courses › Calculus

Using derivatives

level 36 course

Tangent lines, turning points, and best choices.

Pen and paper is fine · no calculator needed why?

Opens at level 36.

Builds on: Derivatives (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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

The slope m or intercept b of the tangent line at x = a.

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

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

Where a curve is flat, like the turning points of a cubic.

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

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

The largest area or product when a total is fixed.

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

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

Tips by skill

TipTangent line: The slope m is f′ at the point. For the intercept, find the point (a, f(a)) and use b = f(a) − m × a.
TipCritical points: Set f′(x) = 0 and factor it. A factor x − r gives x = r. Enter the smaller value first.
TipOptimization: Write the area or product with one variable, set its derivative to 0, then put that x back in.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

Tangent line par 30 s not tried yet
worked example
The tangent line to y = −2x² + 4x − 5 at x = 2 is y = mx + b. What is b?

Answer: 3
1. Slope: f′(x) = −4x + 4, so m = f′(2) = −4.
2. The point is (2, −5), since f(2) = −5.
3. b = y − m·x = −5 − (−4) × 2 = 3.
Critical points par 35 s not tried yet
worked example
f(x) = x³ + 9x² + 24x + 4. What are its critical x-values? Enter both, smaller first.

Answer: (-4, -2)
1. f′(x) = 3x² + 18x + 24 = 3(x + 4)(x + 2).
2. Setting f′(x) = 0 gives x = −4 or x = −2.
Optimization par 60 s not tried yet
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
1. If the width is x, the length is 68 − x and the area is A = x(68 − x).
2. A′(x) = 68 − 2x = 0 at x = 34: the best rectangle is a square.
3. Largest area: 34 × 34 = 1,156 m².

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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