A rectangular garden of area 50 square feet is to be surrounded on three sides by a fence costing $2 per running foot and on one side by a brick wall costing $6 per running foot. Let x be the length of the brick wall side. Which of the following represents the total cost of the material?

2. Solve the second-order linear differential equation y′′ − 2y′ − 3y = −32e−x using the method of variation of parameters.

Solve the following boundary value problem by Laplace Transform. (If you solve with another
method you will NOT get credit. There is nothing wrong about the conditions.)
d2y
dt2 + y = cos(2t); y0(0) = 0; y0(

2
) = ?1:

The U.S. census questionnaire defines kitchens with complete facilities as those having a sink with piped water, a range, and a refrigerator. Homes that lack complete kitchen facilities have been rare in the United States for many years. The first census for which data were tabulated on this subject was in 1970. The table shows the percentage of housing units lacking complete kitchen facilities in the western United States.

Percent of Western U.S. Homes with Incomplete
Kitchens

(a) Use the method of least squares to find the multivariable function f with inputs a and b for the best-fitting line

y = ax + b,

where x is years since 1970.
f(a, b) =

(b) Calculate the minimum value of

f(a, b).

Explain what this minimum value indicates about the relationship between and the best-fitting line.The minimum value of

f(a, b)

is  , which indicates that the line with parameters a =  and b =  passes through each data point.
(c) Write function of the linear model that best fits the data to give the percentage of homes with incomplete kitchens in the Western United States, where x is years since 1970, with data from

0 ≤ x ≤ 20.

h(x) =

(d) In what year does the best-fitting line predict that no housing units will lack complete kitchen facilities?

Integration question: Fluid flows into a tank for 10 minutes. The tank is initially empty. After t minutes, fluid flows in at a rate of 1 + t/2 litres per minute. How much fluid, in litres, flows into the tank?

Prove that:

a) |sinx|<= |x|

b) x = sin x has only one solution in real number using mean value theorem

Find the mass of the solid bounded by the ??-plane, ??-plane, ??-plane, and the plane (?/2)+(?/4)+(?/8)=1, if the density of the solid is given by ?(?,?,?)=?+3?.

(a) Find the limit of the following functions:

-lim as x approaches 0 (1-cos³(x)/x)

-lim as x approaches 0 (sin(x)/2x)

-lim as theta approaches 0 (tan (5theta)/theta)

(b) Find the derivative of the following functions:

-f(x) = cos (3x²-2x)

-f(x) = cos³ (x²/1-x)

(c) Determine the period of the following functions:

-f(x) = 3 cos(x/2)

-f(x)= 21+ 7 sin(2x+3)

6. (a) let f : R → R be a function defined by
f(x) =



x + 4 if x ≤ 1
ax + b if 1 < x ≤ 3
3x x 8 if x > 3
Find the values of a and b that makes f(x) continuous on R. [10 marks]
(b) Find the derivative of f(x) = tann 1
1 ∞x
1 + x

. [15 marks]
(c) Find f
0
(x) using logarithmic differentiation, where f(x) = e
e 3x
√
2x x 5
(6 65x)
4
. [15 marks]
(d) Evaluate the integral Z
(x
3 + 1)1/3x
5
dx.

How are art and science similar?

How are art and science different?

Let f(x,y) = x^2 + y^2 + 8xy + 75y + 10. Find all the critical points and determine if each is a maxima, minima, or saddle points.

Let f(x) = ln(x^2 + 9) Find the first two derivatives of f . Find the critical numbers of f . Find the intervals where f is increasing and decreasing. Determine if the critical numbers of f correspond with local maximums or local minimums. Find the intervals where f is concave up and concave down. Find any inflection points of f

Compute the Maclaurin series for the following functions: (a) f(x) = (2 + x)^5 (b) f(x) = (x^3) * sin(x^2)

Suppose f(x) is a rational function in which the degree of the numerator is greater than the degree of the denominator. To integrate f(x), the first step is

a) Factor the denominator into distinct linear factors

b) Factor the numerator into distinct linear factors

c) Polynomial long division

d) Make a rationalizing substitution

e) None of the above

A company manufactures and sells x television sets per month. The monthly cost and​ price-demand equations are ​C(x)=75,000+40x and p(x)= 300-x/30 ​, 0 less than or equal to x less than or equal to 9000.

​(A) Find the maximum revenue. ​

(B) Find the maximum​ profit, the production level that will realize the maximum​ profit, and the price the company should charge for each television set. ​

(C) If the government decides to tax the company ​$6 for each set it​ produces, how many sets should the company manufacture each month to maximize its​ profit? What is the maximum​ profit? What should the company charge for each​ set? ​