Comprehensive Calculus Study Guide: Essential Derivatives, Integrals, and Theorems

Basic Derivatives

• Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$

• Trigonometric Derivatives:     * $\frac{d}{dx}(\sin(x)) = \cos(x)$     * $\frac{d}{dx}(\cos(x)) = -\sin(x)$     * $\frac{d}{dx}(\tan(x)) = \sec^2(x)$     * $\frac{d}{dx}(\cot(x)) = -\csc^2(x)$     * $\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)$     * $\frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x)$

• Logarithmic Derivative: $\frac{d}{dx}(\ln(u)) = \frac{1}{u}\frac{du}{dx}$

• Exponential Derivative: $\frac{d}{dx}(e^u) = e^u\frac{du}{dx}$

• Note: In these formulas, $u$ represents a function of $x$, and $a$ represents a constant.

Differentiation Rules

• Chain Rule: $\frac{d}{dx}[f(u)] = f'(u)\frac{du}{dx}$ OR $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$

• Product Rule: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$ OR $uv' + vu'$

• Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$ OR $\frac{vu' - uv'}{v^2}$

Fundamental Theorems and Definitions

• Intermediate Value Theorem (IVT): If the function $f(x)$ is continuous on $[a, b]$, and $y$ is a number between $f(a)$ and $f(b)$, then there exists at least one number $x = c$ in the open interval $(a, b)$ such that $f(c) = y$.

• Mean Value Theorem (MVT): If the function $f(x)$ is continuous on $[a, b]$ AND the first derivative exists on the interval $(a, b)$, then there is at least one number $x = c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

• Rolle’s Theorem: If the function $f(x)$ is continuous on $[a, b]$, AND the first derivative exists on the interval $(a, b)$, AND $f(a) = f(b)$, then there is at least one number $x = c$ in $(a, b)$ such that $f'(c) = 0$.

• Extreme Value Theorem (EVT): If the function $f(x)$ is continuous on $[a, b]$, then the function is guaranteed to have an absolute maximum and an absolute minimum on the interval.

• Alternate Definition of the Derivative: $f'(c) = \lim_{x \to c}\frac{f(x) - f(c)}{x - c}$

Curve Sketching and Analysis

• Continuity: $y = f(x)$ must be continuous at each point of analysis.

• Critical point: Occurs where $\frac{dy}{dx} = 0$ or $\frac{dy}{dx}$ is undefined. Always check endpoints.

• Local minimum: $\frac{dy}{dx}$ changes from $-$ to $+$ (via $0$ or $und$), OR $\frac{d^2y}{dx^2} > 0$.

• Local maximum: $\frac{dy}{dx}$ changes from $+$ to $-$ (via $0$ or $und$), OR $\frac{d^2y}{dx^2} < 0$.

• Point of inflection: Occurs where concavity changes. $\frac{d^2y}{dx^2}$ changes signs ($+, 0, -$; $-, 0, +$; $+, und, -$; or $-, und, +$).

• Derivative of an Inverse Function: If $f$ has an inverse function $g$, then $g'(x) = \frac{1}{f'(g(x))}$. Derivatives are reciprocal slopes.

• Implicit Differentiation: Remember to include a $\frac{dy}{dx}$ for every $y$ variable encountered. Isolate $\frac{dy}{dx}$. When finding the second derivative $\frac{d^2y}{dx^2}$, substitute the expression found for the first derivative back into the process.

• Average Rate of Change (AROC): $m_{AROC} = \frac{f(b) - f(a)}{b - a}$

• Instantaneous Rate of Change (IROC): $m_{IROC} = f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$

• Equation of a Tangent Line: Requires a slope ($m$) and a point $(x_1, y_1)$. Formula: $y - y_1 = m(x - x_1)$.

