A) Find the equation of the tangent line to r = 9cos(5θ) when θ =
π/2
B)Find the points on the polar curve r = 1 + cos(θ) where the tangent line is horizontal.

A. Find the indefinite integral.

B. Find the indefinite integral.

C. Find the derivative.

f(x) = x⁶ · log₃(x)

Give your answer using the form below.

x^A(B + C log_D(x))

A =
B =
C =
D =

D. Find the indefinite integral.

E. Find the area under the curve below from x = 1 to x = 2. Give your answer correct to 3 decimal places.

F. Find the area under the curve below from x = 0 to x = 5. Give your answer correct to 3 decimal places.

y = 5x - x²

An airliner flies against the wind on one leg of its round trip in 4.5 hours. It flies back to its starting​ point, with the wind in 4 hours. If the average speed of the wind is 15 miles per​ hour, what is the speed of the airliner in still​ air?

The rate of the airliner in still air is _____ _____

Find the extreme values of f on the region described by the inequality.

f(x, y) = e^−xy;    x² + 4y² ≤ 1

A health club is trying to determine how to allocate funds for advertising. The manager decides to advertise on the radio and in the newspaper. Previous experience with such advertising leads the club to expect

A(r, n) = 0.1r²n responses

when r ads are run on the radio and n ads appear in the newspaper.
Each ad on the radio costs $8, and each newspaper ad costs $4. The manager is currently budgeting $336 for advertising. Therefore the constraint equation is given by

g(r, n) = 8r + 4n = 336 dollars

a) Write the Lagrange system that can be used to find the optimal point of

A(r, n)

subject to the given budget constraint. (Enter your answers as a comma separated list of equations. Use λ to represent the Lagrange multiplier.) You DO NOT need to solve the system.

The solution to the Lagrange system was found to be r=28, n=28, and λ=19.6.

(b) What is the optimal number of responses expected with the given advertising budget? (Round to the nearest whole number.)

(c) What are the units for λ?

(d) Suppose the manager budgeted an additional $26 for advertising. What is the approximate change in the optimal number of responses as a result of this change in the constraint level? (Round your answer to the nearest whole number.)

What is the minimum surface area of a box whose top and bottom is a square if the volume is 40 cubic inches. (Surface area is the area of all the sides of the box).

Schneider's Bakery sells cakes. At a price of $20 per cake, it sells 500 cakes per week. At a price of $40, it sells 300 cakes per week.

(a) (2.5 points) Find a linear demand function ?(?)D(x) that models this situation.

(b) (2.5 points) Find the elasticity of demand.

(c) (2.5 points) Find ?(20)E(20) and ?(50)E(50), and explain what these numbers represent.

(d) (2.5 points) Using the linear demand function from part (a), find the price ?x that results in unit elasticity, and explain what this number represents. Also, determine the weekly revenue at this price.

Solve the linear programming problem. Maximize z=15x+15y , Subject to 9x+7y greater than or equals 153 , 13x-11y greater than or equals 31 , x+y less than or equals 43 , x,y greater than or equals 0

What is the maximum value of​ z?

Select the correct choice below and fill in any answer boxes present in your choice.

A. z=( ? )

​(Type an integer or a​ fraction.)

B. There is no maximum value of z.

At what corner​ point(s) does the maximum value of z​ occur?

Select the correct choice below and fill in any answer boxes present in your choice.

A. The maximum value of z occurs at the corner​ point(s) ( ? )

nothing.​(Type an ordered pair. Use a comma to separate answers as​ needed.)

B. There is no maximum value of z.

Determine whether the series converges absolutely, conditionally, or not at all.

Find the relative extrema, if any, of the function. Use the second derivative test, if applicable. (If an answer does not exist, enter DNE.)

g(x)=x^3-6x

a. Find the​ nth-order Taylor polynomials of the given function centered at the given point​ a, for n = ​0, ​1, and 2.

b. Graph the Taylor polynomials and the function.

f(x) = cos x, a = 2pi / 3

A rectangular tank with a square​ base, an open​ top, and a volume of 1372 ft cubed is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.

a) Find an equation of the plane tangent to the graph of f at the given point P. Write your answer in the form ax + by +cz= d, where a, b, c, and d are integers with no common factor, and a is greater or equal to 0. f(x,y)= 2x³y + 4x-y, P(1,3,7)

b) Use a multivariable chain rule to find a formula for the given derivative or partial derivative. w= f(x,y), x=g(u,v), y=h(u,v);  ∂w/ ∂v

List all of the values of the sine function that you know. Remember that values of sin(x) repeat every 2π radians, so your answer should include infinitely many values.

A plane flying horizontally at an altitude of 5 mi and a speed of 470 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 7 mi away from the station. For full credit I expect to see a well-labeled picture. Show all work. Round your final answer to the nearest whole number. Do not round intermediate values. Include units of measure in your answer.