Study guide and
4 practice problems
on:
Gradient is in direction of steepest ascent
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Let $f(x,y)$ be a function of two (or more) variables.
The direction of steepest ascent of $f$ at $(x,y)$ is given by $\nabla f(x,y)$.
The direction of steepest descent of $f$ at $(x,y)$ is given by $-\nabla f(x,y)$.
Related topics
Gradient
(18 problems)
Functions of several variables
(36 problems)
Multivariable calculus
(147 problems)
Practice problems
Consider the surface $z = 10 - x^2 - 2 y^2$. At $(1,-1,7)$, find a 3d tangent vector that points in the direction of steepest ascent.
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
Here are equispaced level curves of a function $f(x,y)$.
a) Where is $\nabla f$ biggest in magnitude?
b) Where is $\nabla f$ smallest in magnitude?
c) Where is $\partial_x f =0$?
d) Where is the directional derivative $D_{(\mathbf{i}/\sqrt{2} + \mathbf{j}/\sqrt{2})} f = 0$?
Solution