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

The gradient points uphill

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

  1. A differentiable function has gradient (-2, 7) at P. What does the second component tell you?

    • The rate along (1, 0) is 7.
    • The rate along (0, 1) is 7.
    • The rate along (0, 1) is -2.

  2. For f(x, y) = x² + y² at (0, 0), the gradient is (0, 0). Which statement about directional derivatives is correct?

    • Only the positive x direction has rate 0.
    • Every unit direction has a positive rate.
    • Every unit direction has rate 0.
    • Only the positive y direction has rate 0.

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

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

  4. For f differentiable at P and a unit vector u, Kay says D_u f(P) = ∇f(P) dot u. Is Kay right?

    Circle one:   True   False

  5. Let f(x, y) = 2x - y and follow x(t) = 1 + 3t, y(t) = 2 + 4t. What is df/dt at t = 0? Give the rate per unit of t, not per unit distance.

    Answer: ______________

  6. For f(x, y) = 5x + 12y, what is the smallest directional derivative over all unit directions?

    Answer: ______________

  7. For f(x, y) = 5x + 10y, find the directional derivative in the direction of v = (3, 4). Use the rate per unit distance.

    Answer: ______________

  8. For f(x, y) = 6x + 8y, what is the directional derivative in unit direction (3/5, -4/5)? Give a decimal.

    Answer: ______________

  9. Nora follows a level curve f(x, y) = c and says f is increasing along that path. Is Nora right?

    Circle one:   True   False

  10. Which vector is normal to the level curve x² + y² = 25 at (3, 4)?

    • (4, -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. The rate along (0, 1) is 7. · The second component is fy. It gives the rate in the positive y direction, whose unit vector is (0, 1).
  2. Every unit direction has rate 0. · The gradient dotted with any unit vector is 0. Every unit direction ties, so there is no unique fastest direction at this point.
  3. (3, 4) · fx = 3 and fy = 4 everywhere, so the gradient is (3, 4).
  4. True · Yes. Differentiability gives this formula. Using a unit vector makes it a rate per unit distance in the input plane.
  5. 2 · The gradient is (2, -1) and the path velocity is (3, 4). The rate per unit of t is 2(3) - 4 = 2.
  6. -13 · The gradient (5, 12) has length 13. The smallest directional derivative is the negative of this length, -13.
  7. 11 · The direction has length 5, so use (3/5, 4/5). The gradient (5, 10) gives rate 3 + 8 = 11.
  8. -2.8 · The gradient is (6, 8). Its dot product with (3/5, -4/5) is 18/5 - 32/5 = -14/5 = -2.8.
  9. False · No. Every point on this path has function value c, so f does not increase along it.
  10. (6, 8) · For f = x² + y², the gradient at (3, 4) is (6, 8). It is nonzero and normal to this level curve.
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