Solve using Lagrange multipliers. Find a vector in 3-space whose length is 5 and whose components have the largest possible sum.

Sketch a typical level surface for the function.

$$f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$$

Find the zeros of the function

$$f ( x ) = 3 x - 1 + \frac { 1 } { x + 1 }$$

Use the properties of logarithms to write each expression as a single logarithm.

$$\frac { 1 } { 3 } \ln 27 - 3 \ln ( 2 y )$$

Find the directional derivative of f at P in the direction of a vector making the counterclockwise angle θ with the positive x-axis.

$$f ( x , y ) = \frac { x - y } { x + y } ; P ( - 1 , - 2 ) ; \theta = \pi / 2$$

Suppose that u, v, and w are vectors in 3-space such that ||u||=5, $$ u\cdot v=7 $$ , and $$ u\cdot w=-3 $$ . (a) $$ u\codt u= $$ _, (b) $$ v\cdot u= $$ _, (c) $$ u\cdot (v-w)= $$ _, (d) $$ u\cdot (2w)= $$ _

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. The differential equation

$$\frac { d y } { d x } - e ^ { x y } y = 5 x \sqrt { y }$$

is a Bernoulli differential equation.

Find the distance between the given skew lines. x=1+7t, y=3+t, z=5-3t, x=4-t, y=6, z=7+2t

The 2010 Census revealed that 29% of all firms in the United States are owned by women. You call some firms doing business locally, assuming that the national percentage is true in your area. a) What's the probability that the first 3 you call are all owned by women? b) What's the probability that none of your first 4 calls finds a firm that is owned by a woman? c) Suppose none of your first 5 calls found a firm owned by a woman. What's the probability that your next call does?

A lowly high diver pushes off horizontally with a speed of 2.00 m/s from the platform edge 10.0 m above the surface of the water. (a) At what horizontal distance from the edge is the diver 0.800 s after pushing off? (b) At what vertical distance above the surface of the water is the diver just then? (c) At what horizontal distance from the edge does the diver strike the water?

Write an equation of the line in slope-intercept form that passes through the given points. (2, 5), (-2, 8)

Assume the MPC in an economy is 0.8 and the government increases government purchases of goods and services by $60 million. Also assume the absence of taxes, international trade, and changes in the aggregate price level.

What is the value of the spending multiplier?

How the Moon causes tides? Explain.

Solve the equation. Check your solution. r - 23 = -6