Calculus practice

Graphing Differentials worksheets

Compare the exact change on a polynomial or trigonometric curve with the tangent-line differential on an actual reference graph.

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Use the supplied graph and exact function values to find Delta y = f(b) - f(a) and dy = f'(a)(b - a).

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Compare the curve with its tangent

Each problem shows the actual curve, its tangent at P, the curve endpoint Q and the tangent endpoint T. The worked key keeps the function values, slope, Delta y and dy exact.

Learning objective

What this activity builds

Find Delta y = f(b) - f(a) and dy = f'(a)(b - a), then distinguish the curve endpoint Q from the tangent endpoint T.

Worked example

For f(x) = x^2, a = 1, and b = 3/2, compare Delta y and dy.

Delta y = 5/4 and dy = 1.

The actual change is 9/4 - 1 = 5/4. The tangent slope at a is 2, so dy = 2(1/2) = 1.

Teaching notes

Make the practice count

Track the sign of b - a before multiplying.
Use Q for the actual endpoint and T for the tangent endpoint.
A horizontal tangent makes dy zero even when Delta y is nonzero.

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