Find the derivative

a)y=(1+x)(x+3)/2x-1

b)y=(1+x³)(3-4x²)+x+6

Evaluate the surface integral

F · dS

for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation.

F(x, y, z) = x i − z j + y k

S is the part of the sphere

x² + y² + z² = 9

in the first octant, with orientation toward the origin

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 11 ft apart, how far away from the stronger light source should an object be placed on the line between the two sources so as to receive the least illumination? (Round your answer to two decimal places)

Suppose that one factory inputs its goods from two different plants, A and B, with different costs, 4 and 6 each respective. And suppose the price function in the market is decided as p(x,y)= 100−x−y where x and y are the demand functions and 0≤x,y. Then as

x=

y=

the factory can attains the maximum profit,

Find the area of the region enclosed between ?(?)=?2−3?+13 and ?(?)=2?2−3?−3.

Consider the following one-dimensional partial differentiation wave equation. Produce the solution u(x, t) of this equation. 4Uxx = Utt 0 < x 0 Boundary Conditions: u (0, t) = u (2π, t) = 0, Initial Conditions a shown below: consider g(x)= 0 in both cases.

(a) u (x, 0) = f(x) = 3sin 2x +3 sin7x , 0 < x <2π

(b) u (x, 0) = x +2, 0 < x <2π

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.)

1) f(x, y) = x³ + y³ − 3x² − 3y² − 9x
a) local minimum value
b) local maximum value
c) saddle point (x,y,f)

2) f (x, y) = x² + xy + y² + 8y
a) local minimum value
b) local maximum value
c) saddle point (x,y,f)

A real estate office handles a 60-unit apartment complex. When the rent is $530 per month, all units are occupied. For each $25 increase in rent, however, an average of one unit becomes vacant. Each occupied unit requires an average of $30 per month for service and repairs. What rent should be charged to obtain a maximum profit?

Margot is walking in a straight line from a point 40 feet due east of a statue in a park toward a point 34 feet due north of the statue. She walks at a constant speed of 4 feet per second.

(a) Write parametric equations for Margot's position t seconds after she starts walking. (Round your coefficients to four decimal places as needed.)

(b) Write an expression for the distance from Margot's position to the statue at time t. (Round your coefficients to four decimal places as needed.)

(c) Find the times when Margot is 36 feet from the statue. (Round your answers to two decimal places)

1. True or False:

If we know that f(x) is increasing on an interval, then f ''(x) should be positive on that interval.

2. True or False:

If f '(x) is increasing on a certain interval, then f ''(x) is positive on that interval.

3. If f ''(x)<0 on a certain interval, which of the following must be true on that interval? (2 answers)

A. f '(x) is decreasing

B. f(x) is increasing

C. f(x) is concave down

D. f(x) is concave up

E. f(x) is decreasing

F. f '(x) is concave down

G. f(x) is negative

H. f '(x)<0

4. At an inflection point...

A. f(x) has a horizontal tangent line

B. f '(x) = 0

C. f ''(x) = 0

D. f(x) = 0

E. f(x) has a vertical slope

5. If f '(a) = 0 and f ''(a) > 0, then...

A. f(a) is a local minimum

B. f(a) is an inflection point

C. f(a) is a critical point, but we can't determine if it is a max or min

D. f(a) is a local maximum

6. Either the First Derivative or the Second Derivative Test may be used to find local extrema.

True or False

7. Every critical point is a local extrema.

True or False

8. For the function f(x)=x^5-3x^2, x=1 is a local min.

True or False

9. For the function f(x)=x^5-3x^2, x=1 is a global min.

True or False

A man won ​$5 million by scoring 16 goals in 23 seconds in a contest at a sports game that was sponsored by a business. Did the business risk the ​$ ​million? No! The business purchased event insurance from a company specializing in promotions at sporting events. The event insurance company estimates the probability of a contestant winning the contest​ and, for a modest​ charge, insures the event. The promoters pay the insurance premium but take on no added risk as the insurance company will make the large payout in the unlikely event that a contestant wins. To see how it​ works, suppose that the insurance company estimates that the probability a contestant would win the contest is 0.0009 and that the insurance company charges ​$9,500

a. Calculate the expected value of the profit made by the insurance company.

Solve the following equations:

1. 3cos(x)+1=cos(x)3cos⁡(x)+1=cos⁡(x), where x is in the interval [0,2π)[0,2π)
   x=?
2. sin(2x)=−12sin⁡(2x)=−12, where x is in the interval [0,2π)[0,2π).  
   x=?

Solve the following problems:
(a) y'' - 2y' + 5y = 0 with y(0) = 1 and y'(0) = 2.
(b) y(3) - 3y' + 2y = 0 with y(0) = 5, y'(0) = 6, and y''(0) = 11.

Related rates Part A:

1. Consider a right triangle with hypotenuse of (fixed) length 46 cm and variable legs of lengths x and y, respectively. If the leg of length x increases at the rate of 6 cm/min, at what rate is y changing when x = 5 cm? (Round your answer to three decimal places.)

2. Water is flowing into a vertical cylindrical tank of diameter 8 m at the rate of 5 m³/min. Find the rate at which the depth of the water is rising. (Round your answer to three decimal places.)

3. A gas is said to be compressed adiabatically if there is no gain or loss of heat. When such a gas is diatomic (has two atoms per molecule), it satisfies the equation PV ^1.4 = k, where k is a constant, P is the pressure, and V is the volume. At a given instant, the pressure is 23 kg/cm², the volume is 35 cm³, and the volume is decreasing at the rate of 6 cm³/min. At what rate is the pressure changing?