Directional Derivatives and the Gradient · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

The gradient points uphill

Mathematics · Calculus & Analysis · ages 19-20
Name ______________________   Date ____________

  1. What is the gradient of f(x, y) = 3x + 4y at (1, 1)?

    • (3, 4)
    • (4, 3)
    • (1, 1)
    • (7, 7)

  2. In which direction does f increase fastest?

    • along the gradient
    • along the level curve
    • opposite the gradient
    • perpendicular to the gradient

  3. What is the gradient of f(x, y) = 3x + 4y at (1, 1)?

    • (4, 3)
    • (3, 4)
    • (1, 1)

  4. Kay says the directional derivative is the dot product of the gradient with the unit direction. Is Kay right?

    Circle one:   True   False

  5. For f(x, y) = 3x + 4y, what is the directional derivative in unit direction (3/5, 4/5)?

    Answer: ______________

  6. For f(x, y) = 3x + 4y, what is the fastest rate of increase?

    Answer: ______________

  7. For f(x, y) = x^2 + y^2 at (1, 2), what is the directional derivative in direction (1, 0)?

    Answer: ______________

  8. For f(x, y) = 3x + 4y, what is the directional derivative in unit direction (3/5, 4/5)?

    Answer: ______________

  9. Nora walks along a level curve and says f is rising fast. Is Nora right?

    Circle one:   True   False

  10. Which vector is perpendicular to the level curve x squared + y squared = 25 at (3, 4)?

    • (4, minus 3)
    • (1, 0)
    • (6, 8)
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Answer key

For grown-ups. Fold this page away before handing over the rest.

The gradient points uphill W1-mt_LW_KckY5Ad-s1

  1. (3, 4) · fx = 3 and fy = 4 everywhere, so the gradient is (3, 4).
  2. along the gradient · The dot product with a unit vector is largest when it aligns with the gradient.
  3. (3, 4) · fx = 3 and fy = 4 everywhere, so the gradient is (3, 4).
  4. True · Kay is right: that dot product is the definition in unit direction form.
  5. 5 · Gradient (3, 4) dotted with (3/5, 4/5) is 9/5 + 16/5 = 5.
  6. 5 · Gradient (3, 4) has length 5, which is the max rate.
  7. 2 · Gradient is (2, 4); dotting with unit (1, 0) gives 2.
  8. 5 · Gradient (3, 4) dotted with (3/5, 4/5) is 9/5 + 16/5 = 5.
  9. False · Along the curve f does not change, so the rate is 0.
  10. (6, 8) · The gradient (2x, 2y) at (3, 4) is (6, 8), which is perpendicular to the curve.
Worksheet · LightMySky