x' = x − y

y' = 4y − x^2*y

Linearize the system about the point (2, 2). Classify the type and stability of the critical point at (2, 2) by examining the linearized system.

a) A salt-water tank has pure water flowing into it at 5 L/min. The contents of the tank are kept thoroughly mixed, and the contents flow out at 5 L/min. Initially, the tank contains 1 kg of salt in 10L of water. How much salt will be in the tank after 20 minutes? Let ?? represent the amount of salt in the tank at time t and let ?? represent the volume of saltwater in the tank at time t.

b) Now assume that salt is added into the tank at a rate of 0.1 kg/min with pure water still flowing into it at 5 L/min. The contents of the tank are kept thoroughly mixed, and the contents flow out at 5 L/min. Initially, the tank contains 1 kg of salt in 10L of water. How much salt will be in the tank after 20 minutes?

Given f(x) = (x⁴ - 2)(x⁵ - 10x + 1)³

Find the definite integral of f(x) on the closed interval [0, 1].

1) 0.05

2) 204.75

3) 4095

4) None

Use exponential regression to find an exponential function that best fits this data.

f(x) =     

Use linear regression to find an linear function that best fits this data.

g(x) =    

given DE has a regular singular point at x=0 determine two solutions for x>0

x^2y''+3xy'+(1+x)y=0

Let F(x,y,z) = < z tan^-1(y²), z³ ln(x² + 1), z >. Find the flux of F across S, the top part of the paraboloid x² + y² + z = 2 that lies above the plane z = 1 and is oriented upward. Note that S is not a closed surface.

Question : y''=4y'+8y=0 , (y''-4y'+13y)^2=0 , (y''+2y'+2y)^2=0 ,

y''-6y'+13y=0,y(0)=3 , y'(0)=13 , 2y''-6y'+17y=0,y(0)=2, y'(0)=13

Approximently 72% of freshmen entering public high schools in the United States in 2005 graduated with their class in 2009. A random sample of 136 freshmen is chosen. around your answers to four decimal places as needed.

Part 1 of 6

Find the mean u ^p is

Find ??, ?? and ?? of F(x, y, z) = tan(x+y) + tan(y+z) – 1

Consider the function f(x)= 1 + 1/x - 1/x²

^{Find the domain, the vertical and horizontal asymptotes, the intervals of increase or decrease, the local minimum and maximum values, the intervals of concavity and the inflection points.}

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.