방향도함수와 기울기벡터

For the following exercises, find the maximum rate of change of $f$ at the given point and the direction in which it occurs.

$f(x,y)=\sqrt{{x}^{2}+2y},$ $(4,10)$

For the following exercises, find the maximum rate of change of $f$ at the given point and the direction in which it occurs.

$f(x,y)=\cos (3x+2y),\left(\frac{\pi }{6},-\frac{\pi }{8}\right)$

For the following exercises, find equations of

a. the tangent plane and b. the normal line to the given surface at the given point.

The level surface $f(x,y,z)=12$ for $f(x,y,z)=4{x}^{2}-2{y}^{2}+{z}^{2}$ at point $(2,2,2).$

For the following exercises, find equations of

a. the tangent plane and b. the normal line to the given surface at the given point.

$f(x,y,z)=xy+yz+xz=3$ at point $(1,1,1)$

For the following exercises, find equations of

a. the tangent plane and b. the normal line to the given surface at the given point.

$f(x,y,z)=xyz=6$ at point $(1,2,3)$

For the following exercises, find equations of

a. the tangent plane and b. the normal line to the given surface at the given point.

$f(x,y,z)=x{e}^{y}\cos \,z-z=1$ at point $(1,0,0)$

The temperature $T$ in a metal sphere is inversely proportional to the distance from the center of the sphere (the origin: $(0,0,0)).$ The temperature at point $(1,2,2)$ is $120^\circ \text{C}.$

a. Find the rate of change of the temperature at point $(1,2,2)$ in the direction toward point $(2,1,3).$ b. Show that, at any point in the sphere, the direction of greatest increase in temperature is given by a vector that points toward the origin.

The electrical potential (voltage) in a certain region of space is given by the function $V(x,y,z)=5{x}^{2}-3xy+xyz.$

a. Find the rate of change of the voltage at point $(3,4,5)$ in the direction of the vector $\langle 1,1,-1\rangle .$ b. In which direction does the voltage change most rapidly at point $(3,4,5)?$ c. What is the maximum rate of change of the voltage at point $(3,4,5)?$

If the electric potential at a point $(x,y)$ in the xy-plane is $V(x,y)={e}^{-2x}\cos (2y),$ then the electric intensity vector at $(x,y)$ is $E=-\nabla V(x,y).$

a. Find the electric intensity vector at $\left(\frac{\pi }{4},0\right).$ b. Show that, at each point in the plane, the electric potential decreases most rapidly in the direction of the vector $E.$

In two dimensions, the motion of an ideal fluid is governed by a velocity potential $\varphi .$ The velocity components of the fluid $u$ in the *x-*direction and $v$ in the y-direction, are given by $\langle u,v\rangle =\nabla \varphi .$ Find the velocity components associated with the velocity potential $\varphi (x,y)=\sin \,\pi x\,\sin \,2\pi y.$

Explain how the vector $\vec v=\left\langle 1,0,3\right\rangle$ can be thought of as having a “slope” of 3.

Explain how the vector $\vec v=\left\langle 0.6,0.8, -2\right\rangle$ can be thought of as having a “slope” of $-2$.

T/F: Let $z=f(x,y)$ be differentiable at $P$. If $\vec n$ is a normal vector to the tangent plane of $f$ at $P$, then $\vec n$ is orthogonal to $\ell_x$ and $\ell_y$ at $P$.

Explain in your own words why we do not refer to the tangent line to a surface at a point, but rather to directional tangent lines to a surface at a point.

In Exercises 5–8, a function $z=f(x,y)$, a vector $\vec v$ and a point $P$ are given. Give the parametric equations of the following directional tangent lines to $f$ at $P$:

(a) $\ell_x(t)$ (b) $\ell_y(t)$ (c) $\ell_{\vec u\,}(t)$, where $\vec u$ is the unit vector in the direction of $\vec v$.

$f(x,y) = 2x^2y-4xy^2$, $\vec v = \left\langle 1,3\right\rangle$, $P=(2,3)$.

In Exercises 5–8, a function $z=f(x,y)$, a vector $\vec v$ and a point $P$ are given. Give the parametric equations of the following directional tangent lines to $f$ at $P$:

(a) $\ell_x(t)$ (b) $\ell_y(t)$ (c) $\ell_{\vec u\,}(t)$, where $\vec u$ is the unit vector in the direction of $\vec v$.

$f(x,y) = 3\cos x\sin y$, $\vec v = \left\langle 1,2\right\rangle$, $P=(\pi/3, \pi/6)$.

In Exercises 5–8, a function $z=f(x,y)$, a vector $\vec v$ and a point $P$ are given. Give the parametric equations of the following directional tangent lines to $f$ at $P$:

(a) $\ell_x(t)$ (b) $\ell_y(t)$ (c) $\ell_{\vec u\,}(t)$, where $\vec u$ is the unit vector in the direction of $\vec v$.

$f(x,y) = 3x-5y$, $\vec v = \left\langle 1,1\right\rangle$, $P=(4,2)$.

In Exercises 5–8, a function $z=f(x,y)$, a vector $\vec v$ and a point $P$ are given. Give the parametric equations of the following directional tangent lines to $f$ at $P$:

(a) $\ell_x(t)$ (b) $\ell_y(t)$ (c) $\ell_{\vec u\,}(t)$, where $\vec u$ is the unit vector in the direction of $\vec v$.

$f(x,y) = x^2-2x-y^2+4y$, $\vec v = \left\langle 1,1\right\rangle$, $P=(1,2)$.

In Exercises 21–24, an implicitly defined function of $x$, $y$ and $z$ is given along with a point $P$ that lies on the surface. Use the gradient $\nabla F$ to:

(a) find the equation of the normal line to the surface at $P$, and (b) find the equation of the plane tangent to the surface at $P$.

$\displaystyle \frac{x^2}{8}+\frac{y^2}4+\frac{z^2}{16}=1$, at $P = (1,\sqrt{2},\sqrt{6})$

In Exercises 21–24, an implicitly defined function of $x$, $y$ and $z$ is given along with a point $P$ that lies on the surface. Use the gradient $\nabla F$ to:

(a) find the equation of the normal line to the surface at $P$, and (b) find the equation of the plane tangent to the surface at $P$.

$\displaystyle z^2-\frac{x^2}{4} - \frac{y^2}9=0$, at $P = (4,-3,\sqrt{5})$