Using derivatives
Tangent lines, turning points, and best choices.
Pen and paper is fine · no calculator needed why?
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the idea
The derivative is a slope, and that one fact solves three kinds of problem.
The tangent line at x = a touches the curve there, so its slope is f′(a). Where f′(x) = 0 the graph is flat: those x-values are critical points, where it can turn between rising and falling. And the largest area or product sits where the derivative of its formula is 0.
techniques
Tangent line: slope, point, intercept
- Slope: m = f′(a).
- Point: the line touches the curve at (a, f(a)).
- Intercept: b = f(a) − m × a, because that point lies on the line.
worked example
The tangent line to y = x³ − 2x at x = 2 is y = mx + b. What is b?
- m = f′(2) = 3 × 4 − 2 = 10.
- The point: f(2) = 8 − 4 = 4.
- b = 4 − 10 × 2 = −16.
Answer: −16
Critical points: set f′ to 0
- Find f′(x). For a cubic it is a quadratic.
- Take out any common number, then factor the rest as (x − r)(x − s).
- x − r is 0 at x = r, and x + r at x = −r. List both, smaller first.
worked example
f(x) = x³ − 6x² + 9x + 1. What is its larger critical x-value?
- f′(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3).
- That is 0 at x = 1 and x = 3. The larger is 3.
Answer: 3
Best value: formula, then f′ = 0
- Call one length x; write the other from the total.
- Write the area or product in x, set its derivative to 0, and solve.
- Put x back in. The formula is a hill; its flat point is the top.
worked example
You have 80 m of fence for a rectangular pen against a long wall, so only three sides need fence. What is the largest area, in square metres?
- Each end is x, so the far side is 80 − 2x.
- Area: x(80 − 2x) = 80x − 2x². Its derivative is 80 − 4x.
- That is 0 at x = 20, so the area is 20 × 40 = 800 m².
Answer: 800
watch out for
- Giving f(a), the height of the curve, as the intercept. The intercept is f(a) − m × a.
- Reading a root with the wrong sign. The factor x + 1 is 0 at x = −1.
- Fencing a pen against a wall as if it had four sides. The best far side is twice each end.
practice
Tangent line
worked example
The tangent line to y = −2x² + 4x − 5 at x = 2 is y = mx + b. What is b?
Answer: 3
- Slope: f′(x) = −4x + 4, so m = f′(2) = −4.
- The point is (2, −5), since f(2) = −5.
- b = y − m·x = −5 − (−4) × 2 = 3.
Critical points
worked example
f(x) = x³ + 9x² + 24x + 4. What are its critical x-values? Enter both, smaller first.
Answer: (-4, -2)
- f′(x) = 3x² + 18x + 24 = 3(x + 4)(x + 2).
- Setting f′(x) = 0 gives x = −4 or x = −2.
Optimization
worked example
You have 136 m of fence to make a rectangular pen, fenced on all four sides. What is the largest area you can enclose, in square metres?
Answer: 1,156
- If the width is x, the length is 68 − x and the area is A = x(68 − x).
- A′(x) = 68 − 2x = 0 at x = 34: the best rectangle is a square.
- Largest area: 34 × 34 = 1,156 m².