Slopes in the x and y directions

Mathematics · Calculus & Analysis · ages 19-20

Name ______________________ Date ____________

What is fx for f(x, y) = x^3 + 2x^2 y + 5y^2?
- 3x^2 + 4xy
- 3x^2 + 2xy
- 2x + 4xy
- 3x^2 + 4xy + 10y
For f(x, y) = x^2 + 3xy + y^2, what is fx(2, 1)?
Answer: ______________
What does fx(a, b) measure?
- slope along the x direction with y fixed
- slope along the y direction with x fixed
- volume under the surface
- length of the gradient
For f(x, y) = x squared + 3xy + y squared, what is fy(2, 1)?
Answer: ______________
Eli says that for this smooth f, fxy equals fyx everywhere. Is Eli right?
Circle one: True False
Eli says that for a smooth f, fxy equals fyx everywhere. Is Eli right?
Circle one: True False
For g(x, y) = 5x^2 + 4xy - 7y^3, what is gy(1, 1)?
Answer: ______________
What is fx for f(x, y) = x cubed + 2x squared y + 5y squared?
- 3x squared + 2xy
- 3x squared + 4xy
- 3x squared + 4xy + 10y
For f(x, y) = x^2 y^3, what is fxy(1, 2)?
Answer: ______________
For f(x, y) = x squared y cubed, what is fxy(1, 2)?
Answer: ______________

Answer key

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

3x^2 + 4xy · Freezing y: 3x^2 from the first term, 4xy from the second, 0 from the third.
7 · fx = 2x + 3y, so fx(2, 1) = 4 + 3 = 7.
slope along the x direction with y fixed · fx freezes y and gives the slope of the trace in the x direction.
8 · fy = 3x + 2y, so fy(2, 1) = 6 + 2 = 8.
True · Eli is right: mixed partials of a smooth function agree.
True · Eli is right: mixed partials of a smooth function agree.
-17 · gy = 4x - 21y^2, so gy(1, 1) = 4 - 21 = minus 17.
3x squared + 4xy · Freezing y: 3x squared from the first term, 4xy from the second, 0 from the third.
24 · fx = 2x y^3, then fxy = 6x y^2, so fxy(1, 2) = 6 times 1 times 4 = 24.
24 · fx = 2x y cubed, then fxy = 6x y squared, so fxy(1, 2) = 6 times 1 times 4 = 24.