Calculus Cheat Sheet: Comprehensive Study Notes for College Calculus

Limits

Definitions

Limits are foundational to calculus, describing the behavior of functions as inputs approach specific values or infinity.

• Precise Definition: We say $\lim_{x \to a} f(x) = L$ if for every $\epsilon > 0$ there is a $\delta > 0$ such that whenever $0 < |x-a| < \delta$, $|f(x) - L| < \epsilon$.

• Left and Right Hand Limits:

  • Right hand: $\lim_{x \to a^+} f(x)$

  • Left hand: $\lim_{x \to a^-} f(x)$

• Limit at Infinity: $\lim_{x \to \infty} f(x) = L$ means $f(x)$ approaches $L$ as $x$ becomes very large.

• Infinite Limits: $\lim_{x \to a} f(x) = \infty$ means $f(x)$ increases without bound as $x$ approaches $a$.

Relationship Between the Limit and One-Sided Limits

• $\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^-} f(x) = L$ and $\lim_{x \to a^+} f(x) = L$.

Properties

• Assume $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$:

• $\lim_{x \to a} [f(x) + g(x)] = L + M$

• $\lim_{x \to a} [f(x)g(x)] = LM$

• $\lim_{x \to a} \left[\frac{f(x)}{g(x)}\right] = \frac{L}{M}$, provided $M \neq 0$

Basic Limit Evaluations

• $\lim_{x \to 0} \frac{\sin x}{x} = 1$

• $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$

• $\lim_{x \to \infty} \frac{1}{x} = 0$

• $\lim_{x \to 0^+} \ln x = -\infty$

Evaluation Techniques

Continuous Functions

If $f(x)$ is continuous at $a$, then $\lim_{x \to a} f(x) = f(a)$.

L'Hospital's Rule

• If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$.

Polynomials and Rational Expressions

• Factor and cancel common terms to simplify limits.

• Combine rational expressions for easier evaluation.

Intermediate Value Theorem

• If $f(x)$ is continuous on $[a, b]$ and $N$ is any number between $f(a)$ and $f(b)$, there exists $c$ in $[a, b]$ such that $f(c) = N$.

Derivatives

Definition and Notation

The derivative measures the instantaneous rate of change of a function.

• If $f(x)$ is differentiable at $x$, then $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Alternate notations: $\frac{df}{dx}$, $\frac{dy}{dx}$, $D_x f(x)$

Interpretation of the Derivative

• $f'(x)$ is the slope of the tangent line to $f(x)$ at $x$.

• $f'(x)$ is the instantaneous rate of change of $f(x)$ at $x$.

Basic Properties and Formulas

• Sum Rule: $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$

• Product Rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

• Chain Rule: $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$

Common Derivatives

• $\frac{d}{dx}x^n = nx^{n-1}$

• $\frac{d}{dx}\sin x = \cos x$

• $\frac{d}{dx}\cos x = -\sin x$

• $\frac{d}{dx}e^x = e^x$

• $\frac{d}{dx}\ln x = \frac{1}{x}$

Extrema

Absolute and Relative Extrema

Extrema refer to the maximum and minimum values of a function.

• Absolute Extrema: The highest or lowest value of $f(x)$ on its domain.

• Relative (Local) Extrema: The highest or lowest value of $f(x)$ in a neighborhood.

Fermat's Theorem

• If $f$ has a relative extremum at $c$ and is differentiable at $c$, then $f'(c) = 0$.

Finding Absolute Extrema

1. Find critical points by solving $f'(x) = 0$ or where $f'(x)$ is undefined.

2. Evaluate $f(x)$ at critical points and endpoints.

3. Compare values to determine extrema.

Applications of Derivatives

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to another.

• Set up equations relating quantities.

• Differentiate both sides with respect to time.

• Solve for the desired rate.

Optimization

• Find maximum or minimum values of a function subject to constraints.

• Set up the function to be optimized, find critical points, and test endpoints.

Integrals

Definitions

• Definite Integral: $\int_a^b f(x)dx$ is the signed area under $f(x)$ from $a$ to $b$.

• Anti-Derivative: $F(x)$ is an anti-derivative of $f(x)$ if $F'(x) = f(x)$.

Fundamental Theorem of Calculus

• Part 1: If $f$ is continuous on $[a, b]$, then $\int_a^b f(x)dx = F(b) - F(a)$ where $F'(x) = f(x)$.

Properties

• $\int_a^b f(x)dx = -\int_b^a f(x)dx$

• $\int_a^a f(x)dx = 0$

• $\int_a^b [f(x) + g(x)]dx = \int_a^b f(x)dx + \int_a^b g(x)dx$

Techniques of Integration

Substitution

• Let $u = g(x)$, then $\int f(g(x))g'(x)dx = \int f(u)du$

Integration by Parts

• $\int u dv = uv - \int v du$

Trig Substitution

• Use trigonometric identities to simplify integrals involving $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$.

Applications of Integrals

Net Area

• $\int_a^b f(x)dx$ represents the net area between $f(x)$ and the $x$-axis.

Area Between Curves

• $\int_a^b [f(x) - g(x)]dx$ gives the area between $f(x)$ and $g(x)$ from $a$ to $b$.

Volumes of Revolution

• Disk method: $V = \pi \int_a^b [R(x)]^2 dx$

• Shell method: $V = 2\pi \int_a^b r(x)h(x)dx$

Work and Average Value

• Work: $W = \int_a^b F(x)dx$

• Average value: $f_{avg} = \frac{1}{b-a} \int_a^b f(x)dx$

Arc Length & Surface Area

Arc Length

• $L = \int_a^b \sqrt{1 + [f'(x)]^2} dx$

Surface Area

• $S = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} dx$

Improper Integrals

Definition

• If the interval is infinite or the function is unbounded, use limits to define the integral.

• $\int_a^\infty f(x)dx = \lim_{b \to \infty} \int_a^b f(x)dx$

Convergence Tests

• Comparison test: Compare to a known convergent or divergent integral.

Approximating Definite Integrals

Numerical Methods

• Trapezoid Rule: $\int_a^b f(x)dx \approx \frac{b-a}{2n} [f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n)]$

• Simpson's Rule: $\int_a^b f(x)dx \approx \frac{b-a}{3n} [f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-1}) + f(x_n)]$

Table: Common Derivatives and Integrals

Function	Derivative	Integral
$x^n$	$nx^{n-1}$	$\frac{x^{n+1}}{n+1} + C$
$\sin x$	$\cos x$	$-\cos x + C$
$\cos x$	$-\sin x$	$\sin x + C$
$e^x$	$e^x$	$e^x + C$
$\ln x$	$\frac{1}{x}$	$x \ln x - x + C$

Additional info:

• This cheat sheet covers core topics from Calculus I and II, including limits, derivatives, applications of derivatives, integrals, and techniques of integration.

• It is suitable for exam review and as a quick reference for college calculus students.