verify Stokes' thm.Assume that the surface S is oriented upward F = 2zi - 3xj + 4yk ; S that portion of the paraboloid z =16 - x²- y² for z>=0. My primary is how to convert dS into dA

Use Laplace transforms to solve the following system of differential equations.

Evaluate the following limits using l'Hopital's rule.

(a) lim x→0 (sin(x)−x)/(x^2)

(b) lim x→0 (1/x) − (1/e^x−1)

(c) lim x→0+ (x^√ x)

a)

The functions y₁ = x^2 and y₂ = x^5 are two solutions of the equation

x^2 y ″ − 6 x y ′ + 10 y = 0.

Let  y be the solution of the equation x^2 y ″ − 6 x y ′ + 10 y = 3 x^5

satisfyng the conditions y ( 1 ) = 0 and  y ′ ( 1 ) = 1.

Find the value of the function  f ( x ) = y ( x ) / ln ⁡( x ) at  x = 2.

b)

The function y₁ = x^5 is a solution of the equation

x^2 y ″ − 11 x y ′ + 35 y = 0.

Let  y₂ be the solution of this equation satisfying the conditions

y₂ ( 1 ) = 0 and y₂ ′ ( 1 ) = 2.

Find the value of y₂ at x = 2.

c )

The function y₁ = x^7 is a solution of the equation

x^2 y ″ − 8 x y ′ + 14 y = 0.

Let  y₂ be the solution of this equation satisfying the conditions

y₂ ( 1 ) = 1 and y₂ ′ ( 1 ) = 2.

Find the value of y₂ at x = 2.

Use the Laplace transform to solve the given initial value problem.

y⁽⁴⁾ − 4y''' + 6y'' − 4y' + y = 0;

y(0) = 1,

y'(0) = 0,

y''(0) = 0,

y'''(0) = 1

Iodine 131 is a radioactive isotope that decays as time passes. The function M(t) = 200e^-0.093t gives the mass of Iodine 131 remaining in the sample measured in grams in terms of t number of days since the samples mass was originally measured.

a) What was the initial mass of Iodine 131?

b) What is the one days decay factor and what is the one day percent change?

c) How long will it take for the sample to decay such that only half of the original mass of Iodine 131 still remains?

d) How long will it take for only 35 grams of Iodine 131 to still be present in the sample?

e) What is the mass of the remaining Iodine 131 after one week has passed since the mass was originally measured?

f) What is the one week growth factor, and what is the one week percent change?

g) Write a function B that closely approximates the mass of the remaining Iodine 131 in the samplemeasured in grams in terms of the number of weeks w since the samples mass was originally measured?

h) What is the one hour growth factor, and what is the one hour percent change?

g) Write a function C that closely approximates the mass of the remaining Iodine 131 in the sample measured in grams in terms of the number of hours h since the samples mass was originally measured?

Solve the following system of equations using the Substitution Method.
1.x + 3y = – 2
5x + 15y = 0
2. x – 4y = 10
3x – 2y = 10
3. 4a + 7b = 54
2a – 3b = 14
4. 2x – 3y = 1
8x – 12y = 4
5. 3x + 4y = 12
6x + 8y = 24
6.  2a – 5b = 10
3a – b = 2

I am stuck in Austin with a flat tire, and I need to get to my class in Houston within 2 hours. The drive is 180 miles long, but I want to be careful of the Austin cops. I do not usually get caught speeding unless I am seen accelerating too fast, so I do not want to accelerate at a rate of more than 120 mi?/?h². Just to be careful, I?m going to take exactly 2 hours to make the trip. Assuming I accelerate at 120 mi?/?h² for a while, and travel at a constant speed afterwards, what’s the fastest speed I?ll be going during my trip?

2. One week a computer store sold a total of 36 computers and external hard drives. The revenue from these sales was $25600. If computers sold for $1180 per unit and hard drives for $125 per unit, how many of each did the store sell?

How many computers were​ sold?

How many external hard drives were​ sold?

3. You invested $7000 between two accounts paying 3% and 9% annual​ interest, respectively. If the total interest earned for the year was $ 450, how much was invested at each​ rate?

$ ___?__ was invested at 3% and $ ___?___ was invested at 9%.

4. You invested money in two funds. Last​ year, the first fund paid a dividend of​ 9% and the second a dividend of​ 3%, and you received a total of $858. This​ year, the first fund paid a​ 10% dividend and the second only​ 1%, and you received a total of $832. How much money did you invest in each​ fund?

How much money was invested in the first​ fund? $___?__

How much money was invested in the second​ fund? $___?__

5. Things did not go quite as planned. You invested $20,340​, part of it in a stock that paid​ 12% annual interest.​ However, the rest of the money suffered a​ 5% loss. If the total annual income from both investments was $2060​, how much was invested at each​ rate?

