다변수 미분 전체 문제

The volume of a right circular cylinder is calculated by a function of two variables, $V(x,y)=\pi {x}^{2}y,$ where $x$ is the radius of the right circular cylinder and $y$ represents the height of the cylinder. Evaluate $V(2,5)$ and explain what this means.

An oxygen tank is constructed of a right cylinder of height $y$ and radius $x$ with two hemispheres of radius $x$ mounted on the top and bottom of the cylinder. Express the volume of the tank as a function of two variables, $x\,\text{and}\,y,$ find $V(10,2),$ and explain what this means.

A thin plate made of iron is located in the $xy\text{-plane.}$ The temperature $T$ in degrees Celsius at a point $P(x,y)$ is inversely proportional to the square of its distance from the origin. Express $T$ as a function of $x\,\text{and}\,y.$

Refer to the preceding problem. Using the temperature function found there, determine the proportionality constant if the temperature at point $P(1,2)\,is\,50^\circ \text{C}.$ Use this constant to determine the temperature at point $Q(3,4).$

The area of a parallelogram with adjacent side lengths that are $a\,\text{and}\,b,$ and in which the angle between these two sides is $\theta ,$ is given by the function $A(a,b,\theta )=ba\,\sin (\theta ).$ Find the rate of change of the area of the parallelogram with respect to the following:

a. Side a b. Side b c. $\text{Angle}\,\theta$

The function $P(T,V)=\frac{nRT}{V}$ gives the pressure at a point in a gas as a function of temperature $T$ and volume $V.$ The letters $n\,\text{and}\,R$ are constants. Find $\frac{\partial P}{\partial V}$ and $\frac{\partial P}{\partial T},$ and explain what these quantities represent.

Suppose the sides of a rectangle are changing with respect to time. The first side is changing at a rate of $2$ in./sec whereas the second side is changing at the rate of $4$ in/sec. How fast is the diagonal of the rectangle changing when the first side measures $16$ in. and the second side measures $20$ in.? (Round answer to three decimal places.)

The apparent temperature index is a measure of how the temperature feels, and it is based on two variables: $h,$ which is relative humidity, and $t,$ which is the air temperature.

$A=0.885t-22.4h+1.20th-0.544.$ Find $\frac{\partial A}{\partial t}$ and $\frac{\partial A}{\partial h}$ when $t=20^\circ \text{F}$ and $h=0.90.$

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.

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 $\frac{da}{a}.)$

The base radius and height of a right circular cone are measured as $10$ in. and $25$ in., respectively, with a possible error in measurement of as much as $0.1$ in. each. Use differentials to estimate the maximum error in the calculated volume of the cone.

The area of an ellipse with axes of length $2a$ and $2b$ is given by the formula

$A=\pi ab.$ Approximate the percent change in the area when $a$ increases by $2\%$ and $b$ increases by $1.5\%.$

The period $T$ of a simple pendulum with small oscillations is calculated from the formula $T=2\pi \sqrt{\frac{L}{g}},$ where $L$ is the length of the pendulum and $g$ is the acceleration resulting from gravity. Suppose that $L$ and $g$ have errors of, at most, $0.5\%$ and $0.1\%,$ respectively. Use differentials to approximate the maximum percentage error in the calculated value of $T.$

The volume of a right circular cylinder is given by $V(x,y)=\pi {x}^{2}y,$ where $x$ is the radius of the cylinder and y is the cylinder height. Suppose $x$ and $y$ are functions of $t$ given by $x=\frac{1}{2}t$ and $y=\frac{1}{3}t$ so that $x\,\text{and}\,y$ are both increasing with time. How fast is the volume increasing when $x=2$ and $y=\frac{4}{3}?$

The pressure $P$ of a gas is related to the volume and temperature by the formula $PV=kT,$ where temperature is expressed in kelvins. Express the pressure of the gas as a function of both $V$ and $T.$ Find $\frac{dP}{dt}$ when $k=1,$ $\frac{dV}{dt}=2$ cm³/min, $\frac{dT}{dt}=\frac{1}{2}$ K/min, $V=20$ cm³, and $T=20^\circ \text{F}.$

The radius of a right circular cone is increasing at $3$ cm/min whereas the height of the cone is decreasing at $2$ cm/min. Find the rate of change of the volume of the cone when the radius is $13$ cm and the height is $18$ cm.

A closed box is in the shape of a rectangular solid with dimensions $x,y,\,\text{and}\,z.$ (Dimensions are in inches.) Suppose each dimension is changing at the rate of $0.5$ in./min. Find the rate of change of the total surface area of the box when $x=2\,\text{in}.,\,y=3\,\text{in.,}\,\text{and}\,z=1\,\text{in}.$

The temperature $T$ at a point $(x,y)$ is $T(x,y)$ and is measured using the Celsius scale. A fly crawls so that its position after $t$ seconds is given by $x=\sqrt{1+t}$ and $y=2+\frac{1}{3}t,$ where $x\,\text{and}\,y$ are measured in centimeters. The temperature function satisfies ${T}_{x}(2,3)=4$ and ${T}_{y}(2,3)=3.$ How fast is the temperature increasing on the fly’s path after $3$ sec?

The $x\,\text{and}\,y$ components of a fluid moving in two dimensions are given by the following functions: $u(x,y)=2y$ and $v(x,y)=-2x;$ $x\ge 0;y\ge 0.$ The speed of the fluid at the point $(x,y)$ is $s(x,y)=\sqrt{u{(x,y)}^{2}+v{(x,y)}^{2}}.$ Find $\frac{\partial s}{\partial x}$ and $\frac{\partial s}{\partial y}$ using the chain rule.