Function Behavior and Second Derivative Test

• First Derivative Behavior:     * $f'(x) > 0$: Function is increasing.     * $f'(x) < 0$: Function is decreasing.     * $f'(x) = 0$ or DNE: Critical Values at $x$.     * Relative Maximum: $f'(x) = 0$ or DNE and sign of $f'(x)$ changes from $+$ to $-$     * Relative Minimum: $f'(x) = 0$ or DNE and sign of $f'(x)$ changes from $-$ to $+$     * Absolute Max or Min: Must check endpoints in addition to relative extrema. The "maximum value" refers to the $y$-value.

• Second Derivative Behavior:     * $f''(x) > 0$: Function is concave up.     * $f''(x) < 0$: Function is concave down.     * $f''(x) = 0$ and signs change: Point of inflection at $x$.     * Relative Maximum: $f''(x) < 0$     * Relative Minimum: $f''(x) > 0$

• Horizontal Asymptotes:     1. If largest exponent in numerator < largest exponent in denominator, $\lim_{x \to \pm \infty} f(x) = 0$.     2. If largest exponent in numerator > largest exponent in denominator, $\lim_{x \to \pm \infty} f(x) = DNE$.     3. If largest exponent in numerator = largest exponent in denominator, the quotient of the leading coefficients is the asymptote: $\lim_{x \to \pm \infty} f(x) = L$.

• Indeterminate Forms and Limits:     * $\frac{0}{0} = IF$ (Indeterminate Form)     * $\frac{\infty}{\infty} = IF$     * $\frac{c}{0} = \infty$     * $\frac{c}{\infty} = 0$

Integrals and The Fundamental Theorem of Calculus (FTC)

• "PLUS A CONSTANT": Always include $+C$ for indefinite integrals.

• The Fundamental Theorem of Calculus: $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F'(x) = f(x)$.

• Corollary to FTC: $\frac{d}{dx}\left[\int_{g(u)}^{h(u)} f(t)\,dt\right] = f(h(u))h'(u) - f(g(u))g'(u)$.

• The Accumulation Function: $F(x) = f(a) + \int_a^x f'(t)\,dt$. The total amount $F(x)$ at time $x$ is the initial amount $f(a)$ plus the change from $t = a$ to $t = x$.

Kinematics: Distance, Velocity, and Acceleration

• Functions:     * $x(t)$ = position function     * $v(t)$ = velocity function     * $a(t)$ = acceleration function

• Relationships:     * The derivative of position ($ft$) is velocity ($ft/sec$).     * The derivative of velocity ($ft/sec$) is acceleration ($ft/sec^2$).     * The integral of acceleration ($ft/sec^2$) is velocity ($ft/sec$).     * The integral of velocity ($ft/sec$) is position ($ft$).

• Dynamics:     * Speed = $|v(t)|$     * Increasing Speed: Acceleration and velocity have the same sign (moving right).     * Decreasing Speed: Acceleration and velocity have different signs (moving left).     * Displacement = $\int_{t_1}^{t_2} v(t)\,dt$     * Total Distance = $\int_{t_1}^{t_2} |v(t)|\,dt$     * Average Velocity = $\frac{\text{final position} - \text{initial position}}{\text{total time}} = \frac{\Delta x}{\Delta t}$

Calculator Usage, Logarithms, and Growth

• Four Tasks Requiring No Work Shown on Calculator:     1. Graphing a function within an arbitrary viewing window.     2. Finding the zeros of a function.     3. Computing the derivative of a function numerically.     4. Computing the definite integral of a function numerically.

• Logarithm Properties:     * $\ln(N) = p \leftrightarrow e^p = N$     * $\ln(e) = 1$     * $\ln(1) = 0$     * $\ln(MN) = \ln(M) + \ln(N)$     * $\ln\left(\frac{M}{N}\right) = \ln(M) - \ln(N)$     * $p \cdot \ln(M) = \ln(M^p)$

• Exponential Growth and Decay: Use $y = Ce^{kt}$ when text says "rate of change of $y$ is proportional to $y$" or "$y$ is a differentiable function of $t$ such that $y > 0$ and $y' = ky$".

• Solving Differential Equations:     1. Separate variables first.     2. Integrate both sides.     3. Add $+C$ to one side.     4. Use initial conditions to find $C$.     5. Write the final equation in the form $y = f(x)$.

