Search & glossary

Every rule and concept in Calculus, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (204)

$p$ -Series: $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{p}}$ converges if $p>1$ and diverges if $p\le 1$ .; Infinite Sequences and Series › The Integral Test & $p$ -Series
$u$ -substitution, average value, net change: $u$ -sub reverses the chain rule: let $u=g(x)$ , $du=g'(x)\,dx$ , and change the limits for a definite integral.; Course Review › Integration & the Fundamental Theorem
1. A proof-flavored MVT application: Claim: If $f'(x)=0$ for every $x$ in an interval $I$ , then $f$ is constant on $I$ .; Analytical Applications of Differentiation › Going Deeper: Advanced Analysis
1. Derivative of $\sqrt{x}$ from the definition: Multiply by the conjugate to clear the root: \[ f'(x)=_h→0 $\sqrt{x+h}$ - $\sqrt{x}$ h· $\sqrt{x+h}$ + $\sqrt{x}$ $\sqrt{x+h}$ + $\sqrt{x}$ =_h→0(x+h)-xh( $\sqrt{x+h}$ + $\sqrt{x}$ ) =_h→01 $\sqrt{x+h}$ + $\sqrt{x}$ =…; The Derivative › Going Deeper: Advanced Derivative Ideas
1. Newton's Method (root-finding by tangent lines): To approximate a root of $f(x)=0$ , start with a guess $x_0$ and iterate $ $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.$ $Each step follows the tangent line at$ x_n $down to the$ x$-axis.; Contextual Applications of Differentiation › Going Deeper: Advanced Applications
1. Reduction formulas: Repeated integration by parts produces a formula that lowers a power by a fixed step.; Techniques of Integration › Going Deeper: Advanced Techniques
1. Three-layer nested chain rule: For a triple composition, multiply one derivative factor per layer, outside to inside.; Chain, Implicit and Inverse Differentiation › Going Deeper: Advanced Techniques
2. Accuracy of the linear approximation: The error $E(x)=f(x)-L(x)$ of a tangent-line estimate is controlled by concavity: its sign is opposite the estimate's bias (concave up $\Rightarrow L$ under $\Rightarrow E>0$ ).; Contextual Applications of Differentiation › Going Deeper: Advanced Applications
2. Derivative of $\dfrac{1}{x}$ from the definition: Combine the fractions in the numerator first: \[ f'(x)=_h→0 $\frac{1}{x+h}$ - $\frac{1}{x}$ h =_h→0 $\frac{x-(x+h)}{x(x+h)}$ h =_h→0 $\frac{-h}{h\,x(x+h)}$ =_h→0 $\frac{-1}{x(x+h)}$ =- $\frac{1}{x^2}$ .; The Derivative › Going Deeper: Advanced Derivative Ideas
3. Indeterminate powers via logarithms: For $1^\infty,\ 0^0,\ \infty^0$ , let $y$ equal the expression and take $\ln$ .; Contextual Applications of Differentiation › Going Deeper: Advanced Applications
4. Multi-step related rates: draining cone: Water drains from an inverted cone (height $12$ ft, top radius $6$ ft) at $\dfrac{dV}{dt}=-4$ ft $^3$ /min.; Contextual Applications of Differentiation › Going Deeper: Advanced Applications
4. One-sided derivatives and a piecewise function: The one-sided derivatives at $a$ are \[ f'_-(a)=_h→0^- $\frac{f(a+h)-f(a)}{h}$ , f'_+(a)=_h→0^+ $\frac{f(a+h)-f(a)}{h}$ .; The Derivative › Going Deeper: Advanced Derivative Ideas
4. The comparison test for improper integrals: If $0\le f(x)\le g(x)$ for all $x\ge a$ , then: \[ ∫_a^∞g dx converges ⇒ ∫_a^∞f dx converges; ∫_a^∞f dx diverges ⇒ ∫_a^∞g dx diverges.; Techniques of Integration › Going Deeper: Advanced Techniques
5. A related-rates preview: Implicit differentiation with respect to time $t$ is exactly the engine behind related rates.; Chain, Implicit and Inverse Differentiation › Going Deeper: Advanced Techniques
5. Proving the Product Rule: Add and subtract $f(x+h)g(x)$ in the numerator:; The Derivative › Going Deeper: Advanced Derivative Ideas
5. Related rates with an angle: A balloon rises straight up at $3$ m/s.; Contextual Applications of Differentiation › Going Deeper: Advanced Applications
5. Why the Candidates' Test works: The EVT guarantees a continuous $f$ attains its max and min on $[a,b]$ .; Analytical Applications of Differentiation › Going Deeper: Advanced Analysis
6. Proving the Quotient Rule: Let $Q=\dfrac{f}{g}$ , so $f=Qg$ .; The Derivative › Going Deeper: Advanced Derivative Ideas
6. Rectilinear motion: the sign-chart method: For a full motion analysis on $[a,b]$ :; Contextual Applications of Differentiation › Going Deeper: Advanced Applications
7. Completing the square before substituting: When a quadratic has no rational roots, complete the square to expose a standard form.; Techniques of Integration › Going Deeper: Advanced Techniques
7. Generalized power rule and $e^{g\ln f}$: Any power with a constant exponent still uses the chain rule: $\dfrac{d}{dx}[g(x)]^n=n[g(x)]^{n-1}g'(x)$ .; Chain, Implicit and Inverse Differentiation › Going Deeper: Advanced Techniques
7. The symmetric difference quotient: Calculators often estimate $f'(x)$ with the symmetric difference quotient \[ f'(x)≈ $\frac{f(x+h)-f(x-h)}{2h}$ .; The Derivative › Going Deeper: Advanced Derivative Ideas
A decision checklist: Infinite Sequences and Series › Which Convergence Test Should I Use?
