Calculus Study Guide: Limits, Continuity, Derivatives, and Applications

Limits and Continuity

Understanding Limits

Limits are fundamental to calculus, describing the behavior of a function as its input approaches a particular value. They are essential for defining derivatives and integrals.

• 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: $\lim_{x \to a} f(x)$

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

• Infinite limits: If f(x) increases or decreases without bound as x approaches a, the limit is infinite.

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

Continuity

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

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

• Piecewise functions: To ensure continuity at a point where the definition changes, set the left and right limits equal and solve for any unknowns.

Example: For $f(x) = \begin{cases} kx, & x \leq 9 \\ 81, & x > 9 \end{cases}$, find k so f(x) is continuous at x = 9: Set $\lim_{x \to 9^-} f(x) = \lim_{x \to 9^+} f(x)$, so $9k = 81 \implies k = 9$.

Derivatives and Differentiation

Definition of the Derivative

The derivative of a function measures its instantaneous rate of change. It is defined as the limit of the average rate of change as the interval approaches zero.

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

• Interpretation: The slope of the tangent line to the curve at a point.

Example: The derivative of $y = \tan^{-1} \frac{4x}{5}$ is $\frac{20}{16x^2 + 25}$.

Rules of Differentiation

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

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

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

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

Example: For $y = x^{x^3}$, use logarithmic differentiation: $\ln y = x^3 \ln x$.

Applications of Derivatives

Tangent Lines

The equation of the tangent line to a curve at a point (a, f(a)) is given by:

• Formula: $y - f(a) = f'(a)(x - a)$

Example: Find the tangent to $y = (x^2 - 3x)\ln x$ at $x = 4$ by computing $f(4)$ and $f'(4)$.

Velocity and Acceleration

In physics, the derivative of position with respect to time is velocity, and the derivative of velocity is acceleration.

• Velocity: $v(t) = s'(t)$

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

Example: If $s(t) = t^4$, then $v(t) = 4t^3$ and $a(t) = 12t^2$.

Graphical Analysis

Interpreting Graphs of Functions and Their Derivatives

Graphs can represent position, velocity, and acceleration. The slope of the position graph at a point gives velocity; the slope of the velocity graph gives acceleration.

• Key Point: The steepness of the curve at a point indicates the magnitude of the derivative.

Example: If a graph shows a curve with a positive slope, the velocity is positive at that point.

Limits Involving Infinity and Asymptotes

Vertical and Horizontal Asymptotes

Asymptotes describe the behavior of functions as x approaches certain critical values or infinity.

• Vertical Asymptotes: Occur where the denominator of a rational function is zero and the numerator is nonzero.

• Horizontal Asymptotes: Determined by the degrees of the numerator and denominator polynomials.

Example: For $y = \frac{2x^2 + 8}{2x^2 + 3x - 2}$, set the denominator to zero to find vertical asymptotes, and compare degrees for horizontal asymptotes.

Exponential Growth and Average/Instantaneous Rate of Change

Exponential Functions

Exponential functions model growth or decay processes. The general form is $y = ab^{kt}$ or $y = ae^{kt}$.

• Average Rate of Change: $\frac{f(b) - f(a)}{b - a}$ over interval [a, b]

• Instantaneous Rate of Change: The derivative at a specific point, $f'(a)$

Example: For $c(t) = 114.9 \cdot 0.345^t$, the instantaneous growth rate at $t$ is $c'(t) = 114.9 \cdot 0.345^t \cdot \ln(0.345)$.

Table: Comparison of Asymptotes

Type	How to Find	Example
Vertical Asymptote	Set denominator = 0, solve for x	$2x^2 + 3x - 2 = 0$
Horizontal Asymptote	Compare degrees of numerator and denominator	Degrees equal: $y = \frac{2}{2} = 1$

Additional info:

• Some questions involve interpreting graphs and matching them to derivatives or integrals.

• Piecewise functions require checking continuity at the points where the definition changes.

• Implicit differentiation and logarithmic differentiation are useful for complex functions.