Study guide and
7 practice problems
on:
Gradient is perpendicular to level curves
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The gradient $\nabla f(x,y)$ is perpendicular to the level curve of $f$ that contains $(x,y)$.
Related topics
Gradient
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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
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
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 $\mathbf{r}(t) = x(t) \ \mathbf{i} + y(t) \ \mathbf{j}$ be a path along a level curve of $f(x,y)$.
Let $F(t) = f(x(t), y(t))$. Argue that $dF/dt = 0$.
Use the chain rule to express $dF/dt$ as a dot product of two vectors.
Show that the gradient of $f$ is perpendicular to the level curves of $f$.
Solution