A reliable setup: Analytical Applications of Differentiation › Optimization (Applied Max/Min)
Absolute vs. conditional convergence: $\sum a_n$ converges absolutely if $\sum|a_n|$ converges (this forces $\sum a_n$ to converge).; Infinite Sequences and Series › Alternating Series, Error Bound & Absolute vs. Conditional Convergence
Accumulation & net change: If $F'(t)=r(t)$ is a rate of change, the net change of $F$ over $[a,b]$ is \[ F(b)-F(a)=∫_a^b r(t) dt.; Applications of Integration › Average Value, Accumulation & Net Change
Adding up tiny pieces: An integral $\int_a^b f(x)\,dx$ adds up infinitely many thin slices to get a total — most often the area under the curve between $a$ and $b$ .; Core Ideas in Plain Terms › The Integral = Accumulated Total
Alternating Series Error Bound: If an alternating series meets the AST conditions and converges to $S$ , then the error after $N$ terms is bounded by the first omitted term: \[ |S-S_N|≤ b_N+1.; Infinite Sequences and Series › Alternating Series, Error Bound & Absolute vs. Conditional Convergence
Alternating Series Test (AST): A series $\displaystyle\sum(-1)^{n}b_n$ with $b_n>0$ converges if both hold:; Infinite Sequences and Series › Alternating Series, Error Bound & Absolute vs. Conditional Convergence
Antiderivative (basic) rules: Course Review › Integration & the Fundamental Theorem
Approximating area with rectangles: Partition $[a,b]$ into $n$ subintervals of width $\Delta x=\dfrac{b-a}{n}$ .; Integration and the Fundamental Theorem › Riemann Sums & the Trapezoidal Rule
Arc length: For a smooth curve $x=f(t),\ y=g(t)$ traced once for $a\le t\le b$ , the arc length is \[ L=∫_a^b( $\frac{dx}{dt}$ )^2+( $\frac{dy}{dt}$ )^2 dt .; Parametric, Polar and Vector-Valued Functions › Arc Length of a Parametric Curve
Arc length formula: The length of the smooth curve $y=f(x)$ from $x=a$ to $x=b$ is \[ L=∫_a^b $\sqrt{1+\big[f'(x)\big]^2}$ dx.; Applications of Integration › Arc Length of $y=f(x)$
Arc length of a parametric curve (BC preview): For a curve given by $x=x(t)$ , $y=y(t)$ on $[\alpha,\beta]$ , the arc length is \[ L=∫_α^β( $\tfrac{dx}{dt}$ )^2+( $\tfrac{dy}{dt}$ )^2 dt.; Applications of Integration › Going Deeper: Advanced Applications
Arc length, average value, motion: Arc length of $y=f(x)$ on $[a,b]$ : $\displaystyle L=\int_a^b\sqrt{1+[f'(x)]^2}\,dx$ .; Course Review › Applications of Integration
Area between curves: Between $y=f(x)$ (top) and $y=g(x)$ (bottom) on $[a,b]$ : \[ A=∫_a^b [ f(x)-g(x) ] dx=∫_a^b (top-bottom) dx.; Course Review › Applications of Integration
Area formulas: If $f(x)\ge g(x)$ on $[a,b]$ , the area of the region between the curves is \[ A=∫_a^b[ f(x)_top-g(x)_bottom ] dx.; Applications of Integration › Area Between Two Curves
Area in polar: The area swept by $r=f(\theta)$ from $\theta=\alpha$ to $\theta=\beta$ is \[ A=12∫_α^β r^2 dθ .; Parametric, Polar and Vector-Valued Functions › Slope & Area in Polar
Average value of a function: The average value of $f$ on $[a,b]$ is \[ f_avg = $\frac{1}{b-a}$ ∫_a^b f(x) dx .; Integration and the Fundamental Theorem › Average Value & the Net Change Theorem
Average value of a function: The average value of $f$ on $[a,b]$ is \[ f_avg= $\frac{1}{b-a}$ ∫_a^b f(x) dx.; Applications of Integration › Average Value, Accumulation & Net Change
Average vs. instantaneous rate of change: The average rate of change of $f$ on $[a,b]$ is the slope of the secant line through the endpoints: \[ $\frac{f(b)-f(a)}{b-a}$ .; The Derivative › Rates of Change and the Tangent Line
Basic differentiation rules: Course Review › The Derivative
Beta and Gamma: $\Gamma(n+1) = \int_0^\infty x^ne^{-x}dx = n!$ (integrate by parts $n$ times), and $B(m,n) = \int_0^1x^{m-1}(1-x)^{n-1}dx = \tfrac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$ .; Non-Elementary Integration › The Named Integrals: Frullani, Beta, Gamma, Weierstrass
Building new series from known ones (substitution): Substitute into a known Maclaurin series instead of computing derivatives.; Infinite Sequences and Series › Going Deeper: Advanced Series
Chain Rule: The derivative of a composition $f(g(x))$ is \[ $\frac{d}{dx}$ f(g(x))=f'(g(x))· g'(x) (outside derivative $\times$ inside derivative). \] In Leibniz form: $\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}$ .; Course Review › Chain, Implicit & Inverse Differentiation
Choosing disks/washers vs. shells wisely: Match the slice to the axis:; Applications of Integration › Going Deeper: Advanced Applications
Common Maclaurin series (memorize): \[; Infinite Sequences and Series › Taylor & Maclaurin Series; the Lagrange Error Bound
Common Maclaurin series (memorize): \[; Course Review › Infinite Sequences & Series
Concavity and inflection: Where $f''(x)>0$ , $f$ is concave up ( $\smile$ ); where $f''(x)<0$ , $f$ is concave down ( $\frown$ ).; Analytical Applications of Differentiation › Concavity, the Second Derivative Test & Inflection Points
Concavity, inflection & the Second Derivative Test: $f$ is concave up where $f''>0$ , concave down where $f''<0$ .; Course Review › Analytical Applications of Differentiation