• Squeeze Theorem: If $f(x) \le g(x) \le h(x)$ and $\lim_{x \to c} f(x) = L = \lim_{x \to c} h(x)$, then $\lim_{x \to c} g(x) = L$.

Integral Approximations and Mean Value Theorem for Integrals

• Average Value of a Function: If $f(x)$ is continuous on $[a, b]$, there exists $x = c$ on $(a, b)$ such that: $f_{avg} = \frac{1}{b - a} \int_a^b f(x)\,dx$.

• Riemann Sums: Rectangular approximations. Do not evaluate the integral; add the areas of the rectangles.

• Trapezoidal Rule:     * Uneven Intervals: Calculate area of each trapezoid individually: $A_{trap} = \frac{1}{2}h[b_1 + b_2]$.     * Even Intervals: $\int_a^b f(x)\,dx \approx \frac{b - a}{2n} [y_0 + 2y_1 + 2y_2 + \dots + 2y_{n-1} + y_n]$.

Trigonometry Reference

• Values for Common Angles:     * $0$: $\sin = 0, \cos = 1, \tan = 0$     * $\frac{\pi}{6}$: $\sin = \frac{1}{2}, \cos = \frac{\sqrt{3}}{2}, \tan = \frac{\sqrt{3}}{3}$     * $\frac{\pi}{4}$: $\sin = \frac{\sqrt{2}}{2}, \cos = \frac{\sqrt{2}}{2}, \tan = 1$     * $\frac{\pi}{3}$: $\sin = \frac{\sqrt{3}}{2}, \cos = \frac{1}{2}, \tan = \sqrt{3}$     * $\frac{\pi}{2}$: $\sin = 1, \cos = 0, \tan = \text{undefined}$     * $\pi$: $\sin = 0, \cos = -1, \tan = 0$

• Note: Must know inverse trig values (e.g., $\tan\left(\frac{\pi}{4}\right) = 1$ and $\arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}$.)

• Odd and Even Functions:     * $\sin(-x) = -\sin(x)$ (Odd)     * $\cos(-x) = \cos(x)$ (Even)

• Pythagorean Identities:     * $\sin^2(\theta) + \cos^2(\theta) = 1$     * $1 + \tan^2(\theta) = \sec^2(\theta)$     * $\cot^2(\theta) + 1 = \csc^2(\theta)$

• Double Angle Formulas:     * $\sin(2x) = 2\sin(x)\cos(x)$     * $\cos(2x) = \cos^2(x) - \sin^2(x) = 1 - 2\sin^2(x) = 2\cos^2(x) - 1$

• Power-Reducing Formulas:     * $\cos^2(x) = \frac{1}{2}(1 + \cos(2x))$     * $\sin^2(x) = \frac{1}{2}(1 - \cos(2x))$

• Quotient and Reciprocal Identities:     * $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$; $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$     * $\csc(x) = \frac{1}{\sin(x)}$; $\sec(x) = \frac{1}{\cos(x)}$; $\sin(x)\csc(x) = 1$; $\cos(x)\sec(x) = 1$

Limits and Continuity

• Limit Existence: $\lim_{x \to c} f(x)$ exists only if $\lim_{x \to c^+} f(x) = \lim_{x \to c^-} f(x)$.

• Special Limits:     1. $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$     2. $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$

• Continuity Definition: A function $f(x)$ is continuous at $x = c$ if $f(c) = \lim_{x \to c^+} f(x) = \lim_{x \to c^-} f(x)$.

Area and Solids of Revolution

• Coordinates: $(a, b)$ are $x$-coordinates; $(c, d)$ are $y$-coordinates.