How much money was invested at​ 12% annual​ interest? _$__?__

How much money way invested at a 5% loss? $___?__

1. You are given the graph of a function f. Determine the intervals where f is increasing, constant, or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)

The x y-coordinate plane is given. The curve enters the window in the second quadrant, goes down and right becoming less steep, changes direction at the point (−1, 0), goes up and right becoming more steep, passes through the approximate point (−0.58, 0.44), goes up and right becoming less steep, changes direction at the point (0, 1), goes down and right becoming more steep, passes through the approximate point (0.58, 0.44), goes down and right becoming less steep, changes direction at the point (1, 0), goes up and right becoming more steep, and exits the window in the first quadrant.

increasing=

constant=

decreasing=

2.Solar Panel Power Output

The graph of the function f shown in the accompanying figure gives the average "fixed" solar panel power output over a 15-hr period on a typical day. Determine the interval(s) where f is increasing, the interval(s) where f is constant, and the interval(s) where f is decreasing. Here,

t = 0

corresponds to 5 a.m. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)

The t y-coordinate plane is given. The t-axis is labeled: (hr) and the y-axis is labeled: Solar panel operating capacity (%).

• The curve begins above the t value 0, goes up and right until it reaches the t value 6 and approximate y value of 100.
• The curve continues from the t value 6 horizontally right until it reaches the t value 9.
• The curve continues from the t value 9 going down and right until it reaches the t value 14 and approximate y value of 5.
• The curve continues from the t value 14 horizontally right until it reaches the t value 15.

Source: Solarcity.com/California

increasing=

constant=

decreasing=

3.You are given the graph of a function f. Determine the relative maxima and relative minima, if any. (If an answer does not exist, enter DNE.)

The x y-coordinate plane is given. The function enters the window in the second quadrant, goes down and right, changes direction at the point (−4, 0), goes up and right becoming less steep, changes direction at the point (0, 16), goes down and right becoming more steep, changes direction at the point (4, 0), goes up and right, and exits the window in the first quadrant.

relative minimumsmaller x-value

(x, y)

=

relative minimumlarger x-value

(x, y)

=

(x, y)relative maximum

=

4.You are given the graph of a function f. Determine the relative maxima and relative minima, if any. (If an answer does not exist, enter DNE.)

The x y-coordinate plane is given. The curve with 3 parts enters the window at in the second quadrant, goes down and right becoming more steep, exits in the third quadrant almost vertically just to the left of x = −2, reenters in the second quadrant almost vertically just to the right of x = −2, goes down and right becoming less steep, changes direction at the point (0, 4), goes up and right becoming more steep, exits almost vertically just to the left of x = 2, reenters in the fourth quadrant almost vertically just to the right of x = 2, goes up and right becoming less steep, and exits the window in the first quadrant.

relative minimum

(x, y)

=

  (x, y)relative maximum

=

5.Find the x-value(s) of the relative maxima and relative minima, if any, of the function. (If an answer does not exist, enter DNE.)

f(x) = 1/2x² − 4x + 1
relative maxima:

x =

relative minima:

x =

6.Find the relative maxima and relative minima, if any, of the function. (If an answer does not exist, enter DNE.)

f(x) = x³ − 12x + 10

relative maximum(x, y)=

relative minimum(x, y)=

7.Find the relative maxima and relative minima, if any, of the function. (If an answer does not exist, enter DNE.)

F(t) = 3t⁵ − 5t³ + 12

relative maximum(x, y)=

relative minimum(x, y)=

8.Find the relative maxima and relative minima, if any, of the function. (If an answer does not exist, enter DNE.)

f(x) = X/X+4

relative minimum (x, y)=

relative maximum(x, y)=

9.

You are given the graph of a function f.

The x y-coordinate plane is given. A curve and 2 vertical lines are graphed.

• A vertical line crosses the x-axis at x = −4.
• A vertical line crosses the x-axis at x = 4.
• The curve with 3 parts enters the window just above the x−axis, goes up and right becoming more steep, exits almost vertically just to the left of x = −4, reenters almost vertically just to the right of x = −4, goes up and right becoming less steep, changes direction at the point (0, −0.5), goes down and right becoming more steep, exits almost vertically just to the left of x = 4, reenters almost vertically just to the right of x = 4, goes down and right becoming less steep, and exits the window just above the x−axis.

Determine the intervals where the graph of f is concave upward and where it is concave downward. (Enter your answers using interval notation.)

concave upward=

concave downward=

Find the inflection point of f, if any. (If an answer does not exist, enter DNE.)