Continuity & the Intermediate Value Theorem: $f$ is continuous at $a$ if all three hold: (1) $f(a)$ is defined, (2) $\displaystyle\lim_{x\to a}f(x)$ exists, and (3) $\displaystyle\lim_{x\to a}f(x)=f(a)$ .; Course Review › Limits & Continuity
Cross-section method: If every slice perpendicular to the $x$ -axis has area $A(x)$ , then \[ V=∫_a^b A(x) dx.; Applications of Integration › Volumes by Known Cross-Sections
Decomposing a proper rational function: To integrate $\dfrac{P(x)}{Q(x)}$ with $\deg P<\deg Q$ : factor $Q(x)$ , then write the fraction as a sum of simpler pieces.; Techniques of Integration › Rational Functions by Partial Fractions
Definite integral = limit of Riemann sums = signed area: \[ ∫_a^b f(x) dx = _n→∞∑_i=1^n f(x_i^*) Δ x .; Integration and the Fundamental Theorem › The Definite Integral & Its Properties
Definite integral properties: Course Review › Integration & the Fundamental Theorem
Definition: The derivative of $f$ as a function is \[ f'(x)=_h→0 $\frac{f(x+h)-f(x)}{h}$ , \] provided the limit exists.; The Derivative › The Limit Definition of the Derivative
Definition: $F$ is an antiderivative of $f$ if $F'(x)=f(x)$ .; Integration and the Fundamental Theorem › Antiderivatives & Indefinite Integrals
Derivative of an inverse function: If $f$ is one-to-one and differentiable with $f(a)=b$ (so $f^{-1}(b)=a$ ), then \[ (f^-1)'(b)=1f' (f^-1(b))= $\frac{1}{f'(a)}$ .; Course Review › Chain, Implicit & Inverse Differentiation
Derivatives and integrals are opposites: The Fundamental Theorem of Calculus says that integrating and differentiating undo each other.; Core Ideas in Plain Terms › The Fundamental Theorem, Plainly
Derivatives of the elementary functions: Course Review › The Derivative
Differentiability vs. continuity: Differentiable $\Rightarrow$ continuous, but not conversely.; Course Review › The Derivative
Differentiate twice: To get $\dfrac{d^2y}{dx^2}$ implicitly: first find $\dfrac{dy}{dx}$ , then differentiate that expression again with respect to $x$ (using the quotient/product/chain rules), and substitute your formula for $\dfrac{dy{dx}$ wherever it appears.; Chain, Implicit and Inverse Differentiation › Second Derivatives via Implicit Differentiation
Differentiating & integrating a power series: Within the open interval of convergence you may differentiate or integrate term by term, and $R$ stays the same (endpoints may change): \[ f(x)=∑ a_n(x-c)^n ⇒ f'(x)=∑ n a_n(x-c)^n-1, ∫ f dx=C+∑ $\frac{a_n}{n+1}$ (x-c)^n+1.; Infinite Sequences and Series › Power Series: Radius & Interval of Convergence
Direct Comparison Test: Suppose $0\le a_n\le b_n$ for all large $n$ .; Infinite Sequences and Series › Comparison & Limit Comparison Tests
Direction-field analysis of equilibria: An equilibrium solution satisfies $F(x,y)=0$ (a constant solution).; Differential Equations › Going Deeper: Advanced Differential Equations
Dirichlet variants and the tangent bridge: $\int_0^\infty\tfrac{\sin^2 x}{x^2}dx = \int_0^\infty\tfrac{1 - \cos x}{x^2}dx = \tfrac\pi2$ (one integration by parts each lands on $\int_0^\infty\tfrac{\sin x}{x}dx$ ).; Non-Elementary Integration › More Named Integrals, With Proofs
Disk and washer method (slices $\perp$ to the axis): Revolving a region about a horizontal axis and slicing perpendicular to it gives circular cross-sections.; Applications of Integration › Volumes of Revolution: Disks & Washers
Endpoint analysis: the four possibilities: After finding $R$ by the ratio test, each endpoint is tested separately and independently.; Infinite Sequences and Series › Going Deeper: Advanced Series
Error bounds; absolute vs. conditional: Alternating Series Error: if AST holds, $|S-S_N|\le b_{N+1}$ (first omitted term).; Course Review › Infinite Sequences & Series
Euler's Method: Numerically approximate a solution from a starting point using step size $h$ : \[ x_n+1=x_n+h, y_n+1=y_n+h· f(x_n,y_n), \] where $f(x,y)=\frac{dy}{dx}$ .; Course Review › Differential Equations
Euler's method error: Euler's method is first order: the accumulated (global) error is roughly proportional to the step size $h$ .; Differential Equations › Going Deeper: Advanced Differential Equations
Exponential growth & decay: $\dfrac{dP}{dt}=kP$ (rate proportional to amount) solves to \[ P(t)=P_0 e^kt, \] with $k>0$ growth, $k<0$ decay, $P_0=P(0)$ .; Course Review › Differential Equations
Extreme Value Theorem (EVT): If $f$ is continuous on a closed interval $[a,b]$ , then $f$ attains an absolute maximum and an absolute minimum on $[a,b]$ .; Analytical Applications of Differentiation › The Extreme Value Theorem & Absolute Extrema (Candidates' Test)
Extreme Value Theorem & absolute extrema: EVT: a function continuous on a closed interval $[a,b]$ attains an absolute max and min there.; Course Review › Analytical Applications of Differentiation
Feynman's trick: differentiate under the integral: Embed the integral in a family $I(a) = \int f(x, a)\,dx$ , compute $I'(a) = \int \partial_a f\,dx$ (usually elementary), integrate back in $a$ , and fix the constant from a value of $a$ where $I$ is known.; Non-Elementary Integration › Integrals With No Antiderivative
