Calculus Study Guide: Continuity, Limits, and Derivatives

2.5 Continuity

Definition and Identification of Continuity

Continuity is a fundamental concept in calculus describing when a function does not have any abrupt changes, jumps, or holes at a given point.

• Removable Discontinuity: A function has a removable discontinuity at $x = a$ if the limit exists but does not equal the function value. This can often be 'fixed' by redefining the function at $x = a$.

• Nonremovable Discontinuity: Occurs when the limit does not exist at $x = a$ (e.g., jump or infinite discontinuity).

• Left/Right Continuity: A function is left/right continuous at $x = a$ if the left/right-hand limits exist and equal the function value.

• Combining Continuous Functions: The sum, difference, product, and quotient (where denominator is nonzero) of continuous functions are also continuous.

• Intermediate Value Theorem (IVT): If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c ext{ in } (a, b)$ such that $f(c) = N$.

Example

Consider $f(x) = \frac{x^2 - 1}{x - 1}$ for $x \neq 1$. At $x = 1$, $f(x)$ has a removable discontinuity because $\lim_{x \to 1} f(x) = 2$, but $f(1)$ is undefined.

2.6 Limits at Infinity; Horizontal Asymptotes

Evaluating Limits at Infinity

Limits at infinity describe the behavior of a function as $x$ approaches $\infty$ or $-\infty$.

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

• Indeterminate Forms: Limits such as $\frac{\infty}{\infty}$ or $\frac{0}{0}$ require algebraic manipulation (e.g., factoring, rationalizing).

• Increasing/Decreasing Without Bound: If $f(x)$ grows arbitrarily large as $x \to \infty$, $\lim_{x \to \infty} f(x) = \infty$.

• Limits of the Form $\infty - \infty$: These are indeterminate and require further analysis.

• Applications: Limits at infinity are used to analyze end behavior and asymptotic properties of functions.

Example

For $f(x) = \frac{2x^2 + 3}{x^2 - 1}$, $\lim_{x \to \infty} f(x) = 2$; thus, $y = 2$ is a horizontal asymptote.

2.7 Derivatives and Rates of Change

Average and Instantaneous Rate of Change

Derivatives measure how a function changes as its input changes, representing rates of change.

• Average Rate of Change: Given by the slope of the secant line: $\frac{f(b) - f(a)}{b - a}$.

• Instantaneous Rate of Change: The derivative at a point, $f'(a)$, is the slope of the tangent line at $x = a$.

• Applications: Derivatives are used to model velocity, growth rates, and other changing quantities.

Example

If $f(x) = x^2$, then the average rate of change from $x = 1$ to $x = 3$ is $\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4$.

2.8 The Derivative as a Function

Graphical Interpretation and Applications

The derivative function $f'(x)$ gives the rate of change of $f(x)$ at every point $x$.

• Sketching the Graph: The graph of $f'(x)$ can be sketched using the slopes of tangent lines to $f(x)$.

• Finding the Derivative Function: If $f(x)$ is differentiable, $f'(x)$ can be found using differentiation rules.

Example

If $f(x) = x^3$, then $f'(x) = 3x^2$.

3.1 Derivatives of Powers and Exponentials

Differentiation Rules and Physical Meaning

Several rules simplify the process of differentiation for common functions.

• Constant Rule: $\frac{d}{dx}c = 0$

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

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

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

• Exponentials: $\frac{d}{dx}e^x = e^x$

• Physical Meaning: Derivatives represent velocity, acceleration, and jerk in motion.

• Tangent and Normal Lines: The tangent line at $x = a$ has slope $f'(a)$; the normal line is perpendicular.

Example

If $s(t) = t^3$, then velocity $v(t) = s'(t) = 3t^2$.

3.2 The Product and Quotient Rules

Rules for Differentiating Products and Quotients

Special rules are used when differentiating products or quotients of functions.

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

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

• Applications: Used in physics, economics, and engineering when functions are multiplied or divided.

Example

If $f(x) = x^2$ and $g(x) = \sin x$, then $\frac{d}{dx}[x^2 \sin x] = 2x \sin x + x^2 \cos x$.

3.4 The Chain Rule

Differentiating Composite Functions

The Chain Rule is used to differentiate compositions of functions.

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

• Combining with Other Rules: The Chain Rule can be used with the Product and Quotient Rules.

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

Example

If $f(x) = \sin(x^2)$, then $f'(x) = \cos(x^2) \cdot 2x$.

3.5 Implicit Differentiation

Differentiating Implicitly Defined Functions

Implicit differentiation is used when functions are defined by equations rather than explicit formulas.

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

• Differentiating $y^n$: Use the Chain Rule: $\frac{d}{dx}y^n = n y^{n-1} \frac{dy}{dx}$

• Differentiating $|x|$: $\frac{d}{dx}|x| = \frac{x}{|x|}$ for $x \neq 0$

Example

Given $x^2 + y^2 = 1$, differentiating both sides gives $2x + 2y \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{x}{y}$.