Analytical Applications of Differentiation

What you'll learn

This guide covers the analytical applications of differentiation tested in AP Calculus AB. You will learn to use derivatives to analyse function behaviour, locate and classify critical points, solve optimisation problems, and work with related rates. These techniques form the foundation of calculus applications in economics, physics, and engineering.

Key terms and definitions

Critical point — A point where f'(x) = 0 or f'(x) does not exist; potential location of local extrema.

First derivative test — A method using the sign of f'(x) before and after a critical point to classify it as a local maximum, local minimum, or neither.

Second derivative test — A method using f''(c) to classify a critical point: if f''(c) > 0, local minimum; if f''(c) < 0, local maximum; if f''(c) = 0, test is inconclusive.

Inflection point — A point where the concavity of a function changes; occurs where f''(x) = 0 or f''(x) does not exist, provided concavity actually changes.

Concavity — The direction a curve bends: concave up when f''(x) > 0 (curve bends upward like ∪); concave down when f''(x) < 0 (curve bends downward like ∩).

Absolute extremum — The largest (absolute maximum) or smallest (absolute minimum) value of a function over its entire domain or a specified interval.

Optimisation — The process of finding maximum or minimum values of a function, typically subject to constraints.

Related rates — Problems involving rates of change of two or more related variables with respect to time.

Core concepts

Intervals of increase and decrease

The first derivative determines where a function is increasing or decreasing.

Sign analysis of f'(x):

• When f'(x) > 0 on an interval, f is increasing on that interval
• When f'(x) < 0 on an interval, f is decreasing on that interval
• When f'(x) = 0, the function has zero rate of change at that point

Process for finding intervals:

1. Find f'(x)
2. Locate critical points by solving f'(x) = 0 and finding where f'(x) is undefined
3. Create a sign chart using test points in each interval
4. State intervals using correct interval notation

Important note: Express your answer as intervals, not individual points. Write "increasing on (-∞, 2)" not "increasing at x = 1".

Classifying critical points using the First Derivative Test

The First Derivative Test examines how f'(x) changes sign around a critical point c.

Classification rules:

• If f'(x) changes from positive to negative at c: local maximum at x = c
• If f'(x) changes from negative to positive at c: local minimum at x = c
• If f'(x) does not change sign at c: neither maximum nor minimum

Steps:

1. Find all critical points
2. Test the sign of f'(x) immediately left and right of each critical point
3. Apply classification rules
4. State both the x-coordinate and y-coordinate of extrema when asked

The First Derivative Test always works when applicable, unlike the Second Derivative Test which can be inconclusive.

Concavity and the Second Derivative Test

The second derivative reveals the concavity of a function and provides an alternative method for classifying critical points.

Concavity determination:

• f''(x) > 0 → concave up on that interval
• f''(x) < 0 → concave down on that interval

Finding inflection points:

1. Calculate f''(x)
2. Solve f''(x) = 0 and find where f''(x) is undefined
3. Test concavity on either side of these points
4. Confirm concavity actually changes (otherwise not an inflection point)
5. State the coordinates (x, f(x)) of inflection points

Second Derivative Test for critical points:

When f'(c) = 0:

• If f''(c) > 0, then f has a local minimum at x = c
• If f''(c) < 0, then f has a local maximum at x = c
• If f''(c) = 0, the test is inconclusive; use First Derivative Test instead

Absolute extrema on closed intervals

The Extreme Value Theorem guarantees that a continuous function on a closed interval [a, b] attains both an absolute maximum and absolute minimum.

Closed interval method:

1. Find all critical points in the open interval (a, b)
2. Evaluate f(x) at each critical point
3. Evaluate f(a) and f(b) (the endpoints)
4. Compare all values; the largest is the absolute maximum, the smallest is the absolute minimum
5. State both the value and the location

Key distinction: An absolute extremum is the actual y-value, while its location is the x-value. Be precise about which you're reporting.

Optimisation problems

Optimisation uses calculus to find maximum or minimum values in real-world contexts.

Standard approach:

1. Identify the quantity to be optimised (this becomes your objective function)
2. Draw a diagram if the problem involves geometry
3. Define variables clearly
4. Write the objective function in terms of one or more variables
5. Use constraints to express the objective function as a function of a single variable
6. Find the domain (physical constraints often limit the domain)
7. Differentiate and find critical points
8. Test critical points and endpoints using appropriate methods
9. Answer the question with correct units and context

Common contexts in AP Calculus AB:

• Maximising area or volume with fixed perimeter or surface area
• Minimising cost or material
• Maximising profit or revenue
• Shortest distance problems
• Optimising dimensions of containers or enclosures

Always check that your answer makes physical sense in the problem context.

Related rates

Related rates problems involve multiple variables changing with respect to time, connected by an equation.

Solution strategy:

1. Draw and label a diagram showing the situation
2. Identify given rates (what rates are you told?) and the unknown rate (what are you asked to find?)
3. Write an equation relating the variables (not the rates)
4. Differentiate implicitly with respect to time t, applying the chain rule to every variable
5. Substitute known values (rates, then specific values of variables)
6. Solve for the unknown rate
7. Include correct units in your final answer

Critical technique: Differentiate the relationship equation before substituting specific values. Substituting too early is the most common error in related rates.