Five moves, in the order to try them: 1.; Core Ideas in Plain Terms › How to Find Limits
Following the tangent line in steps: When a DE cannot be solved in closed form, approximate the solution numerically.; Differential Equations › Euler's Method (BC)
Fresnel, the reflection formula, Wallis: $\int_0^\infty\sin(x^2)\,dx = \tfrac12\sqrt{\tfrac\pi2}$ (the Mellin integral $\int_0^\infty t^{s-1}\sin t\,dt = \Gamma(s)\sin\tfrac{\pi s}{2}$ at $s = \tfrac12$ ).; Non-Elementary Integration › More Named Integrals, With Proofs
Frullani: $\displaystyle\int_0^\infty\frac{f(ax) - f(bx)}{x}dx = \big(f(0) - f(\infty)\big)\ln\frac ba$ .; Non-Elementary Integration › The Named Integrals: Frullani, Beta, Gamma, Weierstrass
FTC Part 1 — accumulation functions: If $f$ is continuous and $g(x)=\displaystyle\int_a^x f(t)\,dt$ , then \[ g'(x)=f(x).; Integration and the Fundamental Theorem › The Fundamental Theorem of Calculus
FTC Part 2 — evaluation: If $F$ is any antiderivative of $f$ on $[a,b]$ , then \[ ∫_a^b f(x) dx = F(b)-F(a) = [F(x)]_a^b .; Integration and the Fundamental Theorem › The Fundamental Theorem of Calculus
Fundamental Theorem of Calculus: Part 1 (accumulation / derivative of an integral): if $\displaystyle g(x)=\int_a^{x} f(t)\,dt$ with $f$ continuous, then $g'(x)=f(x)$ .; Course Review › Integration & the Fundamental Theorem
Geometric & $p$ -series (the benchmarks): Geometric: $\displaystyle\sum_{n=0}^{\infty}ar^{n}$ converges $\iff |r|<1$ , with sum $\dfrac{a}{1-r}$ ( $a=$ first term).; Course Review › Infinite Sequences & Series
Geometric series: $\displaystyle\sum_{n=0}^{\infty}ar^{n}=a+ar+ar^2+\cdots$ converges if and only if $|r|<1$ , and then \[ ∑_n=0^∞ar^n= $\frac{a}{1-r}$ ( $a$ is the first term, $r$ the common ratio).; Infinite Sequences and Series › Series, Partial Sums, Geometric Series & the $n$ th-Term Test
Glasser's master theorem: For any integrable $f$ and constants $a_i > 0$ , $b_i$ : $\displaystyle\int_{-\infty}^\infty f\!\left(x - \sum_i\frac{a_i}{x - b_i}\right)dx = \int_{-\infty}^\infty f(x)\,dx$ .; Non-Elementary Integration › Glasser's Master Theorem and the 1/x Reflection
Implicit differentiation: When $y$ is defined implicitly by an equation, differentiate both sides with respect to $x$ , applying the chain rule to every $y$ term (each contributes a factor $\frac{dy}{dx}$ ), then solve algebraically for $\frac{dy}{dx}$ .; Course Review › Chain, Implicit & Inverse Differentiation
Improper integrals & $p$ -integrals: Handle infinite limits or infinite discontinuities as limits: $\displaystyle\int_a^{\infty}f\,dx=\lim_{b\to\infty}\int_a^{b}f\,dx$ .; Course Review › Techniques of Integration
Increasing/decreasing & the First Derivative Test: $f$ increases where $f'>0$ , decreases where $f'<0$ .; Course Review › Analytical Applications of Differentiation
Indeterminate forms: Direct substitution sometimes yields an indeterminate form — no immediate answer, more work required: \[ 00, $\frac{\infty}{\infty}$ , 0·∞, ∞-∞, 0^0, ∞^0, 1^∞.; Course Review › Limits & Continuity
Integral Test: If $f$ is positive, continuous, and decreasing for $x\ge 1$ and $a_n=f(n)$ , then \[ ∑_n=1^∞a_n and ∫_1^∞f(x) dx either both converge or both diverge. \] The integral's value is not the sum; it only decides convergence.; Infinite Sequences and Series › The Integral Test & $p$ -Series
Integration by parts (LIATE): \[ ∫ u dv=uv-∫ v du.; Course Review › Techniques of Integration
Intermediate Value Theorem (IVT): If $f$ is continuous on $[a,b]$ and $N$ is any value between $f(a)$ and $f(b)$ , then there exists at least one $c$ in $(a,b)$ with $f(c)=N$ .; Limits and Continuity › IVT, Squeeze Theorem & the Special Trig Limit
Interval of convergence — check the endpoints!: The set of $x$ where the series converges is the interval of convergence, $(c-R,\,c+R)$ possibly including one or both endpoints.; Infinite Sequences and Series › Power Series: Radius & Interval of Convergence
Inverse trigonometric derivatives: Course Review › Chain, Implicit & Inverse Differentiation
King's rule (the reflection substitution): $\displaystyle\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$ — substituting $x \to a + b - x$ flips the interval onto itself.; Non-Elementary Integration › King's Rule, Queen's Rule, and Reflection
L'Hôpital preview: If $\lim\frac{f}{g}$ has indeterminate form $\frac00$ or $\frac{\infty}{\infty}$ and $f,g$ are differentiable, then \[ _x→ a $\frac{f(x)}{g(x)}$ =_x→ a $\frac{f'(x)}{g'(x)}$ .; Limits and Continuity › Going Deeper: Advanced Limit Ideas
L'Hôpital's Rule: If $\displaystyle\lim\frac{f}{g}$ is of form $\frac00$ or $\frac{\infty}{\infty}$ (and $g'\neq0$ nearby), then \[ $\frac{f(x)}{g(x)}$ = $\frac{f'(x)}{g'(x)}$ \] provided the right side exists.; Course Review › Contextual Applications of Differentiation
Lagrange Error Bound: If $P_n(x)$ is the $n$ th-degree Taylor polynomial about $c$ , the remainder satisfies \[ |R_n(x)|=|f(x)-P_n(x)|≤ $\frac{M}{(n+1)!}$ |x-c|^ n+1, \] where $M$ is any bound on $|f^{(n+1)}|$ between $c$ and $x$ .; Infinite Sequences and Series › Taylor & Maclaurin Series; the Lagrange Error Bound
