91Ó°ÊÓ

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=x^{2}+2 x y-y^{3}$$

Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum, or a saddle point. If the Second Derivative Test is inconclusive, determine the behavior of the function at the critical points. $$f(x, y)=\frac{x}{1+x^{2}+y^{2}}$$

Consider the following functions and points \(P\). a. Find the unit vectors that give the direction of steepest ascent and steepest descent at \(P\). b. Find a vector that points in a direction of no change in the function at \(P\). $$F(x, y)=e^{-x^{2} / 2-y^{2} / 2} ; P(-1,1)$$

Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum, or a saddle point. If the Second Derivative Test is inconclusive, determine the behavior of the function at the critical points. $$f(x, y)=\frac{x-1}{x^{2}+y^{2}}$$

Partial derivatives Find the first partial derivatives of the following functions. $$f(x, y)=1-\cos (2(x+y))+\cos ^{2}(x+y)$$

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\frac{x y}{x^{2} y^{2}+1}$$

Use Lagrange multipliers in the following problems. When the constraint curve is unbounded, explain why you have found an absolute maximum or minimum value. Maximum volume cylinder in a sphere Find the dimensions of the right circular cylinder of maximum volume that can be inscribed in a sphere of radius 16

Graph several level curves of the following functions using the given window. Label at least two level curves with their z-values. $$z=x^{2}+y^{2} ;[-4,4] \times[-4,4]$$

Consider the following functions and points \(P\). a. Find the unit vectors that give the direction of steepest ascent and steepest descent at \(P\). b. Find a vector that points in a direction of no change in the function at \(P\). $$f(x, y)=2 \sin (2 x-3 y) ; P(0, \pi)$$

Implicit differentiation Use Theorem 15.9 to evaluate dy/dx. Assume each equation implicitly defines y as a differentiable function of \(x\). $$x^{3}+3 x y^{2}-y^{5}=0$$