01 Key Concepts
Definition
For f(x,y), the gradient is grad f = (f_x, f_y) — a vector made up of all the partial derivatives.
Direction of Steepest Ascent
At any point, grad f points in the direction where f increases fastest; the magnitude of grad f tells you how steep that increase is.
Perpendicular to Level Curves
The gradient at a point is always perpendicular to the level curve (contour line) passing through that point.
Three Variables
For f(x,y,z), grad f = (f_x, f_y, f_z) — the idea extends naturally to more variables.
Zero Gradient
Where grad f = (0,0), the function has a flat point — a candidate for a local max, min, or saddle point.
02 Key Formulas
- grad f = (f_x, f_y)
- grad f = (f_x, f_y, f_z) for three variables
03 Solved Examples
Example 1 Find the gradient of f(x,y) = x^2 + y^2.
- Find f_x = 2x and f_y = 2y.
- Combine into a vector.
Answer: grad f = (2x, 2y)
Example 2 Find the gradient of f(x,y) = x^2*y + y^3 at the point (1,2).
- Find f_x = 2xy and f_y = x^2 + 3y^2.
- Substitute x=1, y=2: f_x = 2(1)(2) = 4. f_y = 1 + 3(4) = 13.
Answer: grad f at (1,2) = (4, 13)
Example 3 Where is the gradient of f(x,y) = x^2 + y^2 equal to zero?
- grad f = (2x, 2y). Set both components to zero: 2x=0 and 2y=0.
- This gives x=0 and y=0.
Answer: At the point (0,0), which is the minimum of the bowl-shaped surface
04 Practice Questions
1Find the gradient of f(x,y) = 3x + 4y.
(3, 4)
2Find the gradient of f(x,y) = x^2 - y^2.
(2x, -2y)
3Find the gradient of f(x,y) = xy at (2,3).
(3, 2)
4What does the direction of grad f indicate?
The direction of steepest increase of f
5What geometric relationship does grad f have with level curves?
It is perpendicular to them
📄 Gradient — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.