Limit Comparison Test: If $a_n,b_n>0$ and $\displaystyle\lim_{n\to\infty}\frac{a_n}{b_n}=L$ with $0<L<\infty$ (a finite positive number), then $\sum a_n$ and $\sum b_n$ both converge or both diverge.; Infinite Sequences and Series › Comparison & Limit Comparison Tests
Limit definition of the derivative: The derivative is the instantaneous rate of change / slope of the tangent line: \[ f'(x)=_h→0 $\frac{f(x+h)-f(x)}{h}$ f'(a)=_x→ a $\frac{f(x)-f(a)}{x-a}$ .; Course Review › The Derivative
Limit laws: If $\lim_{x\to a}f(x)$ and $\lim_{x\to a}g(x)$ both exist, then limits distribute over sums, differences, products, quotients (denominator $\neq 0$ ), constant multiples, and powers/roots.; Limits and Continuity › Limit Laws & Evaluating Algebraically
Limit laws: If $\displaystyle\lim_{x\to a}f=L$ and $\displaystyle\lim_{x\to a}g=M$ (both finite), then; Course Review › Limits & Continuity
Linearization & differentials: The tangent-line (linear) approximation near $x=a$ : \[ L(x)=f(a)+f'(a)(x-a)≈ f(x).; Course Review › Contextual Applications of Differentiation
Logarithmic differentiation: For products of many factors, or a variable base with a variable exponent $y=[f(x)]^{g(x)}$ : take $\ln$ of both sides, use log laws to expand, differentiate implicitly, then multiply back by $y$ .; Course Review › Chain, Implicit & Inverse Differentiation
Logistic growth: Growth limited by a carrying capacity $M$ : \[ $\frac{dP}{dt}$ =kP(1- $\frac{P}{M}$ ).; Course Review › Differential Equations
Mean Value Theorem (MVT): If $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$ , then there exists at least one $c$ in $(a,b)$ with \[ f'(c)= $\frac{f(b)-f(a)}{b-a}$ .; Analytical Applications of Differentiation › The Mean Value Theorem & Rolle's Theorem
Mean Value Theorem & Rolle's Theorem: MVT: If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ , then there is $c\in(a,b)$ with \[ f'(c)= $\frac{f(b)-f(a)}{b-a}$ (instantaneous rate = average rate).; Course Review › Analytical Applications of Differentiation
Mixing (tank) problems: Let $y(t)$ be the amount of a substance in a tank.; Differential Equations › Going Deeper: Advanced Differential Equations
Mixing rules: Real problems layer rules together.; Chain, Implicit and Inverse Differentiation › The Chain Rule
Monotonicity and local extrema: On an interval where $f'(x)>0$ , $f$ is increasing; where $f'(x)<0$ , $f$ is decreasing.; Analytical Applications of Differentiation › Increasing/Decreasing & the First Derivative Test
Net Change Theorem: The integral of a rate of change is the net change: \[ ∫_a^b F'(x) dx = F(b)-F(a).; Integration and the Fundamental Theorem › Average Value & the Net Change Theorem
One-sided limits: $\displaystyle\lim_{x\to a^{-}}f(x)$ uses only $x<a$ (approach from the left); $\displaystyle\lim_{x\to a^{+}}f(x)$ uses only $x>a$ (approach from the right).; Limits and Continuity › One-Sided Limits & When a Limit Fails to Exist
One-sided, infinite, and at-infinity limits: One-sided: $\displaystyle\lim_{x\to a^-}f$ (from the left) and $\displaystyle\lim_{x\to a^+}f$ (from the right).; Course Review › Limits & Continuity
Optimization & connecting $f,f',f''$: Optimization: write the quantity to optimize as a function of one variable (use a constraint to eliminate others), find critical points, and justify the extremum.; Course Review › Analytical Applications of Differentiation
Parametric arc length: \[ L=∫_t_1^t_2( $\frac{dx}{dt}$ )^2+( $\frac{dy}{dt}$ )^2 dt.; Course Review › Parametric, Polar & Vector-Valued Functions
Parametric curves and slope: A parametric curve is given by $x=f(t)$ and $y=g(t)$ , where the parameter $t$ (often time) traces out the points $(x,y)$ .; Parametric, Polar and Vector-Valued Functions › Parametric Equations & Their Derivatives
Parametric derivatives: For $x=x(t)$ , $y=y(t)$ : \[ $\frac{dy}{dx}$ = $\frac{dy/dt}{dx/dt}$ ( $\tfrac{dx}{dt}$ ≠0), $\frac{d^2y}{dx^2}$ = $\dfrac{d}{dt}$ ( $\dfrac{dy}{dx}$ )dx/dt.; Course Review › Parametric, Polar & Vector-Valued Functions
Partial fractions: For a proper rational function (numerator degree $<$ denominator), factor the denominator and split into simpler fractions: \[ $\frac{P(x)}{(x-a)(x-b)}$ = $\frac{A}{x-a}$ + $\frac{B}{x-b}$ , \] solve for constants, then integrate each piece (giving logs / arctangents).; Course Review › Techniques of Integration
Polar arc length: For $r=f(\theta)$ on $\alpha\le\theta\le\beta$ , \[ L=∫_α^β r^2+( $\frac{dr}{d\theta}$ )^2 dθ .; Parametric, Polar and Vector-Valued Functions › Arc Length of a Polar Curve
Polar curves: slope, area, arc length: With $x=r\cos\theta$ , $y=r\sin\theta$ and $r=f(\theta)$ : \[ $\frac{dy}{dx}$ = $\dfrac{dr}{d\theta}$ θ+rθ $\dfrac{dr}{d\theta}$ θ-rθ.; Course Review › Parametric, Polar & Vector-Valued Functions
Polar $\leftrightarrow$ rectangular and a curve catalog: A polar point $(r,\theta)$ satisfies \[ x=rθ, y=rθ, r^2=x^2+y^2, θ= $\frac{y}{x}$ .; Parametric, Polar and Vector-Valued Functions › Polar Coordinates & Graphs
