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Cogito

AP Calculus AB · Chapter 2 · Lesson 1

Practice Worksheet

The Derivative as a Limit

Bringing the two points together.

11 problems · about 25 minutes · AP Calculus AB 2.1, 2.2

NameDate

Figure — use these to answer the problems

Bring the second point onto the first. The secant slope becomes the derivative.
-8-8-6-6-4-4-2-222446688
y = 1x² + 0x + 0
  • Point(2, 4)
  • Second point(5, 25)
  • Slope between them7
a1
b0
c0
Point of tangency2
Second point5
Figure 1

Warm Up

Straightforward practice. Get the method working first.

4 problems

  1. f(x) = x². What is f'(7)?

    Answer

  2. f(x) = x². What is f'(1)?

    Answer

  3. f(x) = x². What is f'(0)?

    Answer

  4. f(x) = 5x + 2. What is f'(9)?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Set the two points together at x = 2 so the secant becomes the tangent, and the slope reads 4.

    Set the model in the app, then write the values you used.

  2. The difference quotient for f(x) = x² at a = 4 simplifies to 8 + h. What is the derivative?

    Answer

  3. f(x) = x³. What is f'(2)? The derivative of x³ is 3x².

    Answer

Stretch Yourself

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

4 problems

  1. Is the derivative an average rate or an instantaneous rate? 1 for average, 2 for instantaneous.

    Answer

  2. f(x) = 7. What is f'(3)?

    Answer

  3. From the Definition

    For f(x) = x², what is f'(5)?

    Answer

  4. Reading the Tangent

    The tangent to y = x² at x = 2 is shown. What is its slope?

    The tangent at x = 2.
    -8-8-6-6-4-4-2-222446688
    y = 1x² + 0x + 0
    • Point(2, 4)
    • Slope of the tangent4
    a1
    b0
    c0
    Point of tangency2
    Answer