Chapter 13

Describe the appearance of a smooth surface with a local maximum at a point.

Explain what $$\lim _{(x, y) \rightarrow(a, b)} f(x, y)=$$ \(L\) means.

Suppose \(\mathbf{n}\) is a vector normal to the tangent plane of the surface \(F(x, y, z)=0\) at a point. How is \(\mathbf{n}\) related to the gradient of \(F\) at that point?

Explain how a directional derivative is formed from the two partial derivatives \(f_{x}\) and \(f_{y}\).

Give two pieces of information which, taken together, uniquely determine a plane.

Suppose \(z=f(x, y),\) where \(x\) and \(y\) are functions of \(t .\) How many dependent, intermediate, and independent variables are there?

A function is defined by \(z=x^{2} y-x y^{2} .\) Identify the independent and dependent variables.

Suppose you are standing on the surface \(z=f(x, y)\) at the point \((a, b, f(a, b)) .\) Interpret the meaning of \(f_{x}(a, b)\) and \(f_{y}(a, b)\) in terms of slopes or rates of change.

If \(f(x, y)=x^{2}+y^{2}\) and \(g(x, y)=2 x+3 y-4=0,\) write the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes \(f\) subject to \(g(x, y)=0\).

Describe the usual appearance of a smooth surface at a saddle point.

Explain why \(f(x, y)\) must approach \(L\) as \((x, y)\) approaches \((a, b)\) along all paths in the domain in order for $$\lim _{(x, y) \rightarrow(a, b)} f(x, y)$$ exist.

Write the explicit function \(z=x y^{2}+x^{2} y-10\) in the implicit form \(F(x, y, z)=0\).

How do you compute the gradient of the functions \(f(x, y)\) and \(f(x, y, z) ?\)

Find a vector normal to the plane \(-2 x-3 y+4 z=12\)

Let \(z\) be a function of \(x\) and \(y,\) while \(x\) and \(y\) are functions of \(t\). Explain how to find \(\frac{d z}{d t}\).

What is the domain of \(f(x, y)=x^{2} y-x y^{2} ?\)

Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=3 x^{2} y+x y^{3}.\)

If \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) and \(g(x, y, z)=\) \(2 x+3 y-5 z+4=0,\) write the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes \(f\) subject to \(g(x, y, z)=0\).

What are the conditions for a critical point of a function \(f ?\)

What does it mean to say that limits of polynomials may be evaluated by direct substitution?

Write an equation for the plane tangent to the surface \(F(x, y, z)=0\) at the point \((a, b, c)\)

Interpret the direction of the gradient vector at a point.

Where does the plane \(-2 x-3 y+4 z=12\) intersect the coordinate axes?

Suppose \(w\) is a function of \(x, y,\) and \(z,\) which are each functions of t. Explain how to find \(\frac{d w}{d t}\).

What is the domain of \(g(x, y)=1 /(x y) ?\)

Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=x \cos (x y).\)

If \(f_{x}(a, b)=f_{y}(a, b)=0,\) does it follow that \(f\) has a local maximum or local minimum at \((a, b) ?\) Explain.

Suppose \((a, b)\) is on the boundary of the domain of \(f .\) Explain how you would determine whether $$\lim _{(x, y) \rightarrow(a, b)} f(x, y)$$ exists.

Give an equation of the plane with a normal vector \(\mathbf{n}=\langle 1,1,1\rangle\) that passes through the point (1,0,0)

Let \(z=f(x, y), x=g(s, t),\) and \(y=h(s, t) .\) Explain how to find \(\partial z / \partial t\).

What is the domain of \(h(x, y)=\sqrt{x-y} ?\)

Find the four second partial derivatives of \(f(x, y)=3 x^{2} y+x y^{3}.\)

Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=x+2 y \text { subject to } x^{2}+y^{2}=4$$

What is the discriminant and how do you compute it?

Explain how examining limits along multiple paths may prove the nonexistence of a limit.

Explain how to approximate a function \(f\) at a point near \((a, b)\) where the values of \(f, f_{x},\) and \(f_{y}\) are known at \((a, b)\)

Given a function \(f,\) explain the relationship between the gradient and the level curves of \(f\).

To which coordinate axes are the following cylinders in \(\mathbb{R}^{3}\) parallel: \(x^{2}+2 y^{2}=8, z^{2}+2 y^{2}=8,\) and \(x^{2}+2 z^{2}=8 ?\)

Given that \(w=F(x, y, z),\) and \(x, y,\) and \(z\) are functions of \(r\) and \(s\), sketch a Chain Rule tree diagram with branches labeled with the appropriate derivatives.

Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=x y^{2} \text { subject to } x^{2}+y^{2}=1$$

Explain how the Second Derivative Test is used.

Explain why evaluating a limit along a finite number of paths does not prove the existence of a limit of a function of several variables.

Explain how to approximate the change in a function \(f\) when the independent variables change from \((a, b)\) to \((a+\Delta x, b+\Delta y)\)

The level curves of the surface \(z=x^{2}+y^{2}\) are circles in the \(x y\) -plane centered at the origin. Without computing the gradient, what is the direction of the gradient at (1,1) and (-1,-1) (determined up to a scalar multiple)?

Describe the graph of \(x=z^{2}\) in \(\mathbb{R}^{3}\).

Suppose \(F(x, y)=0\) and \(y\) is a differentiable function of \(x\). Explain how to find \(d y / d x\).

Explain how to graph the level curves of a surface \(z=f(x, y)\).

The volume of a right circular cylinder with radius \(r\) and height \(h\) is \(V=\pi r^{2} h .\) Is the volume an increasing or decreasing function of the radius at a fixed height (assume \(r>0\) and \(h>0\) )?

Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=x+y \text { subject to } x^{2}-x y+y^{2}=1$$

What three conditions must be met for a function \(f\) to be continuous at the point \((a, b) ?\)