Applications of Derivatives — AP Calculus AB
1. Tangent and Normal Lines ★★☆☆☆ ⏱ 3 min
Using point-slope form for linear equations, the equations are:
y - f(a) = f'(a)(x - a)
y - f(a) = -\frac{1}{f'(a)}(x - a) \quad (f'(a) \neq 0)
If $f'(a) = 0$, the tangent line is horizontal ($y = f(a)$) and the normal line is vertical ($x = a$), since the reciprocal of 0 is undefined.
Exam tip: Examiners almost always expect you to simplify line equations to slope-intercept form ($y = mx + b$) for full credit.
2. Linear Approximation ★★☆☆☆ ⏱ 3 min
For values of $x$ very close to $x=a$, the tangent line to $f(x)$ at $a$ is an accurate approximation of the function itself, since the curve changes very little over a small interval. This method lets you estimate values of complex functions without a calculator, using a nearby point where $f(a)$ and $f'(a)$ are easy to compute by hand.
f(x) \approx L(x) = f(a) + f'(a)(x - a)
3. Related Rates ★★★☆☆ ⏱ 4 min
Related rates problems involve two or more quantities that change over time, linked by a known geometric or algebraic relationship. You will be given the rate of change of one quantity and asked to find the rate of change of the other, using implicit differentiation with respect to time $t$.
- Define all variables, write down known rates (with units) and the unknown rate you need to find.
- Write the constraint equation linking the variables (e.g. Pythagorean theorem, volume of a sphere).
- Differentiate both sides of the constraint with respect to $t$, using the chain rule for all time-dependent variables.
- Plug in known values, solve for the unknown rate, and add units. Positive values mean the quantity is increasing; negative values mean it is decreasing.
Exam tip: Always differentiate the full constraint equation before plugging in numerical values. Plugging in early removes the time dependence of changing variables and leads to incorrect results.
4. Optimization (Extrema Problems) ★★★☆☆ ⏱ 4 min
Optimization problems ask you to find the global maximum or minimum value of a function (called the objective function) subject to one or more constraints that limit input values. These are common in free-response questions, often with real-world contexts like minimizing cost or maximizing area.
- Define variables, write the objective function and the constraint equation.
- Use the constraint to rewrite the objective as a function of a single variable, and note the valid domain.
- Find critical points by setting the first derivative equal to 0, or identify points where the derivative is undefined.
- Test critical points and domain endpoints to confirm the global maximum/minimum, using the first or second derivative test.
- State your final answer with units, making sure you answer the exact question asked.
Exam tip: Examiners deduct 1–2 points for missing justification of extrema. Always explicitly state which test you used to confirm your result.
5. Curve Sketching with $f'$ and $f''$ ★★★☆☆ ⏱ 4 min
You can sketch the full shape of a function without plotting dozens of points by analyzing its first and second derivatives to identify key features: increasing/decreasing intervals, local extrema, concavity, and inflection points.
- **First derivative rules**: If $f'(x) > 0$ on an interval, $f(x)$ is increasing; if $f'(x) < 0$, $f(x)$ is decreasing. Local extrema occur where $f'(x)$ changes sign.
- **Second derivative rules**: If $f''(x) > 0$ on an interval, $f(x)$ is concave up ($\cup$); if $f''(x) < 0$, $f(x)$ is concave down ($\cap$). Inflection points occur where $f''(x)$ changes sign.
6. Concept Check
Common Pitfalls
Why: Students mix up tangent/normal definitions and forget division by zero rules.
Why: Students want to simplify early, but this removes the time dependence of changing variables.
Why: Students assume the only critical point is the desired extrema without proof.
Why: Students memorize the inflection point condition without the critical sign change requirement.
Why: Students prioritize easy computation over approximation accuracy.