Position, velocity, acceleration, speed: A moving particle has position $\mathbf r(t)=\langle x(t),\,y(t)\rangle$ .; Parametric, Polar and Vector-Valued Functions › Vector-Valued Functions & Motion
Position, velocity, speed, acceleration: For a particle moving along a line with position $s(t)$ :; Contextual Applications of Differentiation › Straight-Line (Rectilinear) Motion
Power series and its radius: A power series centered at $c$ is $\displaystyle\sum_{n=0}^{\infty}a_n(x-c)^{n}$ .; Infinite Sequences and Series › Power Series: Radius & Interval of Convergence
Power series: radius & interval: $\displaystyle\sum a_n(x-c)^n$ converges on $|x-c|<R$ .; Course Review › Infinite Sequences & Series
Powers of sine and cosine: For $\displaystyle\int \sin^m x\cos^n x\,dx$ :; Techniques of Integration › Trigonometric Integrals
Products and quotients: \[ (fg)'=f'g+fg'_Product Rule ( $\frac{f}{g}$ )'= $\frac{f'g-fg'}{g^2}$ _Quotient Rule \] Higher-order derivatives come from differentiating again: $f''=(f')'$ , $f'''=(f'')'$ , and so on.; The Derivative › Product Rule, Quotient Rule, and Higher-Order Derivatives
Queen's rule (folding a symmetric interval): $\displaystyle\int_0^{2a} f(x)\,dx = 2\int_0^a f(x)\,dx$ if $f(2a - x) = f(x)$ , and $= 0$ if $f(2a - x) = -f(x)$ .; Non-Elementary Integration › King's Rule, Queen's Rule, and Reflection
Rate, total, or slope? Name it first: Before touching symbols, say what the question IS: a rate or slope → differentiate; an accumulated total or area → integrate.; Core Ideas in Plain Terms › Problem-Solving Playbook
Rates of change & units: A derivative $\frac{dy}{dx}$ carries units of “ $y$ -units per $x$ -unit.” In context, $f'(a)$ is the instantaneous rate of change of $f$ at $a$ ; interpret answers with correct units (e.g. “the volume is increasing at $12$ cm $^3$ /sec”).; Course Review › Contextual Applications of Differentiation
Ratio Test: Let $\displaystyle L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$ .; Infinite Sequences and Series › The Ratio Test & the Root Test
Reading a proof: Do not read a proof the way you read a rule.; Infinite Sequences and Series › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Parametric, Polar and Vector-Valued Functions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Limits and Continuity › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; The Derivative › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Chain, Implicit and Inverse Differentiation › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Contextual Applications of Differentiation › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Analytical Applications of Differentiation › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Integration and the Fundamental Theorem › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Techniques of Integration › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Applications of Integration › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Differential Equations › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Non-Elementary Integration › Why It Works
Reading a slope field: A first-order DE $\dfrac{dy}{dx}=F(x,y)$ gives the slope of the solution curve at every point.; Differential Equations › Slope Fields
Related rates: When several quantities change with time and are linked by an equation: differentiate the relating equation implicitly with respect to $t$ , then substitute known values and solve for the unknown rate.; Course Review › Contextual Applications of Differentiation
Reversing the chain rule: If $u=g(x)$ then $du=g'(x)\,dx$ .; Integration and the Fundamental Theorem › $u$ -Substitution
Rewrite as a limit: An integral is improper if a limit of integration is infinite, or if the integrand blows up somewhere on the interval.; Techniques of Integration › Improper Integrals
Riemann sums & the Trapezoidal Rule: A definite integral is a limit of Riemann sums: $\displaystyle\int_a^b f\,dx=\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^*)\,\Delta x$ , $\Delta x=\frac{b-a}{n}$ .; Course Review › Integration & the Fundamental Theorem
Rolle's Theorem (special case): If $f$ is continuous on $[a,b]$ , differentiable on $(a,b)$ , and $f(a)=f(b)$ , then there is a $c$ in $(a,b)$ with $f'(c)=0$ .; Analytical Applications of Differentiation › The Mean Value Theorem & Rolle's Theorem
Root Test: Let $\displaystyle L=\lim_{n\to\infty}\sqrt[n]{|a_n|}=\lim_{n\to\infty}|a_n|^{1/n}$ .; Infinite Sequences and Series › The Ratio Test & the Root Test
Second-order preview: Higher-order DEs involve $y''$ .; Differential Equations › Going Deeper: Advanced Differential Equations
Separable equations: If $\dfrac{dy}{dx}=g(x)h(y)$ , separate and integrate: \[ ∫ $\frac{dy}{h(y)}$ =∫ g(x) dx.; Course Review › Differential Equations
Series and partial sums: A series is the sum $\displaystyle\sum_{n=1}^{\infty}a_n$ .; Infinite Sequences and Series › Series, Partial Sums, Geometric Series & the $n$ th-Term Test
Shell method (slices $\parallel$ to the axis): Slicing parallel to the axis of revolution produces thin cylindrical shells: \[ V=2π∫_a^b (radius)(height) dx.; Applications of Integration › Volumes of Revolution: Cylindrical Shells
