Grade 12 · Math · Calculus

12th grade The derivative as a limit; slope of a tangent worksheet

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The derivative as a limit; slope of a tangent

Grade 12 Math Worksheet - Building Confidence

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Ari finds f′(2) for f(x) = 2x² + 3x as the limit of [f(2 + h) − f(2)]/h as h → 0. What is f′(2)?
Bo finds f′(3) for f(x) = 3x² − 2x as the limit of [f(3 + h) − f(3)]/h as h → 0. What is f′(3)?
Esme finds f′(4) for f(x) = −2x² + 7x as the limit of [f(4 + h) − f(4)]/h as h → 0. What is f′(4)?
Jun finds f′(2) for f(x) = 4x² − 3x as the limit of [f(2 + h) − f(2)]/h as h → 0. What is f′(2)?
Luz estimates f′(3) from the tangent line drawn in the figure. What is the slope of the tangent at x = 3?
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A curve y = f(x) with its tangent line at x = -1 drawn on a grid. The tangent passes through (-1, 2) and (3, 8).
Milo estimates f′(-1) from the tangent line drawn in the figure. What is the slope of the tangent at x = -1?
Noah estimates f′(0) from the tangent line drawn in the figure. What is the slope of the tangent at x = 0?
A limit gives f′(3) = 4, and f(3) = 5. Pia writes the tangent line at x = 3 as y = mx + b. What is b?
A limit gives f′(2) = 5, and f(2) = -1. Ren writes the tangent line at x = 2 as y = mx + b. What is b?
A limit gives f′(4) = -3, and f(4) = 10. Zee writes the tangent line at x = 4 as y = mx + b. What is b?

The derivative as a limit and the slope of a tangent. Every version has 10 questions, and the answer key prints on the second page.

I can find f′(a) as the limit of the difference quotient for quadratics, cubes, k/x and k√x.
I can estimate a derivative from a tangent line drawn on a graph.
I can write the tangent line at a point and say which f and a a limit describes.

Standard: none (beyond CCSS)

difference quotient: [f(a + h) − f(a)]/h: the slope of a secant from a to a + h.

derivative f′(a): The limit of the difference quotient as h → 0: the tangent's slope.

tangent line: The line through (a, f(a)) with slope f′(a).

Lost a factor of 2: (a + h)² has the middle term 2ah, and √a + √a is 2√a.

Gave the function's value instead of its slope. The derivative is the limit of the difference quotient.

Divided the run by the rise. Slope is rise over run.

What is the most common mistake with the derivative as a limit; slope of a tangent?

Lost a factor of 2: (a + h)² has the middle term 2ah, and √a + √a is 2√a.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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