b)^{n-1} \cdot a$
• Logarithms: $\ln(g(x)) \to \frac{1}{g(x)} \cdot g'(x)$
• Exponentials: $e^{g(x)} \to e^{g(x)} \cdot g'(x)$; $b^{g(x)} \to b^{g(x)}\ln(b) \cdot g'(x)$

Combining chain rule with product and quotient rules

• Identify overall structure (product, quotient, or composition) before applying rules
• Product rule: $(fg)' = f'g + fg'$
  • Use chain rule when differentiating composite factors
• Quotient rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$
  • Use chain rule for composite denominators/numerators

Worked examples

• $(2x + 1)^3 \to 6(2x + 1)^2$
• $\ln(3x^2 + x) \to \frac{6x + 1}{3x^2 + x}$
• $e^{-2x} \to -2e^{-2x}$
• $2^{3x+1} \to (3\ln 2)2^{3x+1}$
• $3x(x^2-1)^3 \to 3(x^2-1)^2(7x^2-1)$
• $(e^x\sin x)^2 \to 2e^{2x}\sin x[\cos x + \sin x]$
• $\frac{x}{2(x^2+1)^2} \to \frac{1-3x^2}{2(x^2+1)^3}$

Chain rule with tables

• Use chain rule formula with given function/derivative values
• Substitute table values for $f$, $g$, $f'$, $g'$ as needed

Challenge problems

• $3\sqrt{2\pi - \pi t} \to -\frac{3\pi}{2\sqrt{2\pi - \pi t}}$
• $\frac{e^{x^2}}{e^{\cos x}} \to \frac{e^{x^2}(2x+\sin x)}{e^{\cos x}}$
• $\sin^3(\ln x) \to \frac{3\sin^2(\ln x)\cos(\ln x)}{x}$

Implicit differentiation

• Used when $y$ is not explicitly defined as a function of $x$ (e.g., $x^2 + y^2 = 25$)
• Signs an equation needs it: $x$ and $y$ multiplied together, $y$ raised to a power, $y$ appearing multiple times

How to find dy/dx

• Differentiate both sides with respect to $x$; multiply every $y$-term by $\frac{dy}{dx}$ (chain rule)
• Collect $\frac{dy}{dx}$ terms, then solve algebraically
• Example: $x^2 + y^2 = 25 \Rightarrow \frac{dy}{dx} = -\frac{x}{y}$

Tangent line at a point

• Substitute the given point after differentiating to find slope
• Use point-slope form: $y - y_1 = m(x - x_1)$

Horizontal vs. vertical tangent lines

• Horizontal: numerator of $\frac{dy}{dx} = 0$, denominator $\neq 0$
• Vertical: denominator of $\frac{dy}{dx} = 0$, numerator $\neq 0$
• Both zero: no tangent line (requires further analysis — possible cusp or node)

Finding tangent line locations

• Set numerator or denominator to zero → get a relationship between $x$ and $y$
• Substitute back into the original equation to find exact points
• Verify the other part (numerator/denominator) is nonzero at each candidate point

Second derivative via implicit differentiation

• Second derivative = derivative of the first derivative; notation: $\dfrac{d^2y}{dx^2}$
• Found by implicitly differentiating twice with respect to $x$
• After second differentiation, substitute the expression for $y'$ to simplify

Key technique: substituting $y'$

• After applying quotient/chain rule a second time, replace $y'$ with the first-derivative expression
• Substitute original equation constraints (e.g., $x^2 + y^2 = 25$) to further simplify result
• Example result for circle: $\dfrac{d^2y}{dx^2} = -\dfrac{x^2+y^2}{y^3} = -\dfrac{25}{y^3}$

Evaluating at a point

• Calculate $y'$ at the given point first, then substitute into the $\dfrac{d^2y}{dx^2}$ expression
• Both $x$, $y$, and $y'$ values must be substituted simultaneously
• Use original curve equation to find unknown constants before differentiating, if needed

Logarithmic Differentiation Overview

• Streamlines differentiation of complex products, quotients, and variable exponents
• Steps: take natural log, expand with log properties, differentiate implicitly, solve for $y'$
• Key log properties:
  • Product: $\log_b(xy) = \log_b(x) + \log_b(y)$
  • Quotient: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
  • Power: $\log_b(x^n) = n\log_b(x)$

