Gradient Fields and Normal Lines

The gradient of a whole surface, drawn everywhere at once.

12 problems · about 22 minutes · N-VM.A.1, G-GPE.A.1

What this lesson teaches

The student uses the gradient to find normal directions and tangent lines to level curves.

The gradient evaluated everywhere gives a field of arrows all flowing uphill.
Because it is perpendicular to level curves, the gradient is a ready-made normal vector.
That makes tangent lines and tangent planes fall out with no rearranging.

Straightforward practice. Get the method working first.

5 problems

f = x² + y² at (6, 8). What is the magnitude of ∇f?
Answer 20
Why 20.
Why does the gradient work as a normal vector to a level curve?
Answer Moving along the curve changes nothing, so the gradient must be perpendicular to it.
Why Zero change along the curve forces a right angle.
f = x² + y². What is the x-component of ∇f at (5, 2)?
Answer 10
Why 2x.
Is the gradient perpendicular to the level curve? 1 yes, 0 no.
Answer 1
Why Along the curve nothing changes.
f = 3x + 4y. What is the magnitude of ∇f?
Answer 5
Why 9 + 16 = 25.

The same ideas with more to keep track of.

3 problems

The tangent line 6(x − 3) + 8(y − 4) = 0 expands to 6x + 8y = k. What is k?
Answer 50
Why 18 + 32.
On a sphere, does the gradient point along the radius? 1 yes, 0 no.
Answer 1
Why ⟨2x, 2y, 2z⟩ is twice the position vector.
f = xy. What is the y-component of ∇f at (4, 9)?
Answer 4
Why fᵧ = x.

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

4 problems

Order the steps for finding a tangent line to a level curve.
Answer 1. Write the curve as a level curve of some f 2. Compute the gradient of f 3. Evaluate the gradient at the point 4. Use it as the normal in the line equation
Why The gradient must be found before it can be evaluated.
f = 7. What is the magnitude of ∇f?
Answer 0
Why A constant has no slope.
The Sphere: On x² + y² + z² = 14 at (1, 2, 3), the gradient is ⟨2x, 2y, 2z⟩. What is its z-component there?
Answer 6
Why 6.
The Normal: A level curve has normal ⟨4, 3⟩ at a point. What is the magnitude of that normal?
Answer 5
Why 5.