Typical scenarios:

• Rates involving geometric shapes (expanding circles, filling cones)
• Ladder problems (sliding ladders forming right triangles)
• Shadow problems
• Rates involving Pythagorean relationships

Worked examples

Example 1: Critical points and function analysis

Question: Let f(x) = x³ - 6x² + 9x + 1.

(a) Find all critical points of f. [2 marks]

(b) Determine the intervals where f is increasing and decreasing. [3 marks]

(c) Classify each critical point as a local maximum, local minimum, or neither. [2 marks]

Solution:

(a) Find f'(x): f'(x) = 3x² - 12x + 9 = 3(x² - 4x + 3) = 3(x - 1)(x - 3)

Set f'(x) = 0: 3(x - 1)(x - 3) = 0 x = 1 or x = 3

Critical points: x = 1 and x = 3 ✓ [2 marks]

(b) Test intervals using sign chart:

• For x < 1 (test x = 0): f'(0) = 9 > 0, so increasing
• For 1 < x < 3 (test x = 2): f'(2) = -3 < 0, so decreasing
• For x > 3 (test x = 4): f'(4) = 9 > 0, so increasing

Increasing on (-∞, 1) and (3, ∞) ✓ Decreasing on (1, 3) ✓ [3 marks]

(c) Using First Derivative Test:

• At x = 1: f' changes from positive to negative → local maximum ✓
• At x = 3: f' changes from negative to positive → local minimum ✓ [2 marks]

Example 2: Optimisation

Question: A farmer has 200 metres of fencing to enclose a rectangular field and divide it into two equal parts with a fence parallel to one side. Find the dimensions that maximise the total enclosed area. [7 marks]

Solution:

Let length = x metres and width = y metres, with the dividing fence parallel to the width.

Constraint equation (total fencing): 2x + 3y = 200 ✓ [1 mark]

Therefore: y = (200 - 2x)/3

Objective function (area): A = xy = x · (200 - 2x)/3 = (200x - 2x²)/3 ✓ [1 mark]

Domain: 0 < x < 100 (from physical constraints)

Differentiate: A'(x) = (200 - 4x)/3 ✓ [1 mark]

Find critical points: (200 - 4x)/3 = 0 200 - 4x = 0 x = 50 ✓ [1 mark]

Using second derivative test: A''(x) = -4/3 < 0, confirming maximum ✓ [1 mark]

When x = 50: y = (200 - 100)/3 = 100/3 ≈ 33.33 metres

Dimensions: length = 50 m, width = 33⅓ m ✓ [2 marks]

Example 3: Related rates

Question: A spherical balloon is being inflated. When the radius is 5 cm, the radius is increasing at a rate of 2 cm/s. How fast is the volume increasing at that instant? [Volume of sphere: V = (4/3)πr³] [5 marks]

Solution:

Given information:

• r = 5 cm (at the instant in question)
• dr/dt = 2 cm/s
• Find: dV/dt

Volume equation: V = (4/3)πr³ ✓ [1 mark]

Differentiate with respect to t: dV/dt = (4/3)π · 3r² · dr/dt = 4πr² · dr/dt ✓ [2 marks]

Substitute known values: dV/dt = 4π(5)² · 2 = 4π(25) · 2 = 200π ✓ [1 mark]

The volume is increasing at 200π cm³/s (or approximately 628.3 cm³/s) ✓ [1 mark]

Common mistakes and how to avoid them

• Confusing critical points with extrema. Not every critical point is a maximum or minimum. Always test using the First or Second Derivative Test, or check the sign of f'(x) around the point.

• Stating intervals incorrectly. Write "increasing on (2, 5)" not "increasing at x = 2 and x = 5". Use interval notation and understand that a function increases on an interval, not at isolated points.

• Forgetting to check endpoints in closed interval problems. The Extreme Value Theorem requires testing both critical points AND endpoints. Absolute extrema often occur at endpoints.

• Substituting values before differentiating in related rates. In related rates problems, write the relationship equation, differentiate it completely with respect to time, then substitute the given values. Substituting too early eliminates the variables you need.

• Misidentifying inflection points. A point where f''(x) = 0 is not automatically an inflection point. You must verify that concavity actually changes on either side of that point.

• Forgetting units in applied problems. Optimisation and related rates problems require units in the final answer. State units clearly and ensure they match the question.

Exam technique for Analytical Applications of Differentiation

• Show full working systematically. AP Calculus AB awards partial credit. Write each step: derivative, setting equal to zero, solving, testing, and conclusion. Don't skip steps even if you can do them mentally.

• Use correct notation. Write f'(x) > 0, not just "derivative positive." Use proper inequality and interval notation. State whether points are included or excluded using brackets correctly.

• Answer the actual question asked. If asked for coordinates, give (x, y). If asked for intervals, give intervals. If asked for a rate, include appropriate units and sign (positive/negative matters).

• Justify classifications explicitly. When classifying critical points, state your reasoning: "f' changes from positive to negative, therefore local maximum" or "f''(c) < 0, therefore local maximum."

Quick revision summary

Use the first derivative to find intervals of increase/decrease and to classify critical points via sign changes. The second derivative reveals concavity and provides an alternative test for extrema. Absolute extrema on closed intervals require testing critical points and endpoints. Optimisation problems need a clear objective function, constraints, and domain consideration. Related rates demand implicit differentiation with respect to time before substituting specific values. Always verify answers make sense in context and include appropriate units.