Rucete ✏ AP Calculus In a Nutshell
4. Applications of Differential Calculus
This chapter explains how derivatives can be used to analyze slope, motion, extrema, inflection points, concavity, and related rates. It also covers parametric and polar slopes, vector motion, and optimization.
Critical Points and Slope
• A critical point is any c where f′(c) = 0 or f′(c) is undefined.
• The slope of a curve at P(x₁, y₁) is f′(x₁), the derivative at that point.
Example: f(x) = 4x³ – 6x² – 8 → f′(x) = 12x² – 12x → critical points: x = 0 and x = 1
Average and Instantaneous Rate of Change
• Average rate over [a, a + h] = [f(a + h) – f(a)] / h
• Instantaneous rate = f′(a)
• Used to find velocity, growth, profit rate, etc.
Example: G(t) = 400(15 – t)² → average rate from 0 to 5 = –10,000 gal/min → G′(5) = –8000 gal/min (instantaneous)
Tangents and Slopes
• Tangent line at x = a: y – f(a) = f′(a)(x – a)
• Horizontal tangent: f′(x) = 0 • Vertical tangent: f′(x) undefined
Example: f(x) = x³ – 3x² at (1, –2) → f′(1) = –3 → tangent: y + 2 = –3(x – 1)
Increasing and Decreasing Functions
• f is increasing when f′(x) > 0 • f is decreasing when f′(x) < 0
Example: f(x) = x⁴ – 4x³ → f′(x) = 4x³ – 12x² → critical points: x = 0, x = 3 → f is decreasing on (–∞, 3], increasing on [3, ∞)
Local Extrema and Concavity
• f has local max at x = c if f changes from increasing to decreasing at c • f has local min at x = c if f changes from decreasing to increasing at c
• Second Derivative Test: – f′(c) = 0 and f″(c) > 0 → local min – f′(c) = 0 and f″(c) < 0 → local max – f″(c) = 0 → test fails
• Inflection point at x = c if f″ changes sign at c
Example: f(x) = x³ – 5x² + 3x + 6 → → local max at x = 1, local min at x = 3, inflection at x = 5/3
Global Max/Min (Candidates Test)
• For closed interval [a, b]: – Find all critical points where f′(x) = 0 – Evaluate f at endpoints and critical points – Largest is global max, smallest is global min
Example: f(x) = 2x³ – 3x² – 12x on [–2, 3] → Critical points: x = –1, 2 → Evaluate at –2, –1, 2, 3 → global max = 7 at x = –1, min = –20 at x = 2
Curve Sketching
• Use f′ and f″ to analyze: – Increasing/decreasing intervals – Concavity and inflection points – Local extrema (via 1st or 2nd derivative test)
• Key points to plot: – x-intercepts – y-intercepts – critical points – inflection points – asymptotes (if any)
Optimization
• Strategy:
1. Identify quantity to maximize/minimize
2. Express it as a function of one variable
3. Determine domain (interval)
4. Find critical points (set derivative = 0)
5. Use Candidates Test or logic to find max/min
Example: Find x and y such that a rectangle of perimeter 100 has maximum area. → Max area occurs when x = y = 25 → square
Motion and Particle Problems
• Given s(t) (position), then:
– v(t) = s′(t) → velocity
– a(t) = v′(t) = s″(t) → acceleration
• A particle changes direction when v(t) = 0 and sign changes
• Speed = |v(t)| – Speed increases if v and a have same sign – Speed decreases if v and a have opposite sign
Related Rates
• When multiple quantities change with respect to time:
1. Identify variables and rate quantities (dx/dt, dy/dt, etc.)
2. Write an equation relating the variables
3. Differentiate both sides with respect to time (t)
4. Plug in known values and solve for the unknown rate
Example: A ladder 10 ft long is sliding down a wall. If bottom is moving away at 3 ft/s, how fast is the top sliding down when bottom is 6 ft away? → Use Pythagoras and implicit differentiation
Tangent Line Approximation (Linearization)
• Use tangent line to approximate f(x) near a
• Formula:
L(x) = f(a) + f′(a)(x – a)
• Works well when x is close to a
Example: Approximate √(25.5) using f(x) = √x at a = 25 → L(x) = √25 + (1/2√25)(x – 25) = 5 + (1/10)(0.5) = 5.05
Derivatives of Parametric and Polar (BC)
Parametric: x = f(t), y = g(t)
• First derivative: dy/dx = (dy/dt) / (dx/dt)
• Second derivative: d²y/dx² = [d/dt(dy/dx)] / (dx/dt)
Polar: r = f(θ)
• Use x = r cos(θ), y = r sin(θ)
• dy/dx = (dy/dθ) / (dx/dθ)
In a Nutshell
Derivatives allow us to analyze real-world behavior: motion, shape, rates, and optimization. Applications include sketching curves, solving physics problems, and maximizing/minimizing outcomes. With related rates, tangent approximations, and parametric/polar derivatives (BC), calculus becomes a toolkit for interpreting change in every context.