^{The number of new businesses established in the US since 1990 can be modeled by the function Nx=110.8x^3-5305.5x^2+76,701x+332,892 where x = 0 represents 1990 and the domain is [0, 25].}

5. Use the First Derivative Test to determine the extrema of the function on the given domain. Classify as max/min and absolute/relative. For what intervals does the function increase/decrease? Please label and interpret each answer in the context of this problem.

Estimate the area (A) between the graph of the function F(X)=3/X  and the interval [1,2]. Use an approximation scheme with N= 2, 5 rectangles. Use the right endpoints.

If your calculating utility will perform automatic summations, estimate the specified area using N=50 and N=100 rectangles.

Round your answers to three decimal places.

A2=
A5=
A10=
A50=
A100=

From a thin piece of cardboard 4 in. by 4 in., square corners are cut out so that the sides can be folded up to make an open box. What dimensions will yield a box of maximum volume? What is the maximum volume?

A quantity of air (γ = 1.40) expands adiabatically from an initial pressure of 2 atm and volume of 2 L at temperature 20 ° C to twice its original volume. Find:

a) The final pressure

b) The final temperature

The number of snowplows needed by a town is given by

s(p)=0.5p−0.01p^2. snowplows where p is the population (in thousands).

The projected population of the town t years from now is p(t)=150(1+12e^−0.02t)^-1 thousand people.

(a) Write the model for the rate-of-change of the number of snowplows needed t years from now.

(b) What will be the rate-of-change of the number of snowplows needed five years from now?

Let f(x) = (x² + 8) / 2x

a) Find all local minimum and local maximum values of f.

b) Find all intervals on which f is concave up/concave down. Does f have any infection points?

Consider the function below. (If an answer does not exist, enter DNE.)

f(x) =

x⁴ − 4x² + 6

(a) Find the interval of increase. (Enter your answer using interval notation.)

Find the interval of decrease. (Enter your answer using interval notation.)

(b) Find the local minimum value(s). (Enter your answers as a comma-separated list.)

Find the local maximum value(s). (Enter your answers as a comma-separated list.)

(c) Find the inflection points.

Find the interval where the graph is concave upward. (Enter your answer using interval notation.)

Find the interval where the graph is concave downward. (Enter your answer using interval notation.)

(d) Use the information from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one.

Solve the exponential equations

a) log₄(x-4) + log₄(x-10) = 2

b) (1/4)^x = 18

c) 3^4x-1 = 14

Find the approximate area under the curve by dividing the intervals into n subintervals and then adding up the areas of the inscribed rectangles. The height of each rectangle may be found by evaluating the function for each value of x. Your instructor will assign you n_1 = 4 and n_2 = 8.

1. y=2x√(x^2+1) Between x = 0 and x = 6 for n1 = 4, and n2 = 8

2. Find the exact area under the curve using integration y=2x√(x^2+1) Between x = 0 and x = 6

3. Explain the reason for the difference in answers.

Find the general solution of the differential equation y'' + 4y' + 4y = (e^-2t/t²)

Let ?(?)= (sqrt ?)

The local linearization of ? at 49 is ? (subscript) 49(?) = ? (subscript)0 +?(?−49) is __________
where ?= __________ and ? (subscript) 0= _____________

Using this, we find our approximation (sqrt 49.3) = ___________

) Find the VA / HA / maximum and minimum values of f(x) = x^ 2 + 1/ x^ 2 − 3

Show the feasible region by shading of the following system of linear inequalities and show all the intersection points (label your graph carefully): Hint: Use the guidelines shown in lectures and show all calculations for everything.

4? + ? ≥ 16

5? + 4? ≥ 40

? + 5? ≥ 20