Calculus I: Limits, Derivatives, and Applications – Study Guide

Limits and Continuity

Evaluating Limits

Limits describe the behavior of a function as the input approaches a particular value. They are foundational in calculus for defining derivatives and continuity.

• Definition: The limit of a function f(x) as x approaches a value a is the value that f(x) approaches as x gets arbitrarily close to a.

• Notation: $\lim_{x \to a} f(x)$

• Techniques:

  • Direct substitution

  • Factoring and simplifying

  • Rationalizing

  • Using special limits (e.g., $\lim_{x \to 0} \frac{\sin x}{x} = 1$)

• One-sided limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right)

• Infinite limits: When f(x) increases or decreases without bound as x approaches a value.

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

Example: $\lim_{x \to 2} \frac{x^2 - 7x + 12}{x^2 - 4}$ Factor numerator and denominator, then simplify and substitute.

Continuity

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

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

• Types of discontinuities: removable, jump, infinite

Example: Piecewise functions may have discontinuities at the points where the formula changes.

Differentiation

Definition of the Derivative

The derivative of a function measures the rate at which the function value changes as its input changes. It is the foundation for understanding slopes, rates of change, and motion.

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

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

Basic Differentiation Rules

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

• Constant Rule: $\frac{d}{dx} c = 0$

• Constant Multiple Rule: $\frac{d}{dx} [c f(x)] = c f'(x)$

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

Example: $\frac{d}{dx} (x^2 + 5x + 6) = 2x + 5$

Derivatives of Trigonometric, Exponential, and Logarithmic Functions

• $\frac{d}{dx} \sin x = \cos x$

• $\frac{d}{dx} \cos x = -\sin x$

• $\frac{d}{dx} e^x = e^x$

• $\frac{d}{dx} \ln x = \frac{1}{x}$

Example: $\frac{d}{dx} (e^{2x}) = 2e^{2x}$ (by chain rule)

Implicit Differentiation

Used when y is defined implicitly as a function of x (not solved for y explicitly).

• Differentiating both sides of an equation with respect to x, treating y as a function of x.

• Apply the chain rule to terms involving y.

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

Related Rates

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

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

• Use the chain rule for variables that are functions of t.

Example: If the radius of a circle increases at 0.2 m/s, the rate of change of area A is $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$

Applications of Derivatives

Tangent Lines

The tangent line to a curve at a point is the straight line that just touches the curve at that point and has the same slope as the curve there.

• Equation: $y - y_1 = f'(x_1)(x - x_1)$, where $f'(x_1)$ is the derivative at $x_1$.

Example: Find the tangent to $y = x^2$ at $x = 2$: Slope is $f'(2) = 4$, so $y - 4 = 4(x - 2)$.

Horizontal Tangents

A tangent is horizontal where the derivative is zero.

• Solve $f'(x) = 0$ to find x-values where the tangent is horizontal.

Motion Problems

Describes the position, velocity, and acceleration of objects moving along a line.

• Position function: s(t)

• Velocity: $v(t) = s'(t)$

• Acceleration: $a(t) = v'(t) = s''(t)$

• Maximum height occurs when $v(t) = 0$

• Time to hit the ground: solve $s(t) = 0$

Example: For $s(t) = -16t^2 + 92t$, $v(t) = -32t + 92$

Graphical Analysis

Limits from Graphs

Limits can be estimated or found exactly by analyzing the behavior of a function's graph as x approaches a value.

• Look for the y-value the graph approaches from both sides.

• If the left and right limits differ, the limit does not exist.

Asymptotes

• Vertical asymptotes: Occur where the function grows without bound as x approaches a certain value (often where denominator is zero).

• Horizontal asymptotes: Describe end behavior as $x \to \infty$ or $x \to -\infty$.

Example: $f(x) = \frac{2x^2 - 7x + 12}{x^2 - 4}$ has vertical asymptotes where $x^2 - 4 = 0$, i.e., $x = 2, -2$.

Solving Equations and Quadratics

Quadratic Equations

• Standard form: $ax^2 + bx + c = 0$

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Solve $2x^2 - 14x + 24 = 0$ using the quadratic formula.

Solving Trigonometric Equations

• Find all solutions in a given interval, often using inverse trig functions.

• Example: Solve $\sin \theta = \frac{1}{2}$ for $0 \leq \theta < 2\pi$.

Special Derivatives and Inverse Functions

• Inverse trigonometric derivatives:

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

  • $\frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1 - x^2}}$

• Logarithmic differentiation: Useful for functions of the form $y = f(x)^{g(x)}$.

Summary Table: Key Derivative Rules

Function	Derivative
$x^n$	$n x^{n-1}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$
$\arcsin x$	$\frac{1}{\sqrt{1 - x^2}}$
$\arccos x$	$-\frac{1}{\sqrt{1 - x^2}}$

Additional info:

• This study guide covers the main topics from a Calculus I exam or homework set, including limits, derivatives, applications to motion, related rates, and graphical analysis.

• Some problems involve interpreting graphs, solving equations, and applying calculus to real-world scenarios.