91Ó°ÊÓ

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Suppose you are standing at the center of a sphere looking at a point \(P\) on the surface of the sphere. Your line of sight to \(P\) is orthogonal to the plane tangent to the sphere at \(P\). b. At a point that maximizes \(f\) on the curve \(g(x, y)=0,\) the dot product \(\nabla f \cdot \nabla g\) is zero.

Write the differential \(d w\) in terms of the differentials of the independent variables. $$w=f(x, y, z)=x y^{2}+z x^{2}+y z^{2}$$

Verify that \(f_{x y}=f_{y x}\) for the following functions. $$f(x, y)=e^{x+y}$$

Show that the Second Derivative Test is inconclusive when applied to the following functions at \((0,0) .\) Describe the behavior of the function at the critical point. $$f(x, y)=4+x^{4}+3 y^{4}$$

A volume function The volume of a right circular cone of radius \(r\) and height \(h\) is \(V(r, h)=\pi r^{2} h / 3.\) a. Graph the function in the window \([0,5] \times[0,5] \times[0,150].\) b. What is the domain of the volume function? c. What is the relationship between the values of \(r\) and \(h\) when \(V=100 ?\)

Find the indicated derivative in two ways: a. Replace \(x\) and \(y\) to write \(z\) as a function of \(t\) and differentiate. b. Use the Chain Rule. \(z^{\prime}(t),\) where \(z=\ln (x+y), x=t e^{t},\) and \(y=e^{t}\)

Verify that \(f_{x y}=f_{y x}\) for the following functions. $$f(x, y)=\sqrt{x y}$$

Consider the following cylinders in \(\mathbb{R}^{3}\). a. Identify the coordinate axis to which the cylinder is parallel. b. Sketch the cylinder. $$x^{2}+4 y^{2}=4$$

Write the differential \(d w\) in terms of the differentials of the independent variables. $$w=f(x, y, z)=\sin (x+y-z)$$

Show that the Second Derivative Test is inconclusive when applied to the following functions at \((0,0) .\) Describe the behavior of the function at the critical point. $$f(x, y)=x^{2} y-3$$