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

NameDate

Figure — use these to answer the problems

This is the gradient field of a bowl. Every arrow crosses the level circles at a right angle.

strength1

start x1

start y1

Figure 1

Straightforward practice. Get the method working first.

5 problems

f = x² + y² at (6, 8). What is the magnitude of ∇f?
Answer
2
Why does the gradient work as a normal vector to a level curve?
- Moving along the curve changes nothing, so the gradient must be perpendicular to it.
- Because it is the longest vector available.
f = x² + y². What is the x-component of ∇f at (5, 2)?
Answer
Is the gradient perpendicular to the level curve? 1 yes, 0 no.
Answer
f = 3x + 4y. What is the magnitude of ∇f?
Answer

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
On a sphere, does the gradient point along the radius? 1 yes, 0 no.
Answer
f = xy. What is the y-component of ∇f at (4, 9)?
Answer

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

4 problems

9
Order the steps for finding a tangent line to a level curve.
Write 1 to 4 in the boxes to put these in order.
- Write the curve as a level curve of some f
- Compute the gradient of f
- Evaluate the gradient at the point
- Use it as the normal in the line equation
f = 7. What is the magnitude of ∇f?
Answer
The Sphere
On x² + y² + z² = 14 at (1, 2, 3), the gradient is ⟨2x, 2y, 2z⟩. What is its z-component there?
Answer
The Normal
A level curve has normal ⟨4, 3⟩ at a point. What is the magnitude of that normal?
Answer