Calculus I & II Review: Differentiation, Integration, and Area Between Curves

Review of Differentiation

Definition of the Derivative

The derivative of a function measures its instantaneous rate of change or the slope of the tangent line at any point. For a function $y = f(x)$, the derivative is defined as:

• Notation: $f'(x)$, $\frac{dy}{dx}$

• Limit Definition:

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

• Applications: Used to calculate instantaneous rates of change and slopes of tangent lines.

Review of Derivative Rules

Common differentiation rules are essential for finding derivatives efficiently. Below is a summary of key rules:

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

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

• Trigonometric, Exponential, and Logarithmic Derivatives: See table below.

Function	Derivative
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$
$a^x$	$a^x \ln a$
$\log_a x$	$\frac{1}{x \ln a}$
$\sinh x$	$\cosh x$
$\cosh x$	$\sinh x$

Worked Examples: Differentiation

Practice problems reinforce the application of differentiation rules. For example:

• Example: Find the derivative of $f(x) = x^2 \sin x$.

• Solution: Use the product rule:

$f'(x) = 2x \sin x + x^2 \cos x$

• Example: Find the derivative of $g(x) = \frac{\ln x}{x^2}$.

• Solution: Use the quotient rule:

$g'(x) = \frac{\frac{1}{x} \cdot x^2 - \ln x \cdot 2x}{x^4} = \frac{x - 2x \ln x}{x^4}$

Review of Integration

Definition of the Definite Integral

The definite integral of a function $f(x)$ from $a$ to $b$ represents the signed area under the curve between $x = a$ and $x = b$:

$\int_a^b f(x)\,dx$

• Limit Definition: $\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x$

• Interpretation: Area under the curve, total change in the antiderivative.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration:

$\int_a^b f(x)\,dx = F(b) - F(a)$

where $F(x)$ is any antiderivative of $f(x)$.

Review of Basic Integration

Common integration rules include:

• Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, $n \neq -1$

• Exponential: $\int e^{ax} dx = \frac{1}{a}e^{ax} + C$

• Trigonometric: $\int \sin x dx = -\cos x + C$, $\int \cos x dx = \sin x + C$

Worked Examples: Integration

• Example: $\int (3x^2 + 5x) dx = x^3 + \frac{5}{2}x^2 + C$

• Example: $\int \frac{\sin x}{x^2} dx$ (requires advanced techniques or series expansion)

Integration by Substitution (u-Substitution)

u-Substitution Method

u-Substitution is a technique for evaluating integrals by substituting part of the integrand with a new variable $u$ to simplify the integral.

• Steps:

  1. Let $u = g(x)$, compute $\frac{du}{dx} = g'(x)$

  2. Rewrite the integral in terms of $u$ and $du$

  3. Integrate with respect to $u$

  4. Substitute back $x$ in terms of $u$

• Example: $\int 2x \cos(x^2) dx$

  • Let $u = x^2$, $du = 2x dx$

  • Integral becomes $\int \cos(u) du = \sin(u) + C = \sin(x^2) + C$

Applications: Area Between Curves

Area Between Two Curves (in terms of x)

The area between two curves $f(x)$ and $g(x)$ from $x = a$ to $x = b$ is given by:

$\text{Area} = \int_a^b [f(x) - g(x)] dx$

• Draw a typical rectangle representing a small area $[f(x) - g(x)]dx$

• Sum all rectangles from $a$ to $b$ using integration

Area Between Curves (in terms of y)

If the functions are given as $x = f(y)$ and $x = g(y)$, the area between curves from $y = c$ to $y = d$ is:

$\text{Area} = \int_c^d [f(y) - g(y)] dy$

• Draw a typical rectangle with width $[f(y) - g(y)]$ and height $dy$

• Sum all rectangles from $c$ to $d$

Keys to Solving Area Problems

• Draw a good picture of the region

• Include a typical 'slice' of area

• Find endpoints using algebra (solve for intersection points)

Additional info: These notes cover core Calculus I & II topics, including differentiation, integration, and applications to area, with worked examples and visual aids for conceptual understanding.