AP Calculus AB/BC Limit, Derivative, and Integral Exhaustive Study Guide

Limits of Functions and Continuity

• Limit of a Continuous Function   - If $f(x)$ is a continuous function for all real numbers, then:   - $\lim_{x o c} f(x) = f(c)$

• Limits of Rational Functions   - A. Given a rational function $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ share no common factors and $c$ is a real number such that $q(c) = 0$:     - I. $\lim_{x o c} f(x)$ does not exist (DNE)     - II. $\lim_{x o c} f(x) = \pm\infty$     - III. $x = c$ is a vertical asymptote   - B. Given a rational function $f(x) = \frac{p(x)}{q(x)}$, such that reducing a common factor between $p(x)$ and $q(x)$ results in the agreeable function $k(x)$, then:     - $\lim_{x o c} f(x) = \lim_{x o c} \frac{p(x)}{q(x)} = \lim_{x o c} k(x) = k(c)$     - This result indicates a Hole at the point $(c, k(c))$

• Limits of a Function as x Approaches Infinity   - For a rational function defined by $f(x) = \frac{p(x)}{q(x)}$, where both are polynomial functions:     - A. If the degree of p(x) > q(x), then $\lim_{x o \infty} f(x) = \infty$     - B. If the degree of p(x) < q(x), then $\lim_{x o \infty} f(x) = 0$, and $y = 0$ is a horizontal asymptote.     - C. If the degree of $p(x) = q(x)$, then $\lim_{x o \infty} f(x) = c$, where $c$ is the ratio of the leading coefficients; $y = c$ is a horizontal asymptote.

• Special Trig Limits   - A. $\lim_{x o 0} \frac{\sin(ax)}{ax} = 1$   - B. $\lim_{x o 0} \frac{ax}{\sin(ax)} = 1$   - C. $\lim_{x o 0} \frac{1 - \cos(ax)}{ax} = 0$

• L'Hospital's Rule   - If $\lim_{x o c} f(x)$ or $\lim_{x o \infty} f(x)$ results in an indeterminate form ($0/0$, $\infty/\infty$, $0 \times \infty$, $0^0$, $1^{\infty}$, $\infty^0$) and $f(x) = \frac{p(x)}{q(x)}$, then:   - $\lim_{x o c} \frac{p(x)}{q(x)} = \lim_{x o c} \frac{p'(x)}{q'(x)}$   - $\lim_{x o \infty} \frac{p(x)}{q(x)} = \lim_{x o \infty} \frac{p'(x)}{q'(x)}$

• Definition of Continuity and Types of Discontinuities   - Definition of Continuity: A function $f(x)$ is continuous at $c$ if:     - I. $\lim_{x o c} f(x)$ exists     - II. $f(c)$ exists     - III. $\lim_{x o c} f(x) = f(c)$   - Removable Discontinuities (Holes):     - I. $\lim_{x o c} f(x) = L$ (the limit exists)     - II. $f(c)$ is undefined   - Non-Removable Discontinuities:     - Jumps: $\lim_{x o c} f(x) = \text{DNE}$ because $\lim_{x o c^-} f(x) \neq \lim_{x o c^+} f(x)$     - Asymptotes (Infinite Discontinuities): $\lim_{x o c} f(x) = \pm\infty$

Derivatives: Definitions and Basic Rules

• Intermediate Value Theorem (IVT)   - If $f$ is a continuous function on the closed interval $[a, b]$ and $k$ is any number between $f(a)$ and $f(b)$, then there exists at least one value of $c$ on $[a, b]$ such that $f(c) = k$.   - On a continuous function, if f(a) < f(b), any y-value greater than $f(a)$ and less than $f(b)$ is guaranteed to exist on the function $f$.

