6E: Chapter Review Excersies

Exercise \(\PageIndex{1}\)

Evaluate the indicated limit or explain why it does not exist.

1) \(\displaystyle \lim_{(x,y)\to(0,0)} {2\sqrt{x^2+y^2}}\)

2) \(\displaystyle \lim_{(x,y)\to(0,0)} \frac{3x}{x^2+y^2}\)

3) \(\displaystyle \lim_{(x,y)\to(0,0)} \frac{2\sin(xy)}{x^2+y^2}\)

4) \(\displaystyle \lim_{(x,y)\to(0,0)} \frac{4x^2y^2}{x^2+y^4}\)

Answer

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Exercise \(\PageIndex{2}\)

Define \(f(0,0)\) in a way that extends \[f(x,y)=2xy \frac{x^2-y^2}{x^2+y^2}\[ to be continuous at the origin.

Answer

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Exercise \(\PageIndex{3}\)

Find the first partial derivative of \[f(x,y,z)=3x^{(y\ln z)}\[ at \((e,2,e)\).

Answer

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Exercise \(\PageIndex{5}\)

1) Find the equations of the tangent plane and normal line to the graph of \[f(x,y)=\tan^{-1}(y/x)\[ at \((1,-1).\)

2) Given \(f(x,y)=\ln(x^2+y^2).\)

a) Find an equation of the plane tangent to the graph of \(f\) at \((1,-2).\)

b) Find an equation of the tangent line at \((1,-2)\) to the level curve of \(f\) that passes through \((1,-2).\)

Answer

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Exercise \(\PageIndex{6}\)

1) Find and classify all the critical points of \(g(x,y)=2xye^{-x+y}.\)

2) Find the maximum and minimum values of \(f(x,y)=3xy\) on the closed disk \(x^2+y^2 \leq 9.\)

Answer

1) \((0,0),(1-1) local minimum\)

Exercise \(\PageIndex{7}\)

Find the Jacobian matrix \(D{\bf f}(x,y,z)\) for the transformation of \(\Re^2\) to \(\Re^3\)

given by \[{\bf f}(x,y)=(xe^y+cos(\pi y),x^2 z,x-e^y).\[ Use it to find an approximate value for \({\bf f}(1.02,0.01).\)

Answer

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Exercise \(\PageIndex{8}\)

The temperature at position \((x,y)\) in a region of the \(xy-\)plane is \(T^{\circ} C,\) where \(T(x,y)= x^2 -x+y+2y^2.\)

1) In what direction should an ant at position \((3,-2)\) move if it wishes to cool off as quickly as possible?

2) If the ant moves in that direction at speed \(k\) ( units distance per unit time), at what rate does it experience a decrease of temperature?

3) At what direction should an ant move from \((1,-1)\) to experience zero temperature change

Answer

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Exercise \(\PageIndex{9}\)

By using Lagrange multipliers solve the following:

A rectangular box having no top and having a prescribed volume \(V \,m^3\) is to be constructed using two different materials. The material used for the bottom and front of the box is five times as costly (per square metre) as the material used for the back and the other two sides. what should be the dimensions of the box to minimize the cost of materials?

Answer

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Exercise \(\PageIndex{10}\)

By using Least square approximation to approximate \(g(x)=x^3\) over the interval \([0,1]\) by a linear function \(f(x)=px^2+q.\)

Answer

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