Calculus Exam 1 & 2 Review: Limits, Differentiation, and Exponential/Logarithmic Functions

Limits and Continuity

/ Numerical, Algebraic, and Graphical Approaches to Limits

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

• Numerical Approach: Evaluating limits by substituting values increasingly close to the point of interest.

• Algebraic Approach: Manipulating expressions to simplify and find limits analytically.

• Graphical Approach: Observing the behavior of a function's graph near the point of interest.

• Continuity: A function is continuous at a point if the limit exists and equals the function's value at that point.

Key Formula: $\lim_{x \to a} f(x) = L$ Example: Find $\lim_{x \to 2} (x^2 - 4)/(x - 2)$.

• Algebraic simplification: $(x^2 - 4)/(x - 2) = ((x - 2)(x + 2))/(x - 2) = x + 2$ for $x \neq 2$.

• Thus, $\lim_{x \to 2} (x^2 - 4)/(x - 2) = 4$.

Average Rates of Change

Definition and Application

The average rate of change of a function over an interval quantifies how the function's output changes per unit input.

• Definition: For a function $f(x)$ over $[a, b]$, the average rate of change is:

$\frac{f(b) - f(a)}{b - a}$

• Application: Used to approximate instantaneous rates and analyze trends.

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

Differentiation Using Limits of Difference Quotients

Definition of the Derivative

The derivative measures the instantaneous rate of change of a function. It is defined as the limit of the difference quotient.

• Difference Quotient: $\frac{f(x + h) - f(x)}{h}$

• Derivative Definition:

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

• Interpretation: Slope of the tangent line to the curve at point $x$.

Example: For $f(x) = x^2$: $\lim_{h \to 0} \frac{(x + h)^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} = 2x$

Leibniz Notation and Differentiation Rules

Leibniz Notation, Power Rule, and Sum-Difference Rule

Leibniz notation expresses derivatives as $\frac{dy}{dx}$, facilitating clear communication of differentiation.

• Power Rule: For $f(x) = x^n$, $\frac{d}{dx}x^n = nx^{n-1}$

• Sum-Difference Rule: The derivative of a sum/difference is the sum/difference of the derivatives.

$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$ Example: $\frac{d}{dx}(x^3 + 2x) = 3x^2 + 2$

Product and Quotient Rules

Rules for Differentiating Products and Quotients

When differentiating products or quotients of functions, specific rules apply.

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

The Chain Rule

Differentiating Composite Functions

The chain rule is used to differentiate functions composed of other functions.

• Chain Rule: If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

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

Higher Order Derivatives

Second and Higher Derivatives

Higher order derivatives represent the rate of change of previous derivatives, such as acceleration (second derivative of position).

• Notation: $f''(x)$ for the second derivative, $f^{(n)}(x)$ for the $n$th derivative.

• Application: Used in physics, engineering, and mathematical modeling.

Example: For $f(x) = x^3$, $f'(x) = 3x^2$, $f''(x) = 6x$

Exponential and Logarithmic Functions

Properties and Definitions

Exponential and logarithmic functions are essential in modeling growth, decay, and many natural phenomena.

• Exponential Function: $f(x) = a^x$

• Natural Exponential Function: $f(x) = e^x$

• Logarithmic Function: $f(x) = \log_a x$

• Natural Logarithm: $f(x) = \ln x$

Example: $e^{2x}$, $\ln(x^2)$

Derivatives of Exponential and Logarithmic Functions

Base-e Exponential and Natural Logarithm Derivatives

Special rules apply for differentiating exponential and logarithmic functions, especially those with base $e$.

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

• Derivative of $a^x$: $\frac{d}{dx}a^x = a^x \ln a$

• Derivative of $\ln x$: $\frac{d}{dx}\ln x = \frac{1}{x}$

• Derivative of $\log_a x$: $\frac{d}{dx}\log_a x = \frac{1}{x \ln a}$

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

Summary Table: Differentiation Rules

Rule	Formula	Example
Power Rule	$\frac{d}{dx}x^n = nx^{n-1}$	$\frac{d}{dx}x^3 = 3x^2$
Sum-Difference Rule	$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$	$\frac{d}{dx}(x^2 + \sin x) = 2x + \cos x$
Product Rule	$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$	$\frac{d}{dx}(x^2 \sin x) = 2x \sin x + x^2 \cos x$
Quotient Rule	$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$	$\frac{d}{dx}\left[\frac{x^2}{\sin x}\right] = \frac{2x \sin x - x^2 \cos x}{\sin^2 x}$
Chain Rule	$\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$	$\frac{d}{dx}\sin(x^2) = \cos(x^2) \cdot 2x$
Exponential Rule	$\frac{d}{dx}e^x = e^x$	$\frac{d}{dx}e^{2x} = 2e^{2x}$
Logarithmic Rule	$\frac{d}{dx}\ln x = \frac{1}{x}$	$\frac{d}{dx}\ln(x^2) = \frac{2}{x}$

Additional info: Some topic headings were fragmented or repeated; they have been logi/Ically grouped and expanded for clarity. The summary table was inferred to provide a concise reference for differentiation rules.