• Average Rate of Change   - The average rate of change, $m$, of a function $f$ on the interval $[a, b]$ is given by the slope of the secant line:   - $m = \frac{f(b) - f(a)}{b - a}$

• Definition of the Derivative   - The derivative, or instantaneous rate of change, converts the slope of the secant line to the slope of a tangent line by letting the change in $x$ ($\Delta x$ or $h$) approach zero:   - $f'(x) = \lim_{h o 0} \frac{f(x+h) - f(x)}{h}$   - Alternate Definition: $f'(c) = \lim_{x o c} \frac{f(x) - f(c)}{x - c}$

• Differentiability and Continuity Properties   - A. If $f(x)$ is differentiable at $x = c$, then $f(x)$ is continuous at $x = c$.   - B. If $f(x)$ is not continuous at $x = c$, then $f(x)$ is not differentiable at $x = c$.   - C. The graph of $f$ is continuous, but not differentiable at $x = c$ if:     - I. The graph has a cusp or sharp point at $x = c$     - II. The graph has a vertical tangent line at $x = c$     - III. The graph has an endpoint at $x = c$

• Basic Derivative Rules   - Considering $c$ as a constant:   - 1. Constant Rule: $\frac{d}{dx}[c] = 0$   - 2. Constant Multiple Rule: $\frac{d}{dx}[cf(x)] = cf'(x)$   - 3. Sum Rule: $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$   - 4. Difference Rule: $\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$   - 5. Product Rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$   - 6. Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$   - 7. Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$

• Derivatives of Trig and Inverse Trig Functions   - Trigonometric Functions:     - 1. $\frac{d}{dx}[\sin(x)] = \cos(x)$     - 2. $\frac{d}{dx}[\cos(x)] = -\sin(x)$     - 3. $\frac{d}{dx}[\tan(x)] = \sec^2(x)$     - 4. $\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x)$     - 5. $\frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)$     - 6. $\frac{d}{dx}[\cot(x)] = -\csc^2(x)$   - Inverse Trigonometric Functions:     - 1. $\frac{d}{dx}[\sin^{-1}(x)] = \frac{1}{\sqrt{1 - x^2}}$     - 2. $\frac{d}{dx}[\cos^{-1}(x)] = \frac{-1}{\sqrt{1 - x^2}}$     - 3. $\frac{d}{dx}[\tan^{-1}(x)] = \frac{1}{1 + x^2}$     - 4. $\frac{d}{dx}[\sec^{-1}(x)] = \frac{1}{|x|\sqrt{x^2 - 1}}$     - 5. $\frac{d}{dx}[\csc^{-1}(x)] = \frac{-1}{|x|\sqrt{x^2 - 1}}$     - 6. $\frac{d}{dx}[\cot^{-1}(x)] = \frac{-1}{1 + x^2}$

Advanced Differentiation and Theorems

• Derivatives of Exponential and Logarithmic Functions   - 1. $\lim_{x o \infty} (1+x) = e$ (Note: likely transcript notation for $\lim_{n \to \infty} (1 + 1/n)^n = e$)   - 2. $\frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)}$ (for a > 0 and $a \neq 1$)   - 3. $\frac{d}{dx}[e^x] = e^x$   - 4. $\frac{d}{dx}[\ln|x|] = \frac{1}{x}$   - 5. $\frac{d}{dx}[\log_a|x|] = \frac{1}{x \ln(a)}$   - 6. $\frac{d}{dx}[e^u] = e^u \cdot u'$   - 7. $\frac{d}{dx}[a^x] = a^x \ln(a)$

• Explicit and Implicit Differentiation   - Explicit Functions: Function $y$ is written only in terms of $x$ ($y = f(x)$). Derivatives rules apply normally.   - Implicit Differentiation: Expression involving both $x$ and $y$.     - I. Differentiate both sides with respect to $x$. Differentiate terms with $x$ normally; multiply terms with $y$ by $\frac{dy}{dx}$ per the chain rule.     - II. Group all $\frac{dy}{dx}$ terms on one side.     - III. Factor $\frac{dy}{dx}$ and solve in terms of $x$ and $y$.

