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Cogito

Multivariable Calculus · Chapter 5 · Lesson 1

Practice Worksheet

The Gradient Vector

Both partial derivatives, packed into one arrow.

12 problems · about 23 minutes · N-VM.A.1, F-IF.B.6

NameDate

Figure — use these to answer the problems

Every arrow points straight uphill on a bowl. Release a point and watch it climb the steepest way.
  • Released from(1, 2)
a, the steepness of the bowl1
start x1
start y2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. f = x² + 3y. What is the x-component of ∇f at (5, 1)?

    Answer

  2. Which way does the gradient point?

    • In the direction of steepest increase.
    • Along the level curve.

  3. f = 3x + 5y. What is the x-component of ∇f?

    Answer

  4. f = x² + y². What is the y-component of ∇f at (1, 4)?

    Answer

  5. ∇f = ⟨5, 12⟩. What is the steepest rate of increase?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. ∇f = ⟨0, 0⟩ at a point. Is it a critical point? 1 yes, 0 no.

    Answer

  2. f = xy. What is the x-component of ∇f at (2, 7)?

    Answer

  3. f = 10 for every point. What is the magnitude of ∇f?

    Answer

Stretch Yourself

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

4 problems

  1. Match each gradient fact to its meaning.

    Draw a line from each item on the left to its match on the right.

    • The direction of ∇f
    • The magnitude of ∇f
    • The direction of −∇f
    • Steepest ascent
    • How steep that ascent is
    • Steepest descent

  2. f = 4x. What is the y-component of ∇f?

    Answer

  3. The Hillside

    A hill has ∇h = ⟨3, 4⟩ at your feet. How steep is the steepest climb from there?

    Answer

  4. The Descent

    ∇f = ⟨6, −8⟩. What is the x-component of the steepest descent direction?

    Answer