Calculus I: Limits, Derivatives, and Introduction to Integration – Study Guide

Limits and Continuity

The Idea of Limits

Limits are foundational to calculus, describing the behavior of functions as inputs approach a specific value. Understanding limits allows us to rigorously define derivatives and integrals.

• Limit of a Function: The value that a function approaches as the input approaches a certain point.

• Notation: $\lim_{x \to a} f(x) = L$ means as $x$ approaches $a$, $f(x)$ approaches $L$.

• Example: $\lim_{x \to 2} (3x + 1) = 7$

Definitions of Limits

The formal (epsilon-delta) definition of a limit provides mathematical rigor to the concept.

• Epsilon-Delta Definition: For every $\varepsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \varepsilon$.

• One-Sided Limits: Limits can be taken from the left ($x \to a^-$) or right ($x \to a^+$).

• Example: $\lim_{x \to 0^+} \frac{1}{x} = +\infty$

Techniques for Computing Limits

Several algebraic and analytic techniques are used to evaluate limits.

• Direct Substitution: Plugging in the value of $x$ directly if the function is continuous at that point.

• Factoring: Simplifying expressions to cancel terms causing indeterminate forms.

• Rationalization: Multiplying by a conjugate to eliminate radicals.

• Example: $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} \frac{(x-1)(x+1)}{x-1} = 2$

Infinite Limits and Limits at Infinity

Infinite limits describe unbounded behavior, while limits at infinity analyze function behavior as $x$ grows large.

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

• Limits at Infinity: $\lim_{x \to \infty} f(x)$ describes the end behavior of $f(x)$.

• Horizontal Asymptotes: If $\lim_{x \to \infty} f(x) = L$, then $y = L$ is a horizontal asymptote.

• Example: $\lim_{x \to \infty} \frac{2x^2 + 1}{x^2 - 3} = 2$

Continuity

A function is continuous at a point if its limit exists and equals the function value at that point.

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

• Types of Discontinuities: Removable, jump, and infinite discontinuities.

• Example: The function $f(x) = \frac{x^2 - 1}{x - 1}$ is discontinuous at $x = 1$ but can be made continuous by defining $f(1) = 2$.

Derivatives and Their Applications

Introducing the Derivative

The derivative measures the instantaneous rate of change of a function, representing the slope of the tangent line at a point.

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

• Interpretation: The derivative at $x$ gives the slope of the tangent to $f(x)$ at $x$.

• Example: For $f(x) = x^2$, $f'(x) = 2x$

The Derivative as a Function

The derivative itself is a function, mapping each $x$ to the slope of $f$ at $x$.

• Notation: $f'(x)$, $\frac{dy}{dx}$, $D_x f(x)$

• Graphical Meaning: The graph of $f'(x)$ shows where $f(x)$ is increasing or decreasing.

Rules of Differentiation

Several rules simplify the process of finding derivatives.

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

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

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

• Example: $\frac{d}{dx}[3x^2 + 2x] = 6x + 2$

Product and Quotient Rules

These rules are used to differentiate products and quotients of functions.

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

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

• Example: $\frac{d}{dx}[x^2 \sin x] = 2x \sin x + x^2 \cos x$

Derivatives of Trigonometric Functions

Trigonometric functions have specific derivative formulas.

• $\frac{d}{dx}[\sin x] = \cos x$

• $\frac{d}{dx}[\cos x] = -\sin x$

• $\frac{d}{dx}[\tan x] = \sec^2 x$

• Example: $\frac{d}{dx}[\sin x + \cos x] = \cos x - \sin x$

Derivatives as Rates of Change

Derivatives model rates of change in various contexts, such as velocity and growth.

• Physical Interpretation: If $s(t)$ is position, $s'(t)$ is velocity.

• Example: If $s(t) = t^3$, then $v(t) = s'(t) = 3t^2$

The Chain Rule

The chain rule is used to differentiate composite functions.

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

• Example: $\frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot 2x$

Implicit Differentiation

Implicit differentiation is used when functions are defined implicitly rather than explicitly.

• Method: Differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$.

• Example: For $x^2 + y^2 = 1$, $2x + 2y \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{x}{y}$

Derivatives of Logarithmic, Exponential, and Inverse Trigonometric Functions

• $\frac{d}{dx}[e^x] = e^x$

• $\frac{d}{dx}[\ln x] = \frac{1}{x}$

• $\frac{d}{dx}[a^x] = a^x \ln a$

• $\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1 - x^2}}$

• $\frac{d}{dx}[\arctan x] = \frac{1}{1 + x^2}$

• Example: $\frac{d}{dx}[\ln(\sin x)] = \cot x$

Related Rates

Related rates problems involve finding the rate at which one quantity changes by relating it to other changing quantities.