• Tangent Lines and Normal Lines   - Equation of the tangent line at point $(a, f(a))$: $y - f(a) = f'(a)(x - a)$   - Equation of the normal line at point $(a, f(a))$: $y - f(a) = -\frac{1}{f'(a)}(x - a)$

• Mean Value Theorem (MVT) for Derivatives   - If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists at least one $c \in (a, b)$ such that:   - $f'(c) = \frac{f(b) - f(a)}{b - a}$   - Geometrically: The slope of the tangent line at some point is equal to the slope of the secant line.

• Rolle's Theorem   - A special case of MVT. If $f$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a, b)$ such that:   - $f'(c) = 0$

• Particle Motion   - Position: $x(t)$   - Velocity: $v(t) = x'(t)$   - Speed: $|v(t)|$   - Acceleration: $a(t) = v'(t) = x''(t)$   - Rules:     - A. If v(t) > 0, the particle moves right or up. If v(t) < 0, it moves left or down.     - B. If $v(t)$ and $a(t)$ have the same sign, speed is increasing. If they have opposite signs, speed is decreasing.     - C. If $v(t) = 0$ and the sign of $v(t)$ changes, the particle changes direction.

• Related Rates   - A. Identify known variables and rates of change ($\frac{d\text{variable}}{dt}$). Construct a relating equation.   - B. Implicitly differentiate both sides with respect to time ($t$). Do not substitute changing variable values before differentiating unless the value is constant.   - C. Substitute known values and solve for the required rate.   - Note: Often uses Pythagorean Theorem, geometric shapes, or similar triangles.

Applications of Derivatives

• Extrema of a Function   - Absolute Extrema: The highest/lowest y-value on a given interval or domain.   - Relative Extrema:     - Relative Maximum: Where the function changes from increasing to decreasing (or $f'$ changes from positive to negative).     - Relative Minimum: Where the function changes from decreasing to increasing (or $f'$ changes from negative to positive).   - Critical Value: Values of $x$ where $f(c)$ is defined and $f'(c) = 0$ or $f'(c)$ is undefined.

• Extreme Value Theorem (EVT)   - If $f$ is continuous on $[a, b]$, absolute extrema occur at either the endpoints or the critical values.   - Identify extrema by creating a table comparing y-values at endpoints and critical values.

• Increasing/Decreasing and the First Derivative Test   - Increasing: f'(x) > 0 (tangent has positive slope).   - Decreasing: f'(x) < 0 (tangent has negative slope).   - Constant: $f'(x) = 0$ (tangent is horizontal).   - First Derivative Test: Use a sign chart for $f'$ involving discontinuities and critical values:     - Sign change of $f'$ from $-$ to $+$ at $x=c \implies$ relative minimum.     - Sign change of $f'$ from $+$ to $-$ at $x=c \implies$ relative maximum.     - No sign change $\implies$ shelf point.

• Concavity and the Second Derivative Test   - Concave Up: f''(x) > 0 ($f'(x)$ is increasing).   - Concave Down: f''(x) < 0 ($f'(x)$ is decreasing).   - Second Derivative Test (for continuous $f(x)$ at $c$):     - If $f'(c) = 0$ and f''(c) > 0 \implies relative minimum.     - If $f'(c) = 0$ and f''(c) < 0 \implies relative maximum.     - If $f'(c) = 0$ and $f''(c) = 0 \implies$ test is inconclusive; use First Derivative Test.

• Point of Inflection   - If $f$ is continuous at $x=c$ and either $f''(c) = 0$ or $f''(c)$ is undefined, a sign change in $f''(x)$ at $x=c$ indicates a point of inflection.

• Optimization   - A. Define primary equation (variable to maximize/minimize) and feasible domain.   - B. Use a secondary equation to relate variables and substitute so the primary equation has one variable.   - C. Take the derivative and find critical values.   - D. Check endpoints and critical values for the optimal solution.

• Derivative of an Inverse   - If $f$ and its inverse $g$ ($f^{-1}$) are differentiable, and $(c, f(c))$ exists on $f$ (meaning $(f(c), c)$ exists on $g$):   - $\frac{d}{dx}[g(x)] = \frac{1}{f'(f^{-1}(x))} = \frac{1}{f'(g(x))}$

Integration and Antiderivatives

• Antiderivatives   - If $F'(x) = f(x)$, then $\int f(x) \, dx = F(x) + C$ (the Indefinite Integral).

