극값과 안장점

Find the maximum volume of a rectangular box with three faces in the coordinate planes and a vertex in the first octant on the plane $x+y+z=1.$

The sum of the length and the girth (perimeter of a cross-section) of a package carried by a delivery service cannot exceed $108$ in. Find the dimensions of the rectangular package of largest volume that can be sent.

A cardboard box without a lid is to be made with a volume of $4$ ft³. Find the dimensions of the box that requires the least amount of cardboard.

Find the point on the surface $f(x,y)={x}^{2}+{y}^{2}+10$ nearest the plane $x+2y-z=0.$ Identify the point on the plane.

Find the point in the plane $2x-y+2z=16$ that is closest to the origin.

A company manufactures two types of athletic shoes: jogging shoes and cross-trainers. The total revenue from $x$ jogging shoes and $y$ cross-trainers is given by $R(x,y)=-5{x}^{2}-8{y}^{2}-2xy+42x+102y,$ where $x$ and $y$ are in thousands of units. Find the values of x and y to maximize the total revenue.

A shipping company handles rectangular boxes provided the sum of the length, width, and height of the box does not exceed $96$ in. Find the dimensions of the box that meets this condition and has the largest volume.

Find the maximum volume of a cylindrical soda can such that the sum of its height and circumference is $120$ cm.

Let a level curve of $z=f(x,y)$ be described by $x=g(t)$, $y = h(t)$. Explain why $\frac{dz}{dt}=0$.

Fill in the blank: The single variable Chain Rule states $\displaystyle \frac{d}{dx}\Big(f\big(g(x)\big)\Big) = f\,'\big(g(x)\big)\cdot$_____.

Fill in the blank: The Multivariable Chain Rule states

$\displaystyle \frac{df}{dt}=\frac{\partial f}{\partial x}\cdot\text{\underline{}}+\text{\underline{}}\cdot\frac{dy}{dt}$.

If $z=f(x,y)$, where $x=g(t)$ and $y=h(t)$, we can substitute and write $z$ as an explicit function of $t$.

T/F: Using the Multivariable Chain Rule to find $\frac{dz}{dt}$ is sometimes easier than first substituting and then taking the derivative.

T/F: The Multivariable Chain Rule is only useful when all the related functions are known explicitly.

The Multivariable Chain Rule allows us to compute implicit derivatives easily by just computing two _____ derivatives.

In Exercises 7–12, functions $z=f(x,y)$, $x=g(t)$ and $y=h(t)$ are given.

(a) Use the Multivariable Chain Rule to compute $\displaystyle \frac{dz}{dt}$. (b) Evaluate $\displaystyle \frac{dz}{dt}$ at the indicated $t$-value.

$z=3x+4y$, $x=t^2$, $y=2t$; $t=1$

In Exercises 7–12, functions $z=f(x,y)$, $x=g(t)$ and $y=h(t)$ are given.

(a) Use the Multivariable Chain Rule to compute $\displaystyle \frac{dz}{dt}$. (b) Evaluate $\displaystyle \frac{dz}{dt}$ at the indicated $t$-value.

$\displaystyle z=x^2-y^2$, $x=t$, $y=t^2-1$; $t=1$

In Exercises 7–12, functions $z=f(x,y)$, $x=g(t)$ and $y=h(t)$ are given.

(a) Use the Multivariable Chain Rule to compute $\displaystyle \frac{dz}{dt}$. (b) Evaluate $\displaystyle \frac{dz}{dt}$ at the indicated $t$-value.

$\displaystyle z=5x+2y$, $x=2\cos t+1$, $y=\sin t-3$; $t=\pi/4$

In Exercises 7–12, functions $z=f(x,y)$, $x=g(t)$ and $y=h(t)$ are given.

(a) Use the Multivariable Chain Rule to compute $\displaystyle \frac{dz}{dt}$. (b) Evaluate $\displaystyle \frac{dz}{dt}$ at the indicated $t$-value.

$\displaystyle z=\frac{x}{y^2+1}$, $x=\cos t$, $y=\sin t$; $t=\pi/2$

In Exercises 7–12, functions $z=f(x,y)$, $x=g(t)$ and $y=h(t)$ are given.

(a) Use the Multivariable Chain Rule to compute $\displaystyle \frac{dz}{dt}$. (b) Evaluate $\displaystyle \frac{dz}{dt}$ at the indicated $t$-value.

$\displaystyle z=x^2+2y^2$, $x=\sin t$, $y=3\sin t$; $t=\pi/4$

In Exercises 7–12, functions $z=f(x,y)$, $x=g(t)$ and $y=h(t)$ are given.

(a) Use the Multivariable Chain Rule to compute $\displaystyle \frac{dz}{dt}$. (b) Evaluate $\displaystyle \frac{dz}{dt}$ at the indicated $t$-value.

$z=\cos x\sin y$, $x=\pi t$, $y=2\pi t+\pi/2$; $t=3$