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Cogito

Calculus · Chapter 5 · Lesson 1

Practice Worksheet

Rates of Change and Related Rates

The derivative as a speed.

10 problems · about 22 minutes · CHA-3.A, CHA-3.B

NameDate

Figure — use these to answer the problems

If this is position against time, the tangent slope is the velocity.
-8-8-6-6-4-4-2-222446688
y = 1x² + 0x + 0
  • Point(2, 4)
  • Slope of the tangent4
a1
b0
c0
Time2
Figure 1

Warm-Up Review

From earlier lessons. Loosen up before the new work.

2 problems

  1. From “The Chain Rule”

    For (6x + 1)³, what is the inner derivative?

    Answer

  2. From “The Product and Quotient Rules”

    y = x² (3x + 1), so y′ = 9x² + 2x. What is y′ at x = 2?

    Answer

Warm Up

Straightforward practice. Get the method working first.

4 problems

  1. A square with side 7 growing at 2 cm/s. Area rate?

    Answer

  2. What does a derivative measure in a real situation?

    • How fast one quantity changes as another does.
    • The total amount.

  3. v(t) = 2t. Velocity at t = 9?

    Answer

  4. A square with side 6 growing at 2 cm/s. Area rate?

    Answer

Build It Up

The same ideas with more to keep track of.

2 problems

  1. A square with side 10 growing at 1 cm/s. Area rate?

    Answer

  2. What is the derivative of velocity?

    • Acceleration.
    • Position.

Stretch Yourself

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

2 problems

  1. The Velocity

    s(t) = t², so v(t) = 2t. What is the velocity at t = 6?

    Answer

  2. The Square

    A square with side 4 growing at 3 cm/s. How fast is the area growing?

    Answercm² per second