Slope at a single instant: The derivative $f'(x)$ is the slope of the graph right at a point — how fast the output is changing at that exact instant.; Core Ideas in Plain Terms › The Derivative = Instantaneous Rate
Slope fields: A slope field plots short segments with slope $\frac{dy}{dx}$ at grid points, showing the family of solution curves.; Course Review › Differential Equations
Slope of a polar curve: For $r=f(\theta)$ , write $x=r\cos\theta,\ y=r\sin\theta$ and use the parametric rule: \[ $\frac{dy}{dx}$ = $\frac{dy/d\theta}{dx/d\theta}$ = $\frac{r'\sin\theta+r\cos\theta}{\,r'\cos\theta-r\sin\theta\,}$ .; Parametric, Polar and Vector-Valued Functions › Slope & Area in Polar
Squeeze (Sandwich) Theorem: If $g(x)\le f(x)\le h(x)$ near $a$ and $\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L$ , then $\lim_{x\to a}f(x)=L$ .; Limits and Continuity › IVT, Squeeze Theorem & the Special Trig Limit
Squeeze Theorem and a key special limit: Squeeze: If $g(x)\le f(x)\le h(x)$ near $a$ and $\displaystyle\lim_{x\to a}g=\lim_{x\to a}h=L$ , then $\displaystyle\lim_{x\to a}f=L$ .; Course Review › Limits & Continuity
Straight-line motion: For position $s(t)$ : velocity $v(t)=s'(t)$ ; acceleration $a(t)=v'(t)=s''(t)$ ; speed $=|v(t)|$ .; Course Review › Contextual Applications of Differentiation
Surface area of a solid of revolution: Revolving $y=f(x)\ge 0$ on $[a,b]$ about the $x$ -axis produces a surface of area \[ S=∫_a^b 2π y $\sqrt{1+(y')^2}$ dx=2π∫_a^b f(x) $\sqrt{1+[f'(x)]^2}$ dx.; Applications of Integration › Going Deeper: Advanced Applications
Take a log first: For messy products, quotients, and powers—especially a variable base raised to a variable power like $x^x$ —take the natural log of both sides, use log laws to break it apart, then differentiate implicitly: \[ y = (simpler expression) …; Chain, Implicit and Inverse Differentiation › Logarithmic Differentiation
Tangent-line approximation: Near $x=a$ a differentiable function is close to its tangent line.; Contextual Applications of Differentiation › Linear Approximation & Differentials
Taylor & Maclaurin series; Lagrange error: Taylor about $c$ : $\displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(c)}{n!}(x-c)^n$ .; Course Review › Infinite Sequences & Series
Taylor and Maclaurin series: The Taylor series of $f$ centered at $c$ is \[ f(x)=∑_n=0^∞f^(n)(c)n!(x-c)^n=f(c)+f'(c)(x-c)+ $\frac{f''(c)}{2!}$ (x-c)^2+… \] A Maclaurin series is the special case $c=0$ .; Infinite Sequences and Series › Taylor & Maclaurin Series; the Lagrange Error Bound
The $n$ th-Term Test (Test for Divergence): If $\displaystyle\lim_{n\to\infty}a_n\neq 0$ (or does not exist), then $\displaystyle\sum a_n$ diverges.; Infinite Sequences and Series › Series, Partial Sums, Geometric Series & the $n$ th-Term Test
The $x \to 1/x$ reflection on $(0, \infty)$: On $(0, \infty)$ the substitution $x \to \tfrac1x$ maps the interval to itself.; Non-Elementary Integration › Glasser's Master Theorem and the 1/x Reflection
The chain rule: rates multiply through: If $y$ changes $3$ times as fast as $u$ , and $u$ changes $2$ times as fast as $x$ , then $y$ changes $6$ times as fast as $x$ .; Core Ideas in Plain Terms › Going Deeper: The Chain Rule and Why Integrals Are Areas
The convergence tests: Course Review › Infinite Sequences & Series
The formal $\varepsilon$ – $\delta$ definition: $\displaystyle\lim_{x\to a}f(x)=L$ means: for every $\varepsilon>0$ there exists $\delta>0$ such that \[ 0<|x-a|<δ |f(x)-L|<.; Limits and Continuity › Going Deeper: Advanced Limit Ideas
The formula and LIATE: Integration by parts reverses the product rule: \[ ∫ u dv = uv-∫ v du.; Techniques of Integration › Integration by Parts
The growth/decay law: The single most important DE in the course is \[ $\frac{dy}{dt}$ =ky y=y_0 e^kt, \] where $y_0=y(0)$ is the initial amount.; Differential Equations › Exponential Growth & Decay
The idea: If $y=f(g(x))$ is a composition (an “outside” function of an “inside” function), then \[ $\frac{d}{dx}$ f(g(x)) = f'(g(x))· g'(x).; Chain, Implicit and Inverse Differentiation › The Chain Rule
The idea: reverse the chain rule: If you can write an integrand in the form $f(g(x))\,g'(x)$ , let $u=g(x)$ so that $du=g'(x)\,dx$ .; Techniques of Integration › $u$ -Substitution Review
The logistic model: When growth is limited by a carrying capacity $M$ , use \[ $\frac{dP}{dt}$ =kP (1- $\frac{P}{M}$ ).; Differential Equations › Logistic Growth (BC)
The meaning of $f'(a)$: If $y=f(x)$ describes a real quantity, then $f'(x)=\dfrac{dy}{dx}$ is the instantaneous rate of change of $y$ with respect to $x$ .; Contextual Applications of Differentiation › Interpreting the Derivative in Context
The reciprocal-slope rule: If $f$ is one-to-one and differentiable, its inverse satisfies \[ (f^-1)'(a)=1f' (f^-1(a)).; Chain, Implicit and Inverse Differentiation › Derivatives of Inverse Functions
The separation method: A DE is separable if it can be written $\dfrac{dy}{dx}=g(x)\,h(y)$ .; Differential Equations › Separable Differential Equations & IVPs
