Application of Derivatives: common misconceptions

If your child says… “I'll put the radius in first, then differentiate.”

Ask together

Does replacing the changing radius with one number preserve how the area changes?

What’s true

Differentiate before inserting the current radius; a shrinking radius has a negative time rate.

If your child says… “The square is positive, so the function increases.”

Ask together

As a negative input approaches zero, does its square grow or shrink?

What’s true

Direction depends on the slope, not whether the output is positive.

If your child says… “Sine and cosine stay positive throughout any interval.”

Ask together

Where does the cosine curve lie below its horizontal axis?

What’s true

Cosine becomes negative after half pi within this interval.

If your child says… “The minimum value is zero because x is zero.”

Ask together

Which number does the function produce when you put zero in?

What’s true

The input locates the minimum; the output gives its value.

If your child says… “Every zero slope must give a maximum or minimum.”

Ask together

What do the slopes on either side tell us that zero alone cannot?

What’s true

A zero slope alone does not establish either kind of extremum.

If your child says… “A zero second derivative means there cannot be a minimum.”

Ask together

If slopes still change from negative to positive, what does that show?

What’s true

A zero second derivative leaves this test undecided.

If your child says… “Only the zero-slope point needs checking.”

Ask together

Which listed outputs are smaller or larger than the output at zero?

What’s true

An endpoint can beat every interior critical point.

If your child says… “Both sides can keep growing while the boundary stays fixed.”

Ask together

If both sides grew, what would happen to their fixed sum?

What’s true

With a fixed perimeter, lengthening one side forces the other to shorten.