Find the 5th Taylor polynomial of f(x) = 1+x+2x^5 +sin(x^2) based at b = 0.

Draw the graph, solid of revolution, one representative disk/ washer.

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region formed by

a) when revolved about y-axis, the volume is ?

b) when revolved about x-axis, the volume is ?

c) when revolved about the line y=8, the volume is ?

d) when revolved about the line x=2, the volume is ?

a. Verify that the given point lies on the curve.

b. Determine an equation of the line tangent to the curve at the given point.

9 (x² y²)² =100xy² ; (1,3)

a)

Let y be the solution of the equation  y ′ = (y/x)+1+(y^2/x^2) satisfying the condition  y ( 1 ) = 0. Find the value of the function  f ( x ) = (y ( x ))/x

at  x = e^(pi/4) .

b)

Let y be the solution of the equation  y ′ = (y/x) − (y^2/x^2) satisfying the condition  y ( 1 ) = 1. Find the value of the function  f ( x ) = x/(y(x))

at  x = e  .

c)

Let y be the solution of the equation

y ′ + (3x^2*y)/(1+x^3)=e^x/(1+x^3) satisfying the condition  y ( 0 ) = 1.

Find  ln ⁡ ( 2 y ( 1 ) ).

Consider the function on the interval (0, 2π). f(x) = sin(x)/ 2 + (cos(x))^2 (a) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.) increasing decreasing (b) Apply the First Derivative Test to identify the relative extrema. relative maximum (x, y) = relative minimum (x, y) =

Find the dimensions (in inches) of the rectangular package of maximum volume subject to the constraint that the sum of the length and the girth cannot exceed 192 inches (see figure). (Hint: Maximize V = xyz subject to the constraint x + 2y + 2z = 192.)

Let R be a ring with at least two elements. Prove that M2×2(R)is always a ring (with addition and multiplication of matrices defined as usual).

-3,7 11,-15

A. Find Point Slope Form

B. Find Slope Intercept

C. Find X and Y intercepts

Sketch a graph

Show that of all isosceles triangles with a given perimeter , the one with greatest area is equilateral.

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) = x³y + 24x² − 8y

The demand equation for your company's virtual reality video headsets is

p=1000/q^.3

where q is the total number of headsets that your company can sell in a week at a price of p dollars. The total manufacturing and shipping cost amounts to $140 per headset.
(a) What is the greatest profit your company can make in a week? (Give your answer to the nearest whole number.)

How many headsets will your company sell at this level of profit? (Give your answer to the nearest whole number.)

(b) How much, to the nearest $1, should your company charge per headset for the maximum profit?

For the given polynomial, find all zeros of the polynomial algebraically. Factor the polynomial completely. ?(?) = ?^4 − 2?^3 − 2?^2 − 2? − 3

For the given polynomial, find all zeros of the polynomial algebraically. Factor the polynomial completely. ?(?) = 6?^4 − 7?^3 − 12?^2 + 3? + 2

Let f(x) = 1 + x − x² +e^x-1.

(a) Find the second Taylor polynomial T₂(x) for f(x) based at b = 1.

b) Find (and justify) an error bound for |f(x) − T2(x)| on the interval
[0.9, 1.1]. The f(x) - T₂(x) is absolute value.

Please answer both questions cause it will be hard to post them separately.

An investor is considering three types of investments: a high risk venture into oil leases with a potential return of 15%, a medium risk investment in bonds with a 9% return, and a relatively safe stock investment with a 5% return. He has $50,000 to invest. Because of the risk, he will limit his investments in oil leases and bonds to 30% and his investments in oil leases and stock to 50%. How much should he invest in each to maximize his return, assuming investment returns are as expected?

a. Define the variables

b. Clearly state the constraints (all inequalities) related to the feasible region

c. State the objective function

d. Set up the initial simplex matrix needed to solve the linear programming problem using the Simplex Method

e. Perform all pivots necessary using row operations to transform the matrix until the solution is feasible

f. How much should he invest in each to maximize his return, assuming investment returns are as expected?

prove using the definition of derivative that if f(x) and g(x) is differentiable than (f'(x)g(x) - f(x)g'(x))/g^2(x)