Study guide and
15 practice problems
on:
Definition of the gradient
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The definition of the gradient of $f(x,y)$ is $$\nabla f(x,y) = \partial_x f(x,y) \mathbf{i} + \partial_y f(x,y) \mathbf{j}$$
Related topics
Gradient
(18 problems)
Functions of several variables
(36 problems)
Multivariable calculus
(147 problems)
Practice problems
Find a normal vector to the surface $x^3 + y^3 z = 3$ at the point $(1,1,2)$.
Solution
Find the tangent plane at $(1,1,1)$ to the surface $$x^2 + y^2 + z^2 + xy + xz = 5.$$
Solution
Suppose $f(x,y) = g(x^2 + y^2)$ for some single-variable function $g$. Show that the gradient of $f$ at any point $(x,y)$ is always pointing toward or away from the origin.
Solution
Give the equation for the tangent plane to the surface $z x^2 + x y^2 + y z^2 = 5$ at the point $(-1,1,2)$.
Solution
Let $f(x,y) = xy$.
a) Sketch the level curves of $f$.
b) Sketch the path of steepest descent starting at $(1,2)$.
b) Find the path of steepest descent starting at $(1,2)$.
Solution
Consider a surface with height $z(x,y) = 10 - x^2 - 2 y^2$. Find the path of steepest ascent starting at $(2, 1, 4)$. Express your answer as a curve in the $xy$ plane.
Solution
Use 3d level surfaces to show that a normal vector to $z=f(x,y)$ is given by $\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, -1 \rangle$.
Solution
Find a normal vector to the curve $\sqrt{x} + \sqrt{y} = 2$ at $(x,y) = (1,2)$. Use it to find the tangent line at $(1,1)$ expressed in the form $\bfn \cdot \bfx = b$.
Solution
Let $f(x,y) = x^2 + y^2$.
Describe the shape of the $f(x,y)=2$ level curve.
Without calculation, find the directional derivative at $(1,1)$ in the direction $-\bfi+\bfj$.
Hint: consider the level curve at $(1,1).$
By computation, find the directional derivative at $(1,1)$ in the direction of $-\bfi + \bfj$.
Solution
Let $f(x,y) = x^2/y$. Compute the directional derivative of $f$ at $(1,2)$ in the direction of $\mathbf{i} + 3\ \mathbf{j}$.
Solution
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