(x, y) =

10.

You are given the graph of a function f.

The x y-coordinate plane is given. The curve enters the window in the second quadrant nearly horizontal, goes down and right becoming more steep, is nearly vertical at the point (0, 1), goes down and right becoming less steep, crosses the x-axis at approximately x = 1, and exits the window just below the x−axis.

Determine the intervals where the graph of f is concave upward and where it is concave downward. (Enter your answers using interval notation.)

concave upward=

concave downward=

Find the inflection point of f. (If an answer does not exist, enter DNE.)

(x, y) =

11.

Refer to the graph of f shown in the following figure.

The x y-coordinate plane is given. There is 1 curve and 9 dashed lines on the graph.

• The curve starts at the point (0, 1), goes up and right becoming more steep, passes through the approximate point (2, 2.4), goes up and right becoming less steep, changes direction at the approximate point (3, 3), goes down and right becoming more steep, passes through the approximate point (4, 2.2), goes down and right becoming less steep, changes direction at the point (5, 1), goes up and right becoming more steep, passes through the point (6, 2), goes up and right becoming almost horizontal at the point (7, 3), goes up and right becoming more steep, changes direction at the point (9, 6), goes down and right becoming less steep, and exits the window at the point (12, 2).
• The 9 dashed vertical lines extend from the x−axis to the curve at x = 1, 2, 3, 4, 5, 6, 7, 9, and 12.

(a)

Find the intervals where f is concave upward and the intervals where f is concave downward. (Enter your answers using interval notation.)

concave upward=

concave downward=

(b)

Find the inflection points of f. (Order your answers from smallest to largest x, then from smallest to largest y. If an answer does not exist, enter DNE.)

12.

Determine where the function is concave upward and where it is concave downward. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

g(x) = −x² + 9x + 8

concave upward=

concave downward=

13.

Determine where the function is concave upward and where it is concave downward. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

g(x) =

concave upward=

concave downward=

14.

Determine where the function is concave upward and where it is concave downward. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

g(x) =

concave upward=

concave downward=

15.

Determine where the function is concave upward and where it is concave downward. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

f(x) =

concave upward=

concave downward=

16.

Determine where the function is concave upward and where it is concave downward. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

f(x) = (x − 3)^2/3

concave upward=

concave downward=

17.

Find the inflection point(s), if any, of the function. (If an answer does not exist, enter DNE.)

g(x) = 4x⁴ − 8x³ + 1

18.

Find the inflection point, if any, of the function. (If an answer does not exist, enter DNE.)

f(x) = (x − 8)^4/3

(x, y) =

An electronics firm manufactures two types of personal computers (PC), a desktop model and a laptop model. The production of a desktop computer requires a capital expenditure of $400 and 40 hours of labor. The production of a laptop computer requires a capital expenditure of $250 and 30 hours of labor. The firm has $20,000 capital and 2,160 labor-hours available for production of desktop and laptop computers. Each desktop computer contributes a profit of $320 and each laptop computer contributes a profit of $220.

What are the corner points of the feasible region if we want to maximize the profit?

(a) (50, 0),(30, 32),(0, 72) ;

(b) (50, 0),(54, 0),(30, 32) ;

(c) (0, 80),(0, 72),(30, 32) ;

(d) (0, 0),(50, 0),(30, 32),(0, 72) ;

(e) (0, 0),(50, 0),(0, 72) ;

What are the decision variables if we want to maximize the profit?

(a) x1, x2 and x3 ;

(b) number of desktop and number of laptop to be made ;

(c) x and y ;

(d) x, y and z ;

(e) x1 and x

When Cristina opens a bag of white and milk chocolate pieces, 20% of the chocolate pieces are white. After Cristina eats 10 milk chocolate pieces, the ratio of brown chocolate to white chocolate is 2 to 3. How many pieces of chocolate are left in the bag?

This exercise is designed to be solved using technology such as calculators or computer spreadsheets.

You borrow $18,000 with a term of four years at an APR of 8%. Make an amortization table. How much equity have you built up halfway through the term? (Round your answer to two decimal places.)

Data from an independent research company found that the annual cost per worker for insurance​ (health, life,​ liability, etc.) was increasing according to the function f(x)=64.91e^(0.34 ×), where f(x) is the cost in dollars per year at time​ x, and x is the number of years measured from the beginnning of the year 1997. That is x=0 corresponds to the start of 1997. Find the total increase in costs during the next 4 years, beginning in 1997.

Four marbles are drawn simultaneously and at random from an urn containing 8 black, 7 white, and 5 red marbles.

What is the probability that at least one of them is white?