Use Newton's method to find all solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. (Enter your answers as a comma-separated list.)

6e^−x2 sin(x) = x² − x + 1

Find the exact location of all the relative and absolute extrema of the function. HINT [See Example 1.] (Order your answers from smallest to largest x.)

f(x) = 117x⁴ − 78x³ with domain [−1, +∞)

f has  ---Select--- a relative minimum, a relative maximum, an absolute minimum, an absolute maximum, no extremum, at

(x, y) =

f has  ---Select--- a relative minimum, a relative maximum, an absolute minimum, an absolute maximum, no extremum, at

(x, y) =

Find a possible formula for the trigonometric function whose values are in the following table.

y =

For each of the following vector fields F , decide whether it is conservative or not by computing curl F . Type in a potential function f (that is, ∇f=F). If it is not conservative, type N.

A. F(x,y)=(10x+7y)i+(7x+10y)jF(x,y)=(10x+7y)i+(7x+10y)j
f(x,y)=f(x,y)=

B. F(x,y)=5yi+6xjF(x,y)=5yi+6xj
f(x,y)=f(x,y)=

C. F(x,y,z)=5xi+6yj+kF(x,y,z)=5xi+6yj+k
f(x,y,z)=f(x,y,z)=

D. F(x,y)=(5siny)i+(14y+5xcosy)jF(x,y)=(5sin⁡y)i+(14y+5xcos⁡y)j
f(x,y)=f(x,y)=

E. F(x,y,z)=5x2i+7y2j+5z2k

T-SHIRTS

A school organization has found four different places from which they may place order of t-shirts. Each function below could represent the cost of placing a t-shirt order as a function of the number of t-shirts purchased. c represents cost in dollars, t represents # of t-shirts.

A) c = 5t
B) c =3.25t + 55
C) c =3t + 100
D) c =6t - 55

DIRECTIONS FOR THE T-SHIRTS PROBLEM

Graph each function (equation)

• Graph all four functions on the same graph (color code) - by hand

• Graph data for up to 90 t-shirts at least. (use Desmos to get an overall picture of what graphs look like)

• Write a reflection for the graph

~Explanation as what the graph shows. Make sure to include:

• Comparison of costs - when is it cheapest, most expensive, etc.

-:- Give at least 6 pieces of comparison information from the graph

• What the slope represents in each function

• What the y-intercept represents in each function

Newton's Law of Cooling tells us that the rate of change of the temperature of an object is proportional to the temperature difference between the object and its surroundings. This can be modeled by the differential equation d T d t = k ( T − A ) , where T is the temperature of the object after t units of time have passed, A is the ambient temperature of the object's surroundings, and k is a constant of proportionality. Suppose that a cup of coffee begins at 188 degrees and, after sitting in room temperature of 65 degrees for 16 minutes, the coffee reaches 181 degrees. How long will it take before the coffee reaches 168 degrees? Include at least 2 decimal places in your answer.

A symmetrical 2D plate is bounded by the curve ?=?2 and ? =2−?2. Find the first moments and the Centroid of the plate.

Make a neat Sketch of the plate; show the element of area with its dimensions; clearly show the arm length for the element of area; write the differentials of the area and the first moment.

find the solution of the given initial value problem.

y′′−2y′−3y=3te^2t y(0)=1 y′(0)=0

Consider the region R, which is bounded by the curves  y=3x and x=y(4−y).

(a) Set up, but DO NOT SOLVE, an integral to find the area of the region RR.

(b) Set up, but DO NOT SOLVE, an integral to find the volume of the solid resulting from revolving the region RRaround the xx-axis.

(c) Set up, but DO NOT SOLVE, an integral to find the volume of the solid resulting from revolving the region RRaround the line x=−5x=−5.

Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

dx, n = 10

What are the dimensions of the box so that the cost of materials is minimized?

A closed rectangular box has a volume of 16 cubic feet. It’s made of two different materials; the top and bottom are made of a material that costs 10 cents per square foot while the sides are made of material that costs 5 cents per square foot.

Let z = x2y5 , x = sin t + t3 , y = 2t2 – t . Find dz/dt and d2z/dt2.

Using the methods of Lagrange multipliers, find out the point on the plane 2x+y-2z=5 closest to the origin. Please show all work.