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Grade 12 · Cubic derivatives

Cubic derivatives: Reverse the cubic direction

Cubic derivatives: investigate reverse the cubic direction with cubic coefficient a = -1; linear coefficient b = 0.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Cubic derivatives: Reverse the cubic direction. Input x: -2. Function value: 8. First derivative: -12. Second derivative: 12Compare value, slope, and curvature-88-202yx → · labeled axes rescale to this model
This family contains only cubic and linear terms with real coefficients. The tangent is an exact local linearization, not a global approximation. The plot uses labeled, automatically fitted vertical scales.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Input x
-2
Function value
8
First derivative
-12
Second derivative
12

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Understand what you are seeing

The idea behind the motion.

A cubic's height, slope, and change of slope answer different questions. The moving tangent shows why a point with zero height need not have zero slope. Varying the linear term can add or remove stationary points while the second derivative still changes sign at zero when a is nonzero. This investigation starts with Cubic coefficient a = -1; Linear coefficient b = 0. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

f(x)=ax³+bx; f′=3ax²+b; f″=6ax

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Predict the function value at zero and separately calculate its first derivative there. The starting case is “Reverse the cubic direction.”

  2. STEP 2

    Follow the changing quantity

    Trace from x=−2 to x=2 while comparing the tangent's direction with the derivative readout.

  3. STEP 3

    Explain and test the result

    Use both derivatives to distinguish a stationary inflection from an ordinary crossing. If a=0, identify the resulting linear function.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Cubic coefficient a = -1; Linear coefficient b = 0. Pause the timeline at 80%. Given input x = 1.2, calculate function value, first derivative, second derivative. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

At x=1.2, f=(-1)(1.2)³+(0)(1.2)=-1.728. Differentiation gives f′=3(-1)(1.2)²+(0)=-4.32 and f″=6(-1)(1.2)=-7.2. Results: Function value: -1.728; First derivative: -4.32; Second derivative: -7.2. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 12 · Calculus

Derive a power derivative from the difference quotient

Cancel a nonzero increment before taking its limit.