다변수 미분 전체 문제

For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for $z$ in terms of $x$ and $y.)$

$z={x}^{2}-2xy+{y}^{2},P(1,2,1)$

For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, ${P}_{0}({x}_{0,}{y}_{0},{z}_{0}),$ and a vector $n=\langle a,b,c\rangle$ that is parallel to the line. Then the equation of the line is $x-{x}_{0}=at,y-{y}_{0}=bt,z-{z}_{0}=ct.)$

$-3x+9y+4z=-4,P(1,-1,2)$

For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, ${P}_{0}({x}_{0,}{y}_{0},{z}_{0}),$ and a vector $n=\langle a,b,c\rangle$ that is parallel to the line. Then the equation of the line is $x-{x}_{0}=at,y-{y}_{0}=bt,z-{z}_{0}=ct.)$

$z=5{x}^{2}-2{y}^{2},P(2,1,18)$

For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, ${P}_{0}({x}_{0,}{y}_{0},{z}_{0}),$ and a vector $n=\langle a,b,c\rangle$ that is parallel to the line. Then the equation of the line is $x-{x}_{0}=at,y-{y}_{0}=bt,z-{z}_{0}=ct.)$

${x}^{2}-8xyz+{y}^{2}+6{z}^{2}=0,P(1,1,1)$

For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, ${P}_{0}({x}_{0,}{y}_{0},{z}_{0}),$ and a vector $n=\langle a,b,c\rangle$ that is parallel to the line. Then the equation of the line is $x-{x}_{0}=at,y-{y}_{0}=bt,z-{z}_{0}=ct.)$

$z=\ln (3{x}^{2}+7{y}^{2}+1),P(0,0,0)$

For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, ${P}_{0}({x}_{0,}{y}_{0},{z}_{0}),$ and a vector $n=\langle a,b,c\rangle$ that is parallel to the line. Then the equation of the line is $x-{x}_{0}=at,y-{y}_{0}=bt,z-{z}_{0}=ct.)$

$z={e}^{4{x}^{2}+6{y}^{2}},P(0,0,1)$

For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, ${P}_{0}({x}_{0,}{y}_{0},{z}_{0}),$ and a vector $n=\langle a,b,c\rangle$ that is parallel to the line. Then the equation of the line is $x-{x}_{0}=at,y-{y}_{0}=bt,z-{z}_{0}=ct.)$

$z={x}^{2}-2xy+{y}^{2}$ at point $P(1,2,1)$

The length of line segment $AC$ is equal to what mathematical expression?

The length of line segment $BC$ is equal to what mathematical expression?

Using the figure, explain what the length of line segment $AB$ represents.

For the following exercises, complete each task.

Show that $f(x,y)={e}^{xy}x$ is differentiable at point $(1,0).$

For the following exercises, complete each task.

Find the total differential of the function $w={e}^{y}\cos (x)+{z}^{2}.$

Show that $f(x,y)={x}^{2}+3y$ is differentiable at every point. In other words, show that $\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)={f}_{x}\Delta x+{f}_{y}\Delta y+{\varepsilon }_{1}\Delta x+{\varepsilon }_{2}\Delta y,$ where both ${\varepsilon }_{1}$ and ${\varepsilon }_{2}$ approach zero as $(\Delta x,\Delta y)$ approaches $(0,0).$

Find the total differential of the function $z=\frac{xy}{y+x}$ where $x$ changes from $10\,\text{to}\,10.5$ and $y$ changes from $15\,\text{to}\,13.$

Let $z=f(x,y)=x{e}^{y}.$ Compute $\Delta z$ from $P(1,2)$ to $Q(1.05,2.1)$ and then find the approximate change in $z$ from point $P$ to point $Q.$ Recall $\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y),$ and $dz$ and $\Delta z$ are approximately equal.

The volume of a right circular cylinder is given by $V(r,h)=\pi {r}^{2}h.$ Find the differential $dV.$ Interpret the formula geometrically.

See the preceding problem. Use differentials to estimate the volume of aluminum in an enclosed aluminum can with diameter $8.0\,\text{cm}$ and height $12\,\text{cm}$ if the aluminum is $0.04$ cm thick.

Use the differential $dz$ to approximate the change in $z=\sqrt{4-{x}^{2}-{y}^{2}}$ as $(x,y)$ moves from point $(1,1)$ to point $(1.01,0.97).$ Compare this approximation with the actual change in the function.

Let $z=f(x,y)={x}^{2}+3xy-{y}^{2}.$ Find the exact change in the function and the approximate change in the function as $x$ changes from $2.00\,\text{to}\,2.05$ and $y$ changes from $3.00\,\text{to}\,2.96.$

The centripetal acceleration of a particle moving in a circle is given by $a(r,v)=\frac{{v}^{2}}{r},$ where $v$ is the velocity and $r$ is the radius of the circle. Approximate the maximum percent error in measuring the acceleration resulting from errors of $3\%$ in $v$ and $2\%$ in $r.$ (Recall that the percentage error is the ratio of the amount of error over the original amount. So, in this case, the percentage error in $a$ is given by $\frac{da}{a}.)$