courses › Calculus

Tangent lines, turning points, and best choices.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 36.

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Builds on: Derivatives (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial 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.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

your rounds

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