Exercise 2.5.1 Suppose T : R n ? R n is a linear transformation. Prove that T is an isometry if and only if T(v) · T(w) = v · w. Recall that an isometry is a bijection that preserves distance.

A company produces a special new type of TV. The company has fixed costs of $471,000 and it costs $1200 to produce each Tv. The company projects that if it charged $2300 for the TV it will sell 700. If the company wants to sell 750 the price must be $2000. Assume a linear demand.

what price should the company charge to earn a profit of $679,000

It would need to charge..?

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If f(c) = L,then lim x→c f(x) = L.

False. Define f to be the piece-wise function where f(x) = x + 3 when x ≠ −1 and f(x) = 2 when x = −1. Then we have f(−1) = 2 while the limit of f as x approaches −1 is equal to −2.

False. Define f to be the piece-wise function where f(x) = x − 4 when x ≠ 2 and f(x) = 0 when x = 2. Then we have f(2) = 0 while the limit of f as x approaches 2 is equal to −2.

False. If f(c) = L, then the limit of f as x approaches c is equal to L/c.

False. If f(c) = L, then the limit of f as x approaches c is equal to cL.

The statement is true.

The amount of money in an investment is modeled by the function A(t)=650(0.943)t. The variable A represents the investment balance in dollars, and t the number of years since 2006.

(A) In 2006, the balance was $.

(B) The amount of money in the investment is

(C) The annual rate of change in the balance is
r= or r=%.

(D) In the year 2016 the investment balance will equal
$. Round answer to the nearest penny.

(1 point) Please answer the following questions about the function f(x)=5x2x2−9.

Instructions: If you are asked for a function, enter a function. If you are asked to find x- or y-values, enter either a number or a list of numbers separated by commas. If there are no solutions, enter None. If you are asked to find an interval or union of intervals, use interval notation. Enter { } if an interval is empty. If you are asked to find a limit, enter either a number, I for ∞, -I for −∞, or DNE if the limit does not exist.

(a) Calculate the first derivative of f. Find the critical numbers of f, where it is increasing and decreasing, and its local extrema.

f′(x)= ______________

Critical numbers x= ____________________

Union of the intervals where f(x) is increasing ______________________

Union of the intervals where f(x) is decreasing __________________________

Local maxima x= ________________________

Local minima x= _______________________

(b) Find the following left- and right-hand limits at the vertical asymptote x=−3.

limx→−3−5x2x2−9= ? limx→−3+5x2x2−9= ?

Find the following left- and right-hand limits at the vertical asymptote x=3.

limx→3−5x2x2−9=? limx→3+5x2x2−9=?

Find the following limits at infinity to determine any horizontal asymptotes.

limx→−∞5x2x2−9=? limx→+∞5x2x2−9= ?

(c) Calculate the second derivative of f. Find where f is concave up, concave down, and has inflection points.

f′′(x)= __________________________

Union of the intervals where f(x) is concave up ___________________

Union of the intervals where f(x) is concave down _____________________

Inflection points x= _______________________

d) The function f is_____ because_____ for all x in the domain of f, and therefore its graph is symmetric about the ________

(e) Answer the following questions about the function f and its graph.

The domain of f is the set (in interval notation) ______________________________

The range of f is the set (in interval notation) y-intercept x-intercepts _______________________

(f) Sketch a graph of the function f without having a graphing calculator do it for you. Plot the y-intercept and the x-intercepts, if they are known. Draw dashed lines for horizontal and vertical asymptotes. Plot the points where f has local maxima, local minima, and inflection points. Use what you know from parts (a) - (c) to sketch the remaining parts of the graph of f. Use any symmetry from part (d) to your advantage. Sketching graphs is an important skill that takes practice, and you may be asked to do it on quizzes or exams.

1) A tank is full of water. Find the work W required to pump the water out of the spout. (Use 9.8 m/s² for g. Use 1000 kg/m³ as the weight density of water. Assume that a = 4 m, b = 4 m, c = 12 m,and d = 1 m.)
W = ___ J

I got 5017600 then in scientific notation it would be 5.0176 X 10^6, then in Juls in would be 5.02?

I got the answer wrong an would like to know how you do this problem?

2. Suppose that 5 J of work is needed to stretch a spring from its natural length of 32 cm to a length of 44 cm.

(a) How much work is needed to stretch the spring from 40 cm to 42 cm? (Round your answer to two decimal places.)
1.25 J

(b) How far beyond its natural length will a force of 40 N keep the spring stretched? (Round your answer one decimal place.)
?  cm

for B I got 11.52, did I do something wrong?

Find the solution.

y'' + 2y' + 0.75y = 2cosx - 0.25sinx + 0.09x , y(0) = 2.78 , y'(0) = -0.43

Prove that the product of a rotation and a translation is a rotation.

Given the following Axioms of Fano's geometry:

1. There exists at least one line

2. Each line is on exactly 3 points

3. Not all points are on the same line

4. Each pair of points are on exactly one line

5. Each pair of lines are on at least one point

a) Prove every point is on exactly three lines

b) What geometries are possible if you eliminate Axiom 5?

1. Jade has $24,500 to invest. She decides to divide it into the three accounts listed. At the end of the year, she has earned $1,300 in interest. If the amount invested in Hanes Oil was four times the amount of money invested in Gary Games, how much did Jade invest in each account?

1. Write down the system of equations describing this situation. Be sure to define the variables.

b) State the matrix for this system of equations.

c) Write down the reduced-row echelon form (i.e., the “answer matrix”) given by your calculator.

d) Explain in 1-2 sentences the meaning of these results.

create a speaker box with a volume of 14400 cubic inches. The box will be used for speakers of various sizes. The speaker box must have a square base and minimal surface area ( prior to cutting holes for the speakers ). Find the box’s dimensions. Your must construct a diagram [ V = LWH ]

2.) (12 pts.) Show that F = ( xy/(1+x^2y^2) + 1 + arctan(xy))i+ (x^2/(1+x^2y^2-1)j is a conservative vector field. Then use the Fundamental Theorem for Line Integrals to find the Work done by F from point (0,0) to point (2, 1/2).

Find the point of intersection of the lines (x-2)/- 3 = (y-2)/6 = z-3 & (x-3)/2 = y+5 = (z+2)/4. Write the answer as (a, b, c). If they are not cut, write: NO

A) Using a calculus approach, sketch p(x) = -3*x^2 – 300x – 140.

B) What is the largest rectangle that can be inscribed inside the curve, if two of its sides are on the x-axis and the other two lie on the positive portion of the curve.