• Basic Integration Rules (where $C$ is the constant of integration):   - 1. $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$)   - 2. $\int k \, dx = kx + C$   - 3. $\int \sin(x) \, dx = -\cos(x) + C$   - 4. $\int \cos(x) \, dx = \sin(x) + C$   - 5. $\int \sec^2(x) \, dx = \tan(x) + C$   - 6. $\int \sec(x)\tan(x) \, dx = \sec(x) + C$   - 7. $\int \csc^2(x) \, dx = -\cot(x) + C$   - 8. $\int \csc(x)\cot(x) \, dx = -\csc(x) + C$

• Definite Integrals and FTC   - The First Fundamental Theorem of Calculus: If $F(x)$ is the antiderivative of continuous $f(x)$:     - $\int_a^b f(x) \, dx = F(b) - F(a)$   - Properties of Definite Integrals:     - 1. $\int_a^a f(x) \, dx = 0$     - 2. $\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$     - 3. $\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$     - 4. $\int_{-a}^a f(x) \, dx = 2 \int_0^a f(x) \, dx$ if function is even.     - 5. $\int_{-a}^a f(x) \, dx = 0$ if function is odd.

• Riemann Sum (Approximations)   - Dividing the interval $[a, b]$ into $n$ subintervals, each width is $\Delta x = \frac{b-a}{n}$.   - A. Right Riemann Sum: $\text{Area} \approx \Delta x [f(x_1) + f(x_2) + … + f(x_n)]$   - B. Left Riemann Sum: $\text{Area} \approx \Delta x [f(x_0) + f(x_1) + … + f(x_{n-1})]$   - C. Midpoint Riemann Sum: $\text{Area} \approx \Delta x [f(x_{1/2}) + f(x_{3/2}) + …]$   - D. Trapezoidal Sum: $\text{Area} = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + … + 2f(x_{n-1}) + f(x_n)]$   - Approximation Properties:     - Under-approximation occurs when:       - I. Left sum on increasing function.       - II. Right sum on decreasing function.       - III. Trapezoidal sum on concave down function.     - Over-approximation occurs when:       - I. Left sum on decreasing function.       - II. Right sum on increasing function.       - III. Trapezoidal sum on concave up function.

• Riemann Sum (Limit Definition of Area)   - For continuous $f$ on $[a, b]$:   - $\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^n f(c_i) \Delta x$   - Where $c_i$ is either left endpoint ($a + (i-1)\Delta x$) or right endpoint ($a + i\Delta x$).

• Average Value of a Function   - On the interval $[a, b]$: $\text{Average Value} = \frac{1}{b-a} \int_a^b f(x) \, dx$

• Second Fundamental Theorem of Calculus   - A. $\frac{d}{dx} \left[ \int_a^x f(t) \, dt \right] = f(x)$   - B. $\frac{d}{dx} \left[ \int_x^b f(t) \, dt \right] = -f(x)$   - C. $\frac{d}{dx} \left[ \int_a^{u(x)} f(t) \, dt \right] = f(u(x)) \cdot u'(x)$

• Integration of Exponential and Logarithmic Formulas   - 1. $\int \frac{1}{x} \, dx = \ln|x| + C$   - 2. $\int \frac{1}{u} \, du = \ln|u| + C$   - 3. $\int \frac{1}{x-a} \, dx = \ln|x-a| + C$   - 4. $\int a^x \, dx = \frac{a^x}{\ln(a)} + C$   - 5. $\int e^x \, dx = e^x + C$   - 6. $\int a^u \, du = \frac{a^u}{\ln(a) \cdot u'} + C$ (Note: usually requires substitution)   - 7. $\int e^u \, du = e^u + C$

