Applications of Derivatives
Grade 12 · Math · Study Guide
Use derivatives to analyze functions, find maxima and minima, and solve real-world optimization problems.
The first derivative f′(x) reveals where a function is increasing (f′ > 0) or decreasing (f′ < 0). Critical points occur where f′(x) = 0 or f′(x) is undefined. The First Derivative Test classifies these as local maxima, minima, or neither based on sign changes of f′.
The second derivative f″(x) describes concavity: f″ > 0 means concave up (cup shape), f″ < 0 means concave down. Inflection points are where concavity changes, found where f″(x) = 0 and the sign of f″ changes. The Second Derivative Test: if f′(c) = 0 and f″(c) > 0, then c is a local minimum; if f″(c) < 0, it's a local maximum.
Optimization problems use derivatives to find the maximum or minimum value of a quantity. The strategy: express the quantity as a function of one variable, find critical points, and verify using the first or second derivative test.
| Test | Method | Conclusion |
|---|---|---|
| First Deriv. | Check sign change of f′ at critical point | f′: + to − → max; − to + → min |
| Second Deriv. | Evaluate f″ at critical point | f″ < 0 → max; f″ > 0 → min; f″ = 0 → inconclusive |
| Condition | Shape | Meaning |
|---|---|---|
| f″(x) > 0 | Concave up (∪) | Graph curves upward |
| f″(x) < 0 | Concave down (∩) | Graph curves downward |
| f″(x) = 0, sign changes | Inflection point | Concavity switches |
Key terms
- Critical point
- A point where f′(x) = 0 or f′(x) is undefined.
- Inflection point
- Where concavity changes; f″(x) = 0 and f″ changes sign.
- Concavity
- Concave up when f″ > 0; concave down when f″ < 0.
- Optimization
- Finding the maximum or minimum value of a function using calculus.
Practice questions
For f(x) = x³ - 3x, at which x-values are the critical points?
- x = 0
- x = ±1
- x = ±3
- x = 1 only
Show answer
f′(x) = 3x² - 3 = 0 → x² = 1 → x = ±1.
For f(x) = x³ - 3x, classify x = -1 using the Second Derivative Test.
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f″(x) = 6x. f″(-1) = -6 < 0, so x = -1 is a local maximum.
A function has f″(x) = 6x. Where is the inflection point?
- x = 6
- x = 0
- x = -6
- x = 1
Show answer
f″(x) = 0 when 6x = 0 → x = 0. Check: f″ changes sign at x = 0 (negative for x<0, positive for x>0).
If f′(c) = 0 and f″(c) = 0, then c must be an inflection point.
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The Second Derivative Test is inconclusive when f″(c) = 0. The point could be a max, min, or inflection point.
A rectangle has perimeter 40. What dimensions maximize its area?
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Let length = x, width = 20-x. Area = x(20-x). A′ = 20-2x = 0 → x = 10. Square maximizes area.
Fun facts
- Optimization is used everywhere in engineering: engineers minimize material cost, maximize structural strength, and find ideal dimensions all using derivative techniques.
- When a company maximizes profit, it sets marginal revenue equal to marginal cost — an application of setting the derivative equal to zero.
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