• Position s(t): Location relative to origin
• Velocity v(t): $v(t) = s'(t)$ = instantaneous rate of change of position
• Acceleration a(t): $a(t) = v'(t) = s''(t)$ = instantaneous rate of change of velocity
• Speed: $|v(t)|$ (absolute value of velocity)

• Positive velocity: moving in positive direction (right/up)
• Negative velocity: moving in negative direction (left/down)
• Speeding up: velocity and acceleration have same sign
• Slowing down: velocity and acceleration have opposite signs

Strategy:

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

Example: For a circle with radius r and area A: $\frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt}$

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

Linearization formula: $L(x) = f(a) + f'(a)(x-a)$

Used to approximate function values near x = a.

Differential approximation: $\Delta y \approx dy = f'(x) \cdot \Delta x$

• Overestimate: Linear approximation is greater than actual value
• Underestimate: Linear approximation is less than actual value
• For concave up functions: tangent line underestimates
• For concave down functions: tangent line overestimates

L'Hôpital's Rule

Other indeterminate forms (must convert to 0/0 or ∞/∞):

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

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

0/0 or Infinity/Infinity (Applicable Conditions)

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