Example: $y = \dfrac{(2x+1)^2}{x^3\sqrt{x^2 + 2x}}$

• Log expansion: $2\ln(2x+1) - 3\ln(x) - \frac{1}{2}\ln(x^2+2x)$
• Derivative: $y' = \left(\frac{4}{2x+1} - \frac{3}{x} - \frac{x+1}{x^2+2x}\right) \cdot \dfrac{(2x+1)^2}{x^3\sqrt{x^2 + 2x}}$

Variable in Both Base and Exponent (e.g., $y = x^x$)

• Log expansion: $\ln(y) = x\ln(x)$
• Derivative: $y' = x^x (1 + \ln(x))$

Nested Exponents (e.g., $y = x^{x^{x^2}}$)

• Multiple log steps: $\ln(\ln(y)) = x^2\ln(x) + \ln(\ln(x))$
• Derivative: $y' = \left(x + 2x\ln(x) + \frac{1}{x\ln(x)}\right) \cdot x^{x^{x^2}} \ln(x^{x^{x^2}})$

Functions Already in Logarithmic Form

• Expand using log properties before differentiating
• Example: $R(V) = \ln\left[\left(\frac{V^2+1}{V-2}\right)^3 \sqrt{5x+4}\right]$
  • Expansion: $3\ln(V^2+1) - 3\ln(V-2) + \frac{1}{2}\ln(5x+4)$
  • Derivative: $R'(V) = \frac{6V}{V^2+1} - \frac{3}{V-2} + \frac{5}{10x+8}$

• The slope of the tangent line to $f^{-1}(x)$ at $(a, b)$ is the reciprocal of the slope of the tangent line to $f(x)$ at $(b, a)$.

• $f(g(x)) = x$ (or $g(f(x)) = x$ is a key phrase that signals the topic is about inverse functions.

Inverse trig functions

• Reverse trig functions: input a ratio, output an angle
• Notation: $\arcsin(x) = \sin^{-1}(x)$ ≠ $\csc(x)$ (reciprocal)
• Derivatives found via implicit differentiation + right-triangle simplification

Standard inverse trig derivatives

• $(\sin^{-1}x)' = \dfrac{1}{\sqrt{1-x^2}}$, $(\cos^{-1}x)' = \dfrac{-1}{\sqrt{1-x^2}}$
• $(\tan^{-1}x)' = \dfrac{1}{1+x^2}$, $(\cot^{-1}x)' = \dfrac{-1}{1+x^2}$
• $(\sec^{-1}x)' = \dfrac{1}{|x|\sqrt{x^2-1}}$, $(\csc^{-1}x)' = \dfrac{-1}{|x|\sqrt{x^2-1}}$
  • Co-functions (cos, cot, csc) are always the negative of their counterparts

Chain rule with inverse trig

• Replace every $x$ in the formula with the inner argument
• Multiply by the derivative of the inner argument
• Example: $(\sec^{-1}(x^2+1))' = \dfrac{2x}{(x^2+1)\sqrt{(x^2+1)^2-1}}$

Simplifying nested trig/inverse trig expressions

• Use a right triangle: set inner inverse trig expression equal to $\theta$, label sides, apply Pythagorean theorem
• Two valid approaches: simplify first then differentiate, or differentiate first then simplify with triangle
• Both approaches yield the same result

Mean value theorem (MVT)

• Applies if $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$
• Guarantees $\exists\, c \in (a,b)$ such that $f'(c) = \dfrac{f(b)-f(a)}{b-a}$
• Instantaneous rate equals average rate at some point

MVT examples

• Check continuity and differentiability before applying MVT
• Solve $f'(c) = \dfrac{f(b)-f(a)}{b-a}$ for $c$ in $(a,b)$
• Real-world: If average speed exceeds speed limit, MVT guarantees speed limit was broken at some instant

