AP Calculus AB — calculusStandard1 min read

Curve Sketching

Use first and second derivatives to sketch curves for AP Calculus AB. Identify intervals of increase/decrease and concavity.

Tutor AI Team2026-02-25T02:12:59.900Z

Curve sketching combines derivative analysis to fully describe a function's behaviour.

Steps

Domain and symmetry.
Intercepts ( $x$ and $y$ ).
$f'(x)$ : critical points, increasing/decreasing intervals.
$f''(x)$ : inflection points, concavity.
Asymptotes (if applicable).
Sketch using all information.

Sign Charts

$f' > 0$ : increasing. $f' < 0$ : decreasing.
$f'' > 0$ : concave up. $f'' < 0$ : concave down.

Worked Example

$f(x) = x^3 - 3x$ .

$f'(x) = 3x^2 - 3 = 3(x-1)(x+1)$ . Critical: $x = \pm 1$ . $f'$ : + for $x < -1$ , − for $-1 < x < 1$ , + for $x > 1$ . Local max at $(-1, 2)$ , local min at $(1, -2)$ .

$f''(x) = 6x$ . Inflection at $x = 0$ . Concave down for $x < 0$ , up for $x > 0$ .

Practice Problems

1. Sketch $f(x) = x^4 - 4x^2$ .
1. Given $f'(x) = x(x-2)^2$ , describe the behaviour of $f$ .

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Key Takeaways

✓
$f'$ tells increasing/decreasing and extrema.
✓
$f''$ tells concavity and inflection points.
✓
Sign charts are essential tools.

Tags:

#ap#calculus#applications#curve-sketching

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Steps

Sign Charts

Worked Example

Practice Problems

Key Takeaways

Related Topics

Slope Fields

Exponential Growth and Decay Models

Area Between Curves

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