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Graphing with Derivatives

f′ says whether f rises or falls; f″ says which way it bends. Stack all three on one x-axis and every peak, valley and inflection point lines up.

Step 1 of 13

■ f′ > 0 (f rises)■ f′ < 0 (f falls)■ f″ > 0 (bends up)■ f″ < 0 (bends down)

Speed

Three plots on one x-axis: f on top, its slope f′ in the middle, and f″ — the slope of the slope — at the bottom. The top curve is drawn only as far as the cursor. Read the two lower plots to say what f will do next.

f(x) = \tfrac{1}{8}x^4 - \tfrac{3}{4}x^2, \quad f'(x) = \tfrac{1}{2}(x^3 - 3x), \quad f''(x) = \tfrac{3}{2}(x^2 - 1)

f(-2.2) = -0.702, \quad f'(-2.2) = \textcolor{#f87171}{-2.024}, \quad f''(-2.2) = \textcolor{#60a5fa}{5.76}

f′ < 0: f is falling; f″ > 0: bending up.

Graphing with Derivatives

f′ says whether f rises or falls; f″ says which way it bends. Stack all three on one x-axis and every peak, valley and inflection point lines up.

Step 1 of 13

■ f′ > 0 (f rises)■ f′ < 0 (f falls)■ f″ > 0 (bends up)■ f″ < 0 (bends down)

Speed

f(x) = \tfrac{1}{8}x^4 - \tfrac{3}{4}x^2, \quad f'(x) = \tfrac{1}{2}(x^3 - 3x), \quad f''(x) = \tfrac{3}{2}(x^2 - 1)

f(-2.2) = -0.702, \quad f'(-2.2) = \textcolor{#f87171}{-2.024}, \quad f''(-2.2) = \textcolor{#60a5fa}{5.76}

f′ < 0: f is falling; f″ > 0: bending up.

DSAverse

Graphing with Derivatives

See also

Graphing with Derivatives

See also