• Method: Differentiate both sides of an equation with respect to time $t$.

• Example: If a circle's radius increases at 2 cm/s, the area increases at $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$

Applications of the Derivative

Maxima and Minima

Critical points where the derivative is zero or undefined may correspond to local maxima or minima.

• First Derivative Test: Determines increasing/decreasing behavior and local extrema.

• Second Derivative Test: Determines concavity and nature of critical points.

• Example: For $f(x) = x^2$, $f'(x) = 2x = 0$ at $x=0$ (minimum).

Mean Value Theorem

The Mean Value Theorem (MVT) connects the average rate of change to the instantaneous rate of change.

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

• Application: Used to prove properties of functions and inequalities.

What Derivatives Tell Us

Derivatives provide information about function behavior, such as intervals of increase/decrease and concavity.

• Increasing/Decreasing: $f'(x) > 0$ implies $f$ is increasing; $f'(x) < 0$ implies decreasing.

• Concavity: $f''(x) > 0$ implies concave up; $f''(x) < 0$ implies concave down.

Graphing Functions

Using derivatives, one can sketch accurate graphs of functions, identifying key features.

• Critical Points: Where $f'(x) = 0$ or undefined.

• Inflection Points: Where $f''(x) = 0$ and concavity changes.

• Asymptotes: Vertical and horizontal asymptotes from limits.

Optimization Problems

Optimization involves finding maximum or minimum values of functions, often subject to constraints.

• Steps: 1) Identify the quantity to optimize, 2) Express as a function, 3) Find critical points, 4) Test endpoints if necessary.

• Example: Maximizing area with a fixed perimeter.

Linear Approximation and Differentials

Linear approximation uses the tangent line to estimate function values near a point.

• Formula: $f(x) \approx f(a) + f'(a)(x - a)$

• Differential: $dy = f'(x) dx$

• Example: Approximate $\sqrt{4.1}$ using $f(x) = \sqrt{x}$ at $x=4$.

L’Hôpital’s Rule

L’Hôpital’s Rule helps evaluate indeterminate forms like $0/0$ or $\infty/\infty$.

• Statement: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ yields $0/0$ or $\infty/\infty$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ (if the latter limit exists).

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

Introduction to Integration

Antiderivatives

An antiderivative of a function $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$.

• General Form: $F(x) + C$, where $C$ is the constant of integration.

• Example: $\int 2x dx = x^2 + C$

Approximating Areas under Curves

Before formal integration, areas under curves are approximated using sums.

• Riemann Sums: Approximating area by summing areas of rectangles.

• Left, Right, and Midpoint Sums: Different methods for choosing rectangle heights.

• Example: Approximate $\int_0^2 x^2 dx$ using 4 rectangles.

Definite Integrals

The definite integral computes the net area under a curve between two points.

• Notation: $\int_a^b f(x) dx$

• Interpretation: Net signed area between $x=a$ and $x=b$.

• Properties: Linearity, additivity over intervals, etc.

Fundamental Theorem of Calculus

This theorem links differentiation and integration, providing a method for evaluating definite integrals.

• Part 1: If $F(x) = \int_a^x f(t) dt$, then $F'(x) = f(x)$.

• Part 2: $\int_a^b f(x) dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$.

• Example: $\int_1^3 x^2 dx = \left. \frac{1}{3}x^3 \right|_1^3 = 9 - \frac{1}{3}$

Working with Integrals

Techniques for manipulating and evaluating integrals.

• Linearity: $\int (af(x) + bg(x)) dx = a\int f(x) dx + b\int g(x) dx$

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

Substitution Rule

The substitution rule simplifies integration by changing variables.

• Formula: $\int f(g(x))g'(x) dx = \int f(u) du$, where $u = g(x)$

• Example: $\int 2x \cos(x^2) dx$; let $u = x^2$, $du = 2x dx$

Velocity and Net Change

Integrals can be used to compute net change in quantities, such as displacement from velocity.

• Net Change Theorem: $\int_a^b F'(x) dx = F(b) - F(a)$

• Application: If $v(t)$ is velocity, $\int_a^b v(t) dt$ is net displacement.