First-Order Partial Derivatives

1. If $f(x,y) = 3x^3 - 2x^2y^5$, find the partial derivatives $f_x$ and $f_y$.

2. If $f(x,y) = \displaystyle\frac{xy^2}{x+1}$, compute $\partial f/\partial x$ and $\partial f/\partial y$.

3. If $g(r,s) = rs\cos(r)$, find the partial derivatives $g_r$ and $g_s$.

4. Find all possible first-order partial derivatives of $q(x,t,z) = \displaystyle \frac{x2^tz^3}{1+x^2}.$

The speed of sound $C$ traveling through ocean water is a function of temperature, salinity and depth. It may be modeled by the function . Here $C$ is the speed of sound in meters/second, $T$ is the temperature in degrees Celsius, $S$ is the salinity in grams/liter of water, and $D$ is the depth below the ocean surface in meters.

1. State the units for each of the partial derivatives, $C_T$, $C_S$ and $C_D$. For each parti…

The wind chill, as frequently reported, is a measure of how cold it feels outside when the wind is blowing. In the table below, the wind chill $w$ is a function of the wind speed $v$ and the ambient air temperature $T$. Both $w$ and $T$ are measured in degrees Fahrenheit, while $v$ is measured in miles per hour. This means that $w$ can be expressed in the form $w = w(v, T)$.

1. When computing the partial derivative with respec…

The temperature on an unevenly heated metal plate positioned in the first quadrant of the $xy$-plane is given by

Assume that temperature is measured in degrees Celsius and that $x$ and $y$ are each measured in inches. (Note: At no point in the following questions should you expand the denominator of $C(x,y)$.)

1. Determine $\frac{\partial C}{\partial x}\restrict{(x,y)}{}$ and $\frac{\partial C}{\partial y}\restrict{(x,y)}{}$.

2. If an ant is on the metal plate, standing at the point $(2,3)$, and starts walking in the d…

Consider the function $f$ defined by $f(x,y) = 8 - x^2 - 3y^2$.

1. Determine $f_x(x,y)$ and $f_y(x,y)$.

2. Find parametric equations in $\R^3$ for the tangent line to the trace $f(x,1)$ at $x=2$.

3. Find parametric equations in $\R^3$ for the tangent line to the trace $f(2,y)$ at $y=1$.

4. State respective direction vectors for the two lines determined in (b) and (c).

5. Determine the equation of the plane that passes through the point $(2,1,f(2,1))$ whose normal vector is orthogonal to the direction vec…

Recall from single variable calculus that, given the derivative of a single variable function and an initial condition, we can integrate to find the original function. We can sometimes use the same process for functions of more than one variable. For example, suppose that a function $f$ satisfies $f_x(x,y) = \cos(y)e^x+2x+y^2$, $f_y(x,y) = -\sin(y)e^x+2xy+3$, and $f(0,0) = 5$.

1. Find all possible functions $f$ of $x$ and $y$ such that $f_x(x,y) = \cos(y)e^x+2x+y^2$. Your function will have both $x$ and $y$ a…