Comprehensive Calculus Study Notes: Limits, Derivatives, Integrals, and Applications

Limits and Continuity

Definitions and Properties of Limits

Limits are foundational to calculus, describing the behavior of functions as inputs approach specific values. Understanding limits is essential for defining derivatives and integrals.

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

• Working Definition: $\lim_{x \to a} f(x) = L$ if $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ (but not equal to $a$).

• One-Sided Limits: Right-hand limit ($x \to a^+$) and left-hand limit ($x \to a^-$) are defined similarly, but $x$ approaches $a$ from the right or left, respectively.

• Limit at Infinity: $\lim_{x \to \infty} f(x) = L$ means $f(x)$ approaches $L$ as $x$ becomes large and positive; similarly for $x \to -\infty$.

• Infinite Limits: $\lim_{x \to a} f(x) = \infty$ if $f(x)$ grows arbitrarily large as $x$ approaches $a$.

Relationship between Limits and One-Sided Limits:

• If both one-sided limits exist and are equal, the two-sided limit exists and equals that value.

• If the one-sided limits differ, the two-sided limit does not exist.

Properties of Limits

• Linearity: $\lim_{x \to a} [cf(x)] = c \lim_{x \to a} f(x)$

• Addition/Subtraction: $\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$

• Product: $\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$

• Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ (if denominator limit is not zero)

• Powers: $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$

Basic Limit Evaluations at Infinity

• $\lim_{x \to \infty} e^x = \infty$, $\lim_{x \to -\infty} e^x = 0$

• $\lim_{x \to \infty} \ln x = \infty$, $\lim_{x \to 0^+} \ln x = -\infty$

• For polynomials, the highest degree term dominates as $x \to \pm\infty$.

Continuous Functions

• A function $f(x)$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$.

• Common continuous functions: polynomials (all $x$), rational functions (except where denominator is zero), $e^x$, $\ln x$ ($x > 0$), trigonometric functions (with domain restrictions).

Intermediate Value Theorem

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

Derivatives

Definition and Notation

The derivative measures the instantaneous rate of change of a function. It is defined as:

• $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Alternative notations: $\frac{dy}{dx}$, $Df(x)$, $y'$, etc.

Interpretation of the Derivative

• Slope of the tangent line: At $x = a$, the slope is $f'(a)$.

• Instantaneous rate of change: $f'(a)$ gives the rate at which $f(x)$ changes at $x = a$.

• Velocity: If $f(x)$ is position, $f'(x)$ is velocity.

Basic Properties and Formulas

• Constant Rule: $\frac{d}{dx} [c] = 0$

• Power Rule: $\frac{d}{dx} [x^n] = n x^{n-1}$

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

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

Common Derivatives

• $\frac{d}{dx} [e^x] = e^x$

• $\frac{d}{dx} [\ln x] = \frac{1}{x}$, $x > 0$

• $\frac{d}{dx} [\sin x] = \cos x$

• $\frac{d}{dx} [\cos x] = -\sin x$

• $\frac{d}{dx} [\tan x] = \sec^2 x$

Higher Order Derivatives

• The second derivative: $f''(x) = \frac{d^2 f}{dx^2}$, measures concavity.

• The $n$th derivative: $f^{(n)}(x) = \frac{d^n f}{dx^n}$.

Implicit Differentiation

Used when $y$ is defined implicitly as a function of $x$. Differentiate both sides with respect to $x$, applying the chain rule to terms involving $y$.

Applications of Derivatives

Critical Points, Extrema, and Tests

• Critical Point: $x = c$ is a critical point if $f'(c) = 0$ or $f'(c)$ does not exist.

• Absolute Extrema: Maximum or minimum values on the entire domain.

• Relative (Local) Extrema: Maximum or minimum values in a neighborhood.

• First Derivative Test: Determines relative extrema by analyzing sign changes of $f'(x)$ around critical points.

• Second Derivative Test: If $f''(c) > 0$, $f$ has a local minimum at $c$; if $f''(c) < 0$, a local maximum.

Mean Value Theorem

If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

Newton’s Method

An iterative method for approximating roots of $f(x) = 0$:

• $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Related Rates

Used to find the rate at which one quantity changes with respect to another, often time. Differentiate both sides of an equation with respect to time $t$.

Optimization

Finds maximum or minimum values of a function subject to constraints. Steps include expressing the quantity to be optimized in terms of one variable, finding critical points, and verifying extrema.

Integrals

Definitions

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

• Antiderivative: $F(x)$ is an antiderivative of $f(x)$ if $F'(x) = f(x)$.

• Indefinite Integral: $\int f(x) dx = F(x) + C$

Fundamental Theorem of Calculus

• Part I: If $g(x) = \int_a^x f(t) dt$, then $g'(x) = f(x)$.

• Part II: $\int_a^b f(x) dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$.

Properties of Integrals

• Linearity: $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$

• Reversal of limits: $\int_a^b f(x) dx = -\int_b^a f(x) dx$

• Zero width: $\int_a^a f(x) dx = 0$

Common Integrals

• $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, $n \neq -1$

• $\int \frac{1}{x} dx = \ln|x| + C$

• $\int e^x dx = e^x + C$

• $\int \sin x dx = -\cos x + C$

• $\int \cos x dx = \sin x + C$

Standard Integration Techniques

• u-Substitution: For $u = g(x)$, $du = g'(x) dx$, $\int f(g(x))g'(x) dx = \int f(u) du$

• Integration by Parts: $\int u dv = uv - \int v du$

• Trig Substitutions: Used for integrals involving $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$

• Partial Fractions: Decompose rational functions into simpler fractions for integration.

Applications of Integrals

Net Area and Area Between Curves

The definite integral $\int_a^b f(x) dx$ gives the net area between $f(x)$ and the $x$-axis. To find the area between two curves $y = f(x)$ and $y = g(x)$ from $x = a$ to $x = b$, use:

• $A = \int_a^b [f(x) - g(x)] dx$ where $f(x) \geq g(x)$ on $[a, b]$.

Alternatively, for curves defined as $x = f(y)$ and $x = g(y)$ from $y = c$ to $y = d$:

• $A = \int_c^d [f(y) - g(y)] dy$

If the curves cross within the interval, split the integral at the intersection point:

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

Volumes of Revolution

Volumes of solids of revolution can be computed using the disk/washer or cylindrical shell methods.

• Disk/Washer Method: $V = \int_a^b \pi [R_{\text{outer}}^2 - R_{\text{inner}}^2] dx$

• Cylindrical Shell Method: $V = \int_a^b 2\pi (\text{radius})(\text{height}) dx$

For rotation about horizontal or vertical axes, adjust the limits and functions accordingly.

Other Applications

• Work: $W = \int_a^b F(x) dx$ where $F(x)$ is the force as a function of position.

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

• Arc Length: $L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$

• Surface Area (about x-axis): $SA = 2\pi \int_a^b y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$

Improper Integrals

Improper integrals involve infinite limits or discontinuous integrands. They are evaluated as limits:

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

• Convergent if the limit exists and is finite; divergent otherwise.

Approximating Definite Integrals

• Midpoint Rule: $\int_a^b f(x) dx \approx \Delta x [f(x_1^*) + f(x_2^*) + \ldots + f(x_n^*)]$

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

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