The six formulas: Chain, Implicit and Inverse Differentiation › Derivatives of the Inverse Trig Functions
The special limit: \[ _x→0 $\frac{\sin x}{x}$ =1 consequently _x→0 $\frac{1-\cos x}{x}$ =0.; Limits and Continuity › IVT, Squeeze Theorem & the Special Trig Limit
The three conditions for continuity at $x=a$: $f$ is continuous at $a$ if and only if all three hold:; Limits and Continuity › Continuity at a Point & Types of Discontinuity
The three-step setup: Related-rates problems link the rates of change of two or more quantities that vary with time.; Contextual Applications of Differentiation › Related Rates
Trigonometric integrals: Course Review › Techniques of Integration
Trigonometric substitution: Course Review › Techniques of Integration
Vector-valued motion: For position $\mathbf r(t)=\langle x(t),y(t)\rangle$ :; Course Review › Parametric, Polar & Vector-Valued Functions
Vertical vs. horizontal asymptotes: An infinite limit, e.g. $\lim_{x\to a}f(x)=+\infty$ , means $f$ grows without bound near $x=a$ ; the line $x=a$ is a vertical asymptote.; Limits and Continuity › Infinite Limits & Limits at Infinity
Volumes by known cross-sections: If cross-sections perpendicular to the $x$ -axis have area $A(x)$ : \[ V=∫_a^b A(x) dx.; Course Review › Applications of Integration
Volumes of revolution: Course Review › Applications of Integration
Weierstrass substitution: $t = \tan\tfrac x2$ turns $\sin x = \tfrac{2t}{1+t^2}$ , $\cos x = \tfrac{1-t^2}{1+t^2}$ , $dx = \tfrac{2\,dt}{1+t^2}$ : ANY rational function of $\sin$ and $\cos$ becomes a rational function of $t$ , solvable by partial fractions.; Non-Elementary Integration › The Named Integrals: Frullani, Beta, Gamma, Weierstrass
What a differential equation is: A differential equation (DE) is an equation involving an unknown function and its derivatives, such as $\dfrac{dy}{dx}=2y$ or $\dfrac{dP}{dt}=kP$ .; Differential Equations › Verifying Solutions & Modeling
What a limit is: $\displaystyle\lim_{x\to a}f(x)=L$ means $f(x)$ can be made arbitrarily close to $L$ by taking $x$ close enough to $a$ (but $x\neq a$ ).; Course Review › Limits & Continuity
What a limit means: The statement $\lim_{x\to a} f(x) = L$ means: as $x$ gets arbitrarily close to $a$ (from both sides) but $x\neq a$ , the outputs $f(x)$ get arbitrarily close to $L$ .; Limits and Continuity › Limit Notation & the Intuitive Idea
What a sequence limit means: A sequence $\{a_n\}$ is an ordered list $a_1,a_2,a_3,\dots$ We say $\{a_n\}$ converges to $L$ if $\displaystyle\lim_{n\to\infty}a_n=L$ (a finite number); otherwise it diverges.; Infinite Sequences and Series › Sequences: Convergence, Divergence & Limits
When $y$ is not solved for: When a curve is defined by an equation like $x^2+y^2=25$ (you cannot easily write $y=f(x)$ ), treat $y$ as a function of $x$ and differentiate both sides with respect to $x$ .; Chain, Implicit and Inverse Differentiation › Implicit Differentiation
When and how: If $\displaystyle\lim_{x\to c}\dfrac{f(x)}{g(x)}$ gives the indeterminate form $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ , then $ $\lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)}$ $ provided the right-hand limit exists (differentiate top and b…; Contextual Applications of Differentiation › L'Hôpital's Rule
When does the derivative fail to exist?: If $f$ is differentiable at $a$ , then $f$ is continuous at $a$ .; The Derivative › Differentiability and Continuity
When L'Hôpital is NOT allowed: The rule needs an actual $\tfrac00$ or $\tfrac{\infty}{\infty}$ .; Core Ideas in Plain Terms › How to Find Limits
Which substitution to use: When a root of a quadratic appears, trade $x$ for a trig function so the Pythagorean identity collapses the root: \[ $\sqrt{a^2-x^2}$ : x=aθ, $\sqrt{a^2+x^2}$ : x=aθ, $\sqrt{x^2-a^2}$ : x=aθ.; Techniques of Integration › Trigonometric Substitution (a first look)
Which test should I use? (decision order): Course Review › Infinite Sequences & Series
Why a new toolkit is needed: Plenty of definite integrals have NO elementary antiderivative — $\int e^{-x^2}dx$ , $\int \tfrac{\sin x}{x}dx$ , $\int \tfrac{x^a - 1}{\ln x}dx$ — yet their values over specific intervals are clean: $\sqrt\pi$ , $\tfrac\pi2$ , $\ln(a+1)$ .; Non-Elementary Integration › Integrals With No Antiderivative
Why the integral measures area: Slice the region under a curve into thin vertical strips.; Core Ideas in Plain Terms › Going Deeper: The Chain Rule and Why Integrals Are Areas
Why the logistic curve is S-shaped (qualitative): Differentiate $\dfrac{dP}{dt}=kP\!\left(1-\frac{P}{M}\right)$ with respect to $t$ to study $\dfrac{d^2P}{dt^2}$ .; Differential Equations › Going Deeper: Advanced Differential Equations
Work: pumping a fluid: To pump fluid out of a tank, slice the fluid into horizontal layers at height $y$ of area $A(y)$ and thickness $dy$ .; Applications of Integration › Going Deeper: Advanced Applications
Work: variable force and Hooke's law: Work done by a variable force $F(x)$ moving an object from $x=a$ to $x=b$ is \[ W=∫_a^b F(x) dx.; Applications of Integration › Going Deeper: Advanced Applications