• Integration of Trig and Inverse Trig   - 1. $\int \cos(x) \, dx = \sin(x) + C$   - 2. $\int \sec^2(x) \, dx = \tan(x) + C$   - 3. $\int \sec(x)\tan(x) \, dx = \sec(x) + C$   - 4. $\int \sin(u) \, du = -\cos(u) + C$   - 5. $\int \csc^2(u) \, du = -\cot(u) + C$   - 6. $\int \csc(u)\cot(u) \, du = -\csc(u) + C$   - 7. $\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(x) + C$   - 8. $\int \frac{-1}{\sqrt{1-x^2}} \, dx = \arccos(x) + C$   - 9. $\int \frac{1}{1+x^2} \, dx = \arctan(x) + C$   - 10. $\int \frac{-1}{1+x^2} \, dx = \text{arccot}(x) + C$   - 11. $\int \frac{1}{x\sqrt{x^2-1}} \, dx = \text{arcsec}(x) + C$   - 12. $\int \frac{-1}{x\sqrt{x^2-1}} \, dx = \text{arccsc}(x) + C$

Differential Equations and Slope Fields

• Exponential Growth and Decay   - A. Differential Equation: $\frac{dy}{dt} = ky$   - B. General Solution: $y = Ce^{kt}$     - I. k > 0 \implies growth.     - II. k < 0 \implies decay.

• Solving Differential Equations   - Use Separation of Variables:     - 1. Separate variables ($y$ terms with $dy$, $x$ terms with $dx$).     - 2. Integrate both sides.     - 3. Solve for $y$ to find the General Solution.     - 4. For a Particular Solution, use the initial condition to solve for $C$.

• Slope Fields   - A graphical representation where the derivative $\frac{dy}{dx}$ provides the slope at each point $(x, y)$. Sketch small segments of the tangent lines to visualize potential solutions.

Applications of Integration

• Area Between Two Curves   - A. Top vs Bottom: $\text{Area} = \int_a^b [f(x) - g(x)] \, dx$   - B. Right vs Left: $\text{Area} = \int_a^b [f(y) - g(y)] \, dy$

• Volumes of Solids of Revolution: Disk Method   - Region borders the axis of revolution on $[a, b]$:   - A. Around x-axis: $V = \pi \int_a^b (f(x))^2 \, dx$   - B. Around y-axis: $V = \pi \int_a^b (f(y))^2 \, dy$   - C. Around horizontal line $y=k$: $V = \pi \int_a^b (f(x) - k)^2 \, dx$   - D. Around vertical line $x=m$: $V = \pi \int_a^b (f(y) - m)^2 \, dy$

• Volumes of Solids of Revolution: Washer Method   - Region has space between it and the axis of revolution:   - A. Around x-axis: $V = \pi \int_a^b ([f(x)]^2 - [g(x)]^2) \, dx$   - B. Around y-axis: $V = \pi \int_a^b ([f(y)]^2 - [g(y)]^2) \, dy$   - C. Around line $y=k$: $V = \pi \int_a^b ([f(x) - k]^2 - [g(x) - k]^2) \, dx$   - D. Around line $x=m$: $V = \pi \int_a^b ([f(y) - m]^2 - [g(y) - m]^2) \, dy$

• Volumes of Known Cross Sections   - Integrates area formulas $A(x)$ perpendicular to the axis:   - I. Squares: $V = \int_a^b [f(x)]^2 \, dx$   - II. Equilateral Triangles: $V = \frac{\sqrt{3}}{4} \int_a^b [f(x)]^2 \, dx$   - III. Isosceles Right Triangles (leg in base): $V = \frac{1}{2} \int_a^b [f(x)]^2 \, dx$   - IV. Isosceles Right Triangles (hypotenuse in base): $V = \frac{1}{4} \int_a^b [f(x)]^2 \, dx$   - V. Semicircles (diameter in base): $V = \frac{\pi}{8} \int_a^b [f(x)]^2 \, dx$   - VI. Semicircles (radius in base): $V = \frac{\pi}{2} \int_a^b [f(x)]^2 \, dx$