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

NameDate

Circle the best answer, or write your answer on the line.

Ari works out f′(2) for f(x) = 12/x as the limit of [f(2 + h) − f(2)]/h as h → 0. What is f′(2)?
Esme works out f′(4) for f(x) = 8/x as the limit of [f(4 + h) − f(4)]/h as h → 0. What is f′(4)?
Kira expands f(2 + h) for f(x) = 2x³ and takes the limit of [f(2 + h) − f(2)]/h as h → 0. What is f′(2)?
Luz expands f(-2 + h) for f(x) = 4x³ and takes the limit of [f(-2 + h) − f(-2)]/h as h → 0. What is f′(-2)?
Milo expands f(2 + h) for f(x) = -3x³ and takes the limit of [f(2 + h) − f(2)]/h as h → 0. What is f′(2)?
Noah expands f(4 + h) for f(x) = 2x³ and takes the limit of [f(4 + h) − f(4)]/h as h → 0. What is f′(4)?
Pia finds the tangent to f(x) = x² + 3 at x = 2, using a limit for the slope. Written as y = mx + b, what is b?
Ren finds the tangent to f(x) = x² − 1 at x = 3, using a limit for the slope. Written as y = mx + b, what is b?
Tala finds the tangent to f(x) = x² + 5 at x = -2, using a limit for the slope. Written as y = mx + b, what is b?
Zee finds the tangent to f(x) = x² + 4 at x = 1, using a limit for the slope. Written 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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