• Area Between Two Curves:     * Slices $\perp$ to $x$-axis: $A = \int_a^b [f(x) - g(x)]\,dx$     * Slices $\perp$ to $y$-axis: $A = \int_c^d [f(y) - g(y)]\,dy$

• Volume by Disk Method:     * About $x$-axis: $V = \pi \int_a^b [R(x)]^2\,dx$     * About $y$-axis: $V = \pi \int_c^d [R(y)]^2\,dy$

• Volume by Washer Method:     * About $x$-axis: $V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2)\,dx$     * About $y$-axis: $V = \pi \int_c^d ([R(y)]^2 - [r(y)]^2)\,dy$

• Known Cross Sections (Volume): Base is the distance between two curves.     * Squares: $V = \int_a^b (base)^2\,dx$     * Equilateral Triangles: $V = \frac{\sqrt{3}}{4} \int_a^b (base)^2\,dx$     * Isosceles Right Triangles: $V = \frac{1}{4} \int_a^b (base)^2\,dx$     * Rectangles: $V = \int_a^b (base) \cdot h\,dx$ (where $h$ is height).     * Semi-circles: $V = \frac{\pi}{8} \int_a^b (base)^2\,dx$ OR $V = \frac{\pi}{2} \int_a^b (radius)^2\,dx$ (radius is $\frac{1}{2}$ distance between curves).

Basic Integrals Table

• $\int du = u + C$

• $\int u^n\,du = \frac{u^{n+1}}{n+1} + C$ ($n \ne -1$)

• $\int \frac{du}{u} = \ln|u| + C$

• $\int e^u\,du = e^u + C$

• $\int a^u\,du = \frac{a^u}{\ln(a)} + C$

• $\int \sin(u)\,du = -\cos(u) + C$

• $\int \cos(u)\,du = \sin(u) + C$

• $\int \tan(u)\,du = -\ln|\cos(u)| + C$

• $\int \cot(u)\,du = \ln|\sin(u)| + C$

• $\int \sec(u)\,du = \ln|\sec(u) + \tan(u)| + C$

• $\int \csc(u)\,du = -\ln|\csc(u) + \cot(u)| + C$

• $\int \sec^2(u)\,du = \tan(u) + C$

• $\int \csc^2(u)\,du = -\cot(u) + C$

• $\int \sec(u)\tan(u)\,du = \sec(u) + C$

• $\int \csc(u)\cot(u)\,du = -\csc(u) + C$

Advanced Derivatives and Integrals

• Inverse Trig Derivatives:     * $\frac{d}{dx}(\arcsin(u)) = \frac{1}{\sqrt{1 - u^2}}\frac{du}{dx}$     * $\frac{d}{dx}(\arccos(u)) = \frac{-1}{\sqrt{1 - u^2}}\frac{du}{dx}$     * $\frac{d}{dx}(\arctan(u)) = \frac{1}{1 + u^2}\frac{du}{dx}$     * $\frac{d}{dx}(\text{arccot}(u)) = \frac{-1}{1 + u^2}\frac{du}{dx}$     * $\frac{d}{dx}(\text{arcsec}(u)) = \frac{1}{|u|\sqrt{u^2 - 1}}\frac{du}{dx}$     * $\frac{d}{dx}(\text{arccsc}(u)) = \frac{-1}{|u|\sqrt{u^2 - 1}}\frac{du}{dx}$

• Other Derivatives:     * $\frac{d}{dx}(a^u) = a^u \ln(a) \frac{du}{dx}$     * $\frac{d}{dx}(\log_a(x)) = \frac{1}{x \ln(a)}$

• Inverse Trig Integrals:     * $\int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin\left(\frac{u}{a}\right) + C$     * $\int \frac{du}{a^2 + u^2} = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C$     * $\int \frac{du}{u\sqrt{u^2 - a^2}} = \frac{1}{a} \text{arcsec}\left(\frac{|u|}{a}\right) + C$

Essential Parent Function Graphs

• Linear: $y = x$

• Quadratic: $y = x^2$

• Cubic: $y = x^3$

• Absolute Value: $y = |x|$

• Square Root: $y = \sqrt{x}$

• Rational: $y = \frac{1}{x}$ and $y = \frac{1}{x^2}$

• Trigonometric: $y = \sin(x)$, $y = \cos(x)$

• Exponential: $y = e^x$

• Logarithmic: $y = \ln(x)$

• Semicircle: $y = \sqrt{a^2 - x^2}$