Tangent Planes and Linear Approximation · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

The plane that hugs a surface

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

  1. Omar says the tangent plane at (1, 2) is built from fx(1, 2) and fy(1, 2). Is Omar right?

    Circle one:   True   False

  2. For f(x, y) = x^2 + y^2, what is f(1, 2)?

    Answer: ______________

  3. For f(x, y) = x squared + y squared, what is f(1, 2)?

    Answer: ______________

  4. Omar says the tangent plane at (1, 2) is built from fx(1, 2) and fy(1, 2). Is Omar right?

    Circle one:   True   False

  5. Using L(x, y) = 5 + 2(x - 1) + 4(y - 2), estimate f(1.1, 1.9).

    Answer: ______________

  6. A normal vector to the plane z = 5 + 2(x minus 1) + 4(y minus 2) is:

    • (1, 2, 5)
    • (2, 4, minus 1)
    • (2, 4, 1)

  7. What is the tangent plane to z = x^2 + y^2 at (1, 2)?

    • z = 5 + 2(x - 1) + 4(y - 2)
    • z = 5 + 2x + 4y
    • z = 1 + 2(x - 1) + 4(y - 2)
    • z = 5 + 4(x - 1) + 2(y - 2)

  8. A normal vector to the plane z = 5 + 2(x - 1) + 4(y - 2) is:

    • (2, 4, -1)
    • (1, 2, 5)
    • (2, 4, 1)
    • (5, 2, 4)

  9. Priya says a surface can have both partials at a point but still no tangent plane there. Is Priya right?

    Circle one:   True   False

  10. Priya says a surface can have both partials at a point but still no tangent plane there. What makes that possible?

    • partials are never defined at a point
    • every surface has a tangent plane
    • the surface can misbehave between the axis slices
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Answer key

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

The plane that hugs a surface W1-mt_EYMcjChX_m-s1

  1. True · Omar is right: those two slopes are the tilts of the plane.
  2. 5 · f(1, 2) = 1 + 4 = 5, the base height for the plane.
  3. 5 · f(1, 2) = 1 + 4 = 5, the base height for the plane.
  4. True · Omar is right: those two slopes are the tilts of the plane.
  5. 4.8 · 5 + 2(0.1) + 4(minus 0.1) = 5 + 0.2 - 0.4 = 4.8.
  6. (2, 4, minus 1) · Rewrite as 2x + 4y minus z = const, so (2, 4, minus 1) is normal.
  7. z = 5 + 2(x - 1) + 4(y - 2) · Base height 5 with fx = 2 and fy = 4 at (1, 2) gives the first choice.
  8. (2, 4, -1) · Rewrite as 2x + 4y - z = const, so (2, 4, minus 1) is normal.
  9. True · Priya is right: partials only see axis slices, and the surface can misbehave between them.
  10. the surface can misbehave between the axis slices · Partials only see axis slices, and the surface can misbehave between them.
Worksheet · LightMySky