A. Find the region bounded by the curves y = (x−3)^2 and y = 12−4x. Show all of your work.

B. Find the equation of the tangent line to the curve 5x^2 −6xy + 5y^2 = 4 at the point (1,1) Show all of your work. Thanks

You run a small furniture business. You sign a deal with a customer to deliver up to 600 chairs, the exact number to be determined by the customer later. The price will be $ 130 per chair up to 500 chairs, and above 500 , the price will be reduced by $ 0.25 per chair (on the whole order) for every additional chair over 500 ordered.

An investment offers an onetime bonus of 2% of the principal after being invested for 5 years. If $50 000 is invested at 4.75% compounded annually for 10 years, describe how the graph of the investment with the bonus differs from the graph of the investment without the bonus. Include any calculations.

In general, do not use any derivative rules ( as power rule, etc.). Find the derivative of the following functions at the point "a" by computing average rate of changes for h = ±1, ±0.1, ±0.01, ±0.001, ±0.0001 :

a.) f(x) = x² + x³ , a = 1

b.) f(x) = x ²+ x³ , a = 3

c.)  . f(x) = x − x² , a = 2

(a) Let λ be a real number. Compute A − λI.

(b) Find the eigenvalues of A, that is, find the values of λ for which the matrix A − λI is not invertible. (Hint: There should be exactly 2. Label the larger one λ1 and the smaller λ2.)

(c) Compute the matrices A − λ1I and A − λ2I.

(d) Find the eigenspace associated with λ1, that is the set of all solutions v = v1 v2 to (A − λ1I)v = 0.

(e) Find the eigenspace associated with λ2 similarly.

Repeat the process to find the eigenvalues and corresponding eigenspaces for

A = [ 0 2 0

2 0 0

1 1 4 ]

(Note that this matrix has three eigenvalues, not 2.)

Solve the given equation.

(tan²(θ) − 16)(2 cos(θ) + 1) = 0

θ =

x^2y''+2xy'+x^2y=0

Determine y if    

y1=(sin(x))/x

Verifica si las ecuaciones diferenciales son exactas, de no serlo calcula el factor integrante. En ambos casos resuélvelas:

(5t^4 y-15t^2-y)dt+(t^5+3y^2-t)dy=0, y(1)=-2

cos⁡〖(x)〗 dx+(1+2/y) sin⁡x dy=0

For this parametrized curve:

x = e^(2t) sin t , y = cos(4t)

find tangent line to curve when t=1

Find the solution of the given initial value problem:

y^(4)+2y′′′+y′′+8y′−12y=12sint+40e^−t

y(0)=0, y′(0)=38/5,  y′′(0)=4/5,  y′′′(0)=−54/5

Prove: A quadrilateral is a square if and only if its diagonals are congruent and bisect each other at right angles.

Consider the following homogeneous linear system: x1 + 2x2 + 7x3 − 9x4 + 31x5 = 0 2x1 + 4x2 + 7x3 − 11x4 + 34x5 = 0 3x1 + 6x2 + 5x3 − 11x4 + 29x5 = 0 [10p] a) Find the rank of the coefficient matrix. [5p] b) Use part (a) to determine the dimension of the solution space. [10p] c) Find a basis for the solution space.

The population of bacteria​ (in millions) in a certain culture x hours after an experimental

nutrient is introduced into the culture is

​P(x)=25x/6+x^2.

Use the differential to approximate the changes in population for the following changes in x.a.

2

to

2.5

                                                                                                                               b.

3

to 3.25

y=（1-x)e^x

Use the "Guidelines for sketching a curve A-H"

A.) Domain
B.) Intercepts
C.) Symmetry
D.) Asymptotes
E.) Intervals of increase or decrease
F.) Local Maximum and Minimum Values
G.) Concavity and Points of Inflection
H.) Sketch the Curve