Rolle's theorem

• Special case of MVT: $f(a) = f(b)$
• Requires continuity on $[a,b]$, differentiability on $(a,b)$, and $f(a) = f(b)$
• Guarantees $\exists\, c \in (a,b)$ such that $f'(c) = 0$
  • At least one horizontal tangent in $(a,b)$

Rolle's theorem examples

• Must check all three conditions (continuity, differentiability, $f(a)=f(b)$)
• If any condition fails, theorem does not apply
• Find $c$ by solving $f'(c) = 0$ in $(a,b)$

Extreme value theorem (EVT)

• If $f(x)$ is continuous on $[a,b]$, $f$ attains absolute maximum and minimum on $[a,b]$
• Extrema occur at:
  • Critical points ($f'(x)=0$ or undefined in $[a,b]$)
  • Endpoints $a$ and $b$
• Procedure:
  • Verify continuity on $[a,b]$
  • Find all critical points in $[a,b]$
  • Evaluate $f(x)$ at critical points and endpoints
  • Largest value = absolute max; smallest = absolute min

EVT examples

• For polynomials: always continuous, so EVT applies
• For rational functions: check for discontinuities in $[a,b]$
• If no critical points, compare only endpoint values

Connecting graphs of $f$, $f'$, and $f''$

• $f'(x)$: slope of $f(x)$; sign indicates increasing/decreasing; zeros/undefined = critical points
• $f''(x)$: concavity of $f(x)$; sign indicates concave up/down; sign changes = inflection points
• Extrema at critical points where $f'(x)$ changes sign; inflection points where $f''(x)$ changes sign

Analyzing $f'(x)$ to determine $f(x)$

• Critical points where $f'(x) = 0$ or undefined
• 1st derivative test: sign change in $f'(x)$ classifies extrema (min/max)
• 2nd derivative test: $f''(x) > 0$ at critical point = min, $f''(x) < 0$ = max

Comparing $f(x)$, $f'(x)$, and $f''(x)$ at points

• Use graph features to estimate values and order of $f(x)$, $f'(x)$, $f''(x)$
• At extrema: $f'(x) = 0$; at inflection: $f''(x) = 0$
• Concavity and slope inform relative sizes

Curve sketching process

• Step 1: Find domain, intercepts, asymptotes
• Step 2: Find critical points using $f'(x)$
• Step 3: Test intervals for increasing/decreasing ($f'(x)$ sign)
• Step 4: Find inflection points using $f''(x)$
• Step 5: Test intervals for concavity ($f''(x)$ sign)
• Step 6: Combine all info to sketch graph

Example: $f(x) = \frac{x}{x^2-1}$

• Vertical asymptotes: $x = 1$, $x = -1$; horizontal asymptote: $y = 0$
• Intercepts: $(0,0)$; domain excludes $x = \pm1$
• $f'(x) = \frac{-x^2-1}{(x^2-1)^2}$: always negative, so $f(x)$ always decreasing
• No real critical points (no relative extrema)
• $f''(x) = \frac{-2x(x^2-1)(-2x^2-1)}{(x^2-1)^4}$; inflection point at $x = 0$
• Concavity changes at $x = 0$; use sign diagrams for $f'(x)$ and $f''(x)$ to guide sketch

Optimization strategy (4 steps)

• Step 1: Identify objective function (quantity to maximize/minimize)
• Step 2: Identify constraint equation(s)
• Steps 3–4: Substitute constraint into objective, then find critical points via $f'(x) = 0$ and classify with 2nd derivative test

Distinguishing optimization from related rates

• Optimization keywords: maximum, minimum, largest, least
• Related rates keywords: increasing/decreasing at a rate, changing with respect to time
• Time-based optimization still possible if asked for when a function is maximized/minimized (not a rate)

Closed interval optimization (EVT)

• Use EVT when problem asks for an absolute extremum on a closed interval $[a, b]$
• Candidates: all critical points + both endpoints
• Evaluate objective function at each candidate; compare to identify absolute max/min

Motion optimization

• Match the objective function to what's being optimized (e.g., max acceleration → differentiate $v(t)$ to get $a(t)$, then optimize $a(t)$)
• If only one critical point exists on the domain, a relative extremum is automatically the absolute extremum
• Always substitute back into the correct function to find the actual max/min value