Relationships: v(t) = s'(t)

Velocity is the derivative of position: $v(t) = s'(t) = \frac{ds}{dt}$

Relationships: a(t) = v'(t) = s''(t)

Acceleration is the derivative of velocity, second derivative of position: $a(t) = v'(t) = s''(t) = \frac{d^2s}{dt^2}$

Problems where multiple variables change with respect to time, all related through an equation.

1. Draw diagram and label variables
2. Write equation relating variables
3. Differentiate both sides with respect to time t
4. Substitute known values
5. Solve for desired rate

All variables are functions of time, so differentiate with respect to t: $\frac{d}{dt}(x^2 + y^2) = 2x\frac{dx}{dt} + 2y\frac{dy}{dt}$

Related rates problems involve multiple quantities changing with respect to time, all related through an equation.

Common applications:

• Biology: Population growth, drug concentration
• Economics: Cost, revenue, profit optimization
• Physics: Distance, area, volume changes

Linear Approximation Formula: L(x) = f(a) + f'(a)(x - a)

Tangent line approximation near x = a: $L(x) = f(a) + f'(a)(x - a)$

Example: Approximate $\sqrt{4.1}$ Let f(x) = $\sqrt{x}$, a = 4 $f(4) = 2$, $f'(x) = \frac{1}{2\sqrt{x}}$, $f'(4) = \frac{1}{4}$ $L(4.1) = 2 + \frac{1}{4}(0.1) = 2.025$

L'Hôpital's Rule Statement and Application

For $\frac{0}{0}$ or $\frac{\infty}{\infty}$: $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$

Other indeterminate forms:

• $0 \cdot \infty$: Rewrite as fraction
• $\infty - \infty$: Combine fractions
• $0^0, 1^\infty, \infty^0$: Take ln first