Optimization & Related Rates Guide | DerivativeCalculus.com

📈 Optimization Problems: Finding Maximum/Minimum

1

Read & Understand

Carefully read the problem. Identify what needs to be maximized or minimized. Draw a diagram if possible.

Tip:

Underline key quantities and constraints.

2

Define Variables

Assign variables to all relevant quantities. Let x be the quantity you can control.

Tip:

Use descriptive variable names (length = L, width = W, etc.)

3

Write Equations

Write the primary equation (what to optimize) and constraint equations (given conditions).

Example:

Area = L × W (optimize), Perimeter = 2L + 2W = 100 (constraint)

4

Express in One Variable

Use constraint equations to express the primary equation in terms of a single variable.

From example:

If P = 2L + 2W = 100, then W = 50 - L

5

Find Derivative

Differentiate the primary equation with respect to your variable.

Remember:

You're finding where derivative = 0 (critical points)

6

Solve & Verify

Solve derivative = 0 for critical points. Check endpoints and use second derivative test if needed.

Test:

f''(x) > 0 → minimum, f''(x) < 0 → maximum

📐 Example: Maximum Area with Fixed Perimeter Optimization

A farmer has 100 meters of fencing to enclose a rectangular plot along a river (no fencing needed along river). What dimensions maximize the area?

Variables:

L = length (parallel to river)

W = width (perpendicular to river)

A = area to maximize

1

Primary Equation: Area = L × W

2

Constraint: Fencing used: L + 2W = 100 (only three sides needed)

3

Express in one variable: From constraint: L = 100 - 2W
Substitute: A(W) = (100 - 2W) × W = 100W - 2W²

4

Derivative: A'(W) = 100 - 4W

5

Find critical point: Set A'(W) = 0

100 - 4W = 0 \to W = 25 meters

6

Find other dimension: L = 100 - 2(25) = 50 meters

7

Verify maximum: A''(W) = -4 < 0 → maximum

Maximum Area: 50m × 25m = 1250 m²

⚡ Related Rates: How Quantities Change Together

1

Read & Diagram

Identify all quantities and how they're related. Draw a diagram showing relationships.

Tip:

Label all known and unknown rates of change.

2

Identify Rates

Determine what rate is given and what rate you need to find.

Notation:

Use dx/dt, dy/dt, etc. for rates of change.

3

Find Relationship

Write an equation relating the quantities (geometry, trigonometry, etc.)

Common:

Pythagorean theorem, similar triangles, volume formulas.

4

Differentiate Implicitly

Differentiate both sides with respect to time t.

Remember:

Use chain rule: d/dt[f(x)] = f'(x) × dx/dt

5

Substitute Values

Plug in known values and rates. Be careful about units!

Warning:

Don't substitute too early! Differentiate first.

6

Solve & Interpret

Solve for the unknown rate. Include units in your answer.

Check:

Does the sign make sense? Positive = increasing, negative = decreasing.

📏 Example: Ladder Sliding Down Wall Related Rates

A 10-foot ladder leans against a wall. The bottom slides away from the wall at 2 ft/sec. How fast is the top sliding down when the bottom is 6 feet from the wall?

Variables:

x = distance from wall to ladder bottom

y = height of ladder on wall

dx/dt = 2 ft/sec (given)

dy/dt = ? (what we need)

L = 10 ft (constant ladder length)

1

Relationship: Pythagorean theorem

x² + y² = L² = 100

2

Differentiate with respect to t:

2x(dx/dt) + 2y(dy/dt) = 0

3

Simplify:

x(dx/dt) + y(dy/dt) = 0

4

Find y when x = 6:

6² + y² = 100 \to y² = 64 \to y = 8 ft

5

Substitute known values:

6(2) + 8(dy/dt) = 0

6

Solve for dy/dt:

12 + 8(dy/dt) = 0 \to dy/dt = -12/8 = -1.5 ft/sec

Top is sliding down at 1.5 ft/sec (negative means decreasing)

🎯 Problem-Solving Strategies & Tips

🧠 General Strategies for Word Problems

Draw Diagrams

Always sketch the situation. Label all known and unknown quantities.

Identify "What's Changing"

For related rates: What quantities vary with time? For optimization: What can we control?

Write Clear Notation

Use descriptive variable names. Always define your variables.

⚠️ Common Mistakes to Avoid

Substituting too early: Don't substitute known values before differentiating in related rates problems.

Forgetting units: Always include units in your final answer. Check that units make sense.

Ignoring endpoints: In optimization, check endpoints of the domain in addition to critical points.

Sign errors: Pay attention to whether rates are positive (increasing) or negative (decreasing).

Missing constraints: Read carefully for hidden constraints like physical limitations (can't have negative length, etc.).

✅ Success Checklist

Optimization

✓ Diagram drawn
✓ Variables defined
✓ Primary equation identified
✓ Constraints identified
✓ Single-variable function created
✓ Derivative found
✓ Critical points found
✓ Endpoints checked
✓ Maximum/minimum verified

Related Rates

✓ Diagram drawn
✓ Variables defined with units
✓ Given and needed rates identified
✓ Relationship equation written
✓ Differentiated with respect to time
✓ Known values substituted
✓ Unknown rate solved
✓ Units included in answer
✓ Sign interpreted correctly

✏️ Practice Problems

Practice 1: Box Volume

From a 12×12 inch square piece of cardboard, squares are cut from each corner and sides folded up to make a box. What size squares maximize volume?

Hint: Volume = length × width × height. Let x be side length of cut squares.

Practice 2: Inflating Balloon

Air is pumped into a spherical balloon at 100 cm³/sec. How fast is radius increasing when radius is 10 cm?

Hint: Volume of sphere V = (4/3)πr³. You know dV/dt, need dr/dt.

Practice 3: Distance Between Cars

Car A travels north at 60 mph, Car B travels east at 80 mph. Both start from same intersection. How fast is distance between them changing after 2 hours?

Hint: Use Pythagorean theorem: D² = A² + B² where A and B are distances traveled.

Practice 4: Rectangle Area

Find dimensions of rectangle with area 100 m² that has minimum perimeter.

Hint: Area = L×W = 100. Perimeter P = 2L + 2W. Express P in terms of one variable using constraint.

📚 Need More Practice?

Visit our Calculators section for interactive problem solvers, or check our Resources page for more worksheets.