Applications of Derivatives

Related Rates

1. Identify all variables and given rates.
2. Write an equation relating the variables.
3. Differentiate implicitly with respect to time $t$.
4. Substitute known values and solve.

Optimization

1. Define the quantity to maximize/minimize.
2. Write it as a function of one variable (using constraints).
3. Find critical points ($f'(x) = 0$ or DNE).
4. Verify max/min with second derivative test or endpoint analysis.

Curve Sketching

• $f'(x) > 0$: $f$ is increasing.
• $f'(x) < 0$: $f$ is decreasing.
• $f''(x) > 0$: concave up.
• $f''(x) < 0$: concave down.
• Inflection point: where $f''$ changes sign.

Mean Value Theorem (MVT)

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$: $f'(c) = \frac{f(b) - f(a)}{b - a}$ for some $c \in (a,b)$.

L'Hôpital's Rule

For indeterminate forms $0/0$ or $\infty/\infty$: $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$

Linearization

$L(x) = f(a) + f'(a)(x - a)$

Problem: A spherical balloon is inflated at $2\ \text{cm}^3/\text{s}$. How fast is the radius increasing when $r = 5\ \text{cm}$?

Solution:

$V = \frac{4}{3}\pi r^3$. Differentiate: $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$.

$2 = 4\pi(25)\frac{dr}{dt}$

$\frac{dr}{dt} = \frac{2}{100\pi} = \frac{1}{50\pi} \approx 0.00637\ \text{cm/s}$

• Related rates: differentiate equations with respect to time.
• Optimization: find critical points and check endpoints.
• MVT guarantees an instantaneous rate equals the average rate.
• L'Hôpital's Rule handles $0/0$ and $\infty/\infty$ indeterminate forms.