11. Partial Derivatives and Tangent Planes

Homework

1. Find the partial derivatives, \(f_x\) and \(f_y\), of the function \(f(x,y)=y\ln\left(x+e^{xy}\right)\).

2. Find the \(y\)-partial derivative of \(f(x,y)=\sqrt{x^2+y}\) using the limit definition of partial derivatives.

3. For the graph of the function \(f=y\sin(xy)\), compute

   1. the slope of the \(x\)-trace at the point \((x,y)=\left(\pi,\dfrac{1}{3}\right)\).

   2. the tangent line to the \(x\)-trace at the point \((x,y)=\left(\pi,\dfrac{1}{3}\right)\).

   3. the slope of the \(y\)-trace at the point \((x,y)=\left(\pi,\dfrac{1}{3}\right)\).

   4. the tangent line to the \(y\)-trace at the point \((x,y)=\left(\pi,\dfrac{1}{3}\right)\).

   5. the tangent plane at the point \((x,y)=\left(\pi,\dfrac{1}{3}\right)\) and its \(z\)-intercept.

4. Find the equation of the plane tangent to \(z=xy+x^3y^2\) at \((x,y)=(1,2)\). Then find the \(z\)-intercept.

5. Compute the tangent plane to \(z=\cos x\sin y\) at \((x,y)=\left(\dfrac{\pi}{3},\dfrac{\pi}{6}\right)\). Then find the \(z\)-intercept.

6. Find the partial derivatives, \(f_x\), \(f_y\) and \(f_z\), of the function \(f=xe^z+z\ln y\).

7. According to the ideal gas law, \(PV=nRT\) where \(R\) is a constant. Find the partial derivatives of the temperature, \(T\), with respect to the volume, \(V\), the mole number, \(n\) and the pressure, \(P\).

8. The density of salt in a fish tank is \(\delta(x,y,z)=(10-z)\sin(x^2y)\). A fish is located at \((x,y,z)=\left(\dfrac{1}{2},\pi,4\right)\). Find the rate of change of the density in the \(x\), \(y\), and \(z\) directions at the location of the fish.