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Cogito

Calculus · Chapter 3 · Lesson 2

Practice Worksheet

The Derivative as a Function

One formula giving the slope everywhere.

12 problems · about 22 minutes · CHA-2.B, CHA-2.C

NameDate

Figure — use these to answer the problems

Move the point along the parabola and read the tangent slope. It is always 2x.
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y = 1x² + 0x + 0
  • Point(1, 1)
  • Slope of the tangent2
a1
Point of tangency1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. f(x) = x². What is f′(10)?

    Answer

  2. What does the derivative of a function report?

    • The slope of the tangent at each input.
    • The height of the function.

  3. f(x) = x². What is f′(3)?

    Answer

  4. f(x) = x³, so f′(x) = 3x². What is f′(2)?

    Answer

  5. f(x) = 9. What is f′(x)?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. f(x) = −4x. What is f′(x)?

    Answer

  2. f(x) = x². What is f′(−5)?

    Answer

  3. f′(x) is negative at x = 2. Is the function rising or falling there? 1 rising, 2 falling.

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Put the limit definition process in order.

    Write 1 to 4 in the boxes to put these in order.

    • Write f(x + h) − f(x).
    • Expand and cancel the terms without h.
    • Divide every remaining term by h.
    • Let h approach zero.

  2. f(x) = x³. What is f′(3)?

    Answer

  3. The Slope

    f(x) = x², so f′(x) = 2x. What is f′(7)?

    Answer

  4. The Line

    f(x) = 5x + 2. What is f′(x)?

    Answer