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

For problems 1-4 the linear transformation T/; R^n - R^m  is defined by T(v)=AV with

A=[-1 -2 -1 -1

2 1 0 -4

4 6 1 15]

1.   Find the dimensions of R^n and R^m (3 pts)

2.   Find T(<-2, 1, 4, -1>) (5 pts)

3.   Find the preimage of <-6, 12, 4> (8 pts)

4.   Find the Ker(T) (8 pts)

1-The sum of two numbers is 34.
    a)Find the largest possible product of these numbers.
    b)What would be the largest possible product if the sum if the two numbers were "k"?

2-Sixty meters of fencing are used to fence a rectangular garden.
    a)Find the dimensions that will give that maximum area.
    b)What would be the maximum area if "k" feet of fencing were used in terms of "k"?

THANK YOU

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x, y) = 3x^3 − 9x + 9xy^2