MATH 524 Fall 17 Chapter 1
Basic, Intermediate and CAS problems

Find and sketch the domain of the following functions.
 \( f(x,y) = \sqrt{yx2} \)
 \( f(x,y) = \log \big( x^2+y^24 \big) \)
 \( f(x,y) = \dfrac{(x1)(y+2)}{(yx)(yx^3)} \)
 \( f(x,y) = \log (xy+xy1) \)

Find and sketch the level lines \( f(x,y)=c \) on the same set of coordinate axes for the given values of \( c \).
 \( f(x,y) = x+y1, c \in \{ 3, 2, 1, 0, 1, 2, 3 \}. \)
 \( f(x,y) = x^2+y^2, c \in \{ 0, 1, 4, 9, 16, 25 \}. \)
 \( f(x,y) = xy, c \in \{ 9, 4, 1, 0, 1, 4, 9 \}. \)

Use a Computer Algebra System of your choice to produce contour plots of the given functions on the given domains.
 \( f(x,y) = (\cos x)(\cos y) e^{\sqrt{x^2+y^2}/4} \) on \( [2\pi, 2\pi] \times [2\pi, 2\pi]. \)
 \( g(x,y) = \dfrac{xy(x^2y^2)}{x^2+y^2} \) on \( [1,1] \times [1,1]. \)
 \( h(x,y) = y^2  y^4 x^2 \) on \( [1,1]\times[1,1] \).
 \( k(x,y) = e^{y}\cos x \) on \( [2\pi, 2\pi]\times[2,0]. \)

Sketch the curve \( f(x,y)=c \) together with \( \nabla f \) and the tangent line at the given point \( P \). Write an equation for the tangent line.
 \( f(x,y)=x^2+y^2, c=4, P=(\sqrt{2}, \sqrt{2}). \)
 \( f(x,y)=x^2y, c=1, P=(\sqrt{2}, 1). \)
 \( f(x,y)=xy, c=1, P=(2, 1/2). \)
 \( f(x,y)=x^2xy+y^2, c=7, P=(1,2). \)
 For the function \( f(x,y) = \dfrac{xy}{x+y} \) at the point \( P_0 = (1/2, 3/2) \), find the directions \( \boldsymbol{v} \) and the directional derivatives \( D_{\boldsymbol{v}}f(P_0) \) for which
 \( D_{\boldsymbol{v}}f(P_0) \) is largest.
 \( D_{\boldsymbol{v}}f(P_0) \) is smallest.
 \( D_{\boldsymbol{v}}f(P_0) = 0. \)
 \( D_{\boldsymbol{v}}f(P_0) = 1. \)
 \( D_{\boldsymbol{v}}f(P_0) = 2. \)

The derivative of \( f(x,y) \) at \( (1,2) \) in the direction \( \frac{\sqrt{2}}{2}[1,1] \) is \( 2\sqrt{2} \) and in the direction \( [0,1] \) is \( 3 \). What is the derivative of \( f \) in the direction \( \frac{\sqrt{5}}{5}[1,2] \)?

Find the absolute maxima and minima of the function \( f(x,y) = (4xx^2)\cos y \) on the rectangular plate \( 1\leq x \leq 3, \frac{\pi}{4} \leq y \leq \frac{\pi}{4}. \)

Find two numbers \( a \leq b \) such that \( \int_a^b (242xx^2)^{1/3}\, dx \) has its largest value.

Find the points of the hyperbolic cylinder \( x^2z^21=0 \) in \( \mathbb{R}^3 \) that are closest to the origin.

Find the extreme values of the function \( f(x,y,z)=xy+z^2 \) on the circle in which the plane \( yx=0 \) intersects the sphere \( x^2+y^2+z^2=4. \)

Write a routine (in your favorite CAS) that uses symbolic computation to find the minimum of a differentiable realvalued function \( f \colon \mathbb{R} \to \mathbb{R} \) over
 a closed interval \( [a,b] \)
 An interval of the form \( [a,\infty) \), or \( (\infty, b] \)
The routine should accept as input:
 the expression of the function \( f \),
 the endpoints \( a,b. \)