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Cogito

Calculus · Chapter 3 · Lesson 1

Practice Worksheet

From Secant to Tangent

The derivative, built rather than asserted.

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

NameDate

Figure — use these to answer the problems

Move the blue point towards the orange one. Watch the slope settle.
-8-8-6-6-4-4-2-222446688
y = 1x² + 0x − 2
  • Point(2, 2)
  • Second point(5, 23)
  • Slope between them7
a1
b0
c-2
Point of interest2
Second point5
Figure 1

Warm-Up Review

From earlier lessons. Loosen up before the new work.

2 problems

  1. From “The Intermediate Value Theorem”

    f(1) = −3 and f(4) = 6. Does the theorem guarantee a root between them? 1 or 0.

    Answer

  2. From “Types of Discontinuity”

    Left limit 3, right limit 3, function value 8. Which kind? 1, 2 or 3.

    Answer

Warm Up

Straightforward practice. Get the method working first.

4 problems

  1. f′(x) = 2x. Slope at x = 7?

    Answer

  2. What is a derivative?

    • The limit of a slope between two points as they come together.
    • A rule for reducing powers.

  3. f′(x) = 2x. Slope at x = 5?

    Answer

  4. f′(x) = 2x. Slope at x = −4?

    Answer

Build It Up

The same ideas with more to keep track of.

2 problems

  1. f(x) = 3x. What is f′(x)?

    Answer

  2. What does the secant become as the points meet?

    • The tangent.
    • A chord.

Stretch Yourself

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

2 problems

  1. The Slope

    f(x) = x², so f′(x) = 2x. What is the slope at x = 3?

    Answer

  2. The Turning Point

    f(x) = x². At what x is the slope zero?

    Answer