91Ó°ÊÓ

Find all points where the partial derivatives of \(f(x, y)\) are both 0. $$f(x, y)=e^{x^{2}+2 x+y^{2}}$$

Find the points on the surface \(y=4+x^{2}+z^{2}\) where the gradient is parallel to \(\vec{i}+\vec{j}+\vec{k}.\)

A student was asked to find the directional derivative of \(f(x, y)=x^{2} e^{y}\) at the point (1,0) in the direction of \(\vec{v}=4 \vec{i}+3 \vec{j} .\) The student's answer was $$ f_{i i}(1,0)=\operatorname{grad} f(1,0) \cdot \vec{u}=\frac{8}{5} \vec{i}+\frac{3}{5} \vec{j} $$ (a) At a glance, how do you know this is wrong? (b) What is the correct answer?

Find all points where the partial derivatives of \(f(x, y)\) are both 0. $$f(x, y)=x^{3}+3 x^{2}+y^{3}-3 y$$

Are the statements true or false? Give reasons for your answer. If \(f(x, y)\) is a function with the property that \(f_{x}(x, y)\) and \(f_{y}(x, y)\) are both constant, then \(f\) is linear.

give an example of: A surface in three space whose tangent plane at (0,0,3) is the plane \(z=3\)

For the surface \(z+7=2 x^{2}+3 y^{2},\) where does the tangent plane at the point (-1,1,-2) meet the three axes?

Is there a function \(f\) which has the following partial derivatives? If so, what is it? Are there any others? $$\begin{array}{l}f_{x}(x, y)=4 x^{3} y^{2}-3 y^{4} \\\f_{y}(x, y)=2 x^{4} y-12 x y^{3} \end{array}$$.

In Problems \(58-64,\) find the quantity. Assume that \(g\) is a smooth function and that $$ \nabla g(2,3)=-2 \vec{i}+\vec{j} \quad \text { and } \quad \nabla g(2.4,3)=4 \vec{i} $$ $$g_{y}(2.4,3)$$

Are the statements true or false? Give measons for your answer. The tangent plane approximation of \(f(x, y)=y e^{x^{2}}\) at the point (0,1) is \(f(x, y) \approx y\)