Let z = 4 - 4i. find the sixth roots of z and represent them
in the plane.

f(x)=e^x/(3+e^x)

Find the first derivative of f.

Use interval notation to indicate where f(x) is increasing.

List the x coordinates of all local minima of f. If there are no local maxima, enter 'NONE'.

List the x coordinates of all local maxima of f. If there are no local maxima, enter 'NONE'.

Find the second derivative of f:

Use interval notation to indicate the interval(s) of upward concavity of f(x).

Use interval notation to indicate the interval(s) of downward concavity for f(x)

List the x values of the inflection points of f. If there are no inflection points, enter 'NONE'.

For the cost and price functions​ below, find

​(​a) the​ number, q, of units that produces maximum​ profit;

​(​b) the​ price, p, in dollars per unit that produces maximum​ profit; and

​(​c) the maximum​ profit, P, in dollars.

C(q)=100+20qe^-0.01q ; p=60e^-0.01q

​(a) The​ number, q, of units that produces maximum profit is $__

​(Do not round until the final answer. Then round to the nearest whole number as​ needed.)

​(b) The​ price, p, per unit that produces maximum profit is $__

​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

​(c) The maximum profit is $__

​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

The balance in an investment account years t after the account is opened is given by 7500(1.065t). Compute the average rate of change for 2.5 ≤ t ≤ 3.5. (Round your answer to two decimal places.) Δ balance/ Δt=?

Use the model for projectile motion, assuming there is no air resistance and g=32 feet per second per second.

Determine the maximum height and range of a projectile fired at a height of 8 feet above the ground with an initial speed of 300 feet per second and at an angle of 45degrees above the horizontal. (Round your answers to three decimal places.)

(a) maximum height (feet)

(b) maximum range (feet)

Complete the following table: (Use Table 15.1) (Do not round intermediate calculations. Round your answers to the nearest cent.)

A bicyclist pedals at a rate of 120 rpm. The diameter of the pedal sprocket is 20 cm. The diameter of the rear wheel sprocket is 8 cm. The diameter of the rear bicycle tire is 70 cm.

a) What is the angular velocity (in rad) of the rear sprocket?

b) What is the linear velocity of the chain in cm per min

c) What is the speed of the bicycle in cm per min?

Formulate a system of equations for the situation below and solve.

A manufacturer of women's blouses makes three types of blouses: sleeveless, short-sleeve, and long-sleeve. The time (in minutes) required by each department to produce a dozen blouses of each type is shown in the following table.

The cutting, sewing, and packaging departments have available a maximum of 73.5, 150, and 44 labor-hours, respectively, per day. How many dozens of each type of blouse can be produced each day if the plant is operated at full capacity?

Use the given conditions to write an equation for the line in? point-slope form and in? slope-intercept form.

Passing through (-5,-6) and parallel to the line whose equation is y= -5x+1

______________________________________________________________________________________________________________________________________________

Write an equation for the line in? point-slope form.

_______

(Simplify your answer. Use integers or fractions for any numbers in the? equation.)

Write an equation for the line in? slope-intercept form.

________

(Simplify your answer. Use integers or fractions for any numbers in the? equation.)

Find the local maximum and minimum values of the function.

Tell the intervals where the graph of the function is increasing and decreasing

f(x)= 2x^3 +3x^2 - 12x

Use an appropriate comparison test to determine the convergence/divergence of the following series:

a.)∑ n= (1)/(√n−1) (Upper limit of the sigma is ∞ and the lower limit of the sigma is n=2)

b.) ∑ n=n(n+1)/(n^2+1) (n-1) (Upper limit of sigma is ∞ and the lower limit of sigma is n=2)

c.) ∑ n= cos^2(n)/ (n^3/2) (Upper limit of sigma is ∞ and the lower limit of sigma is n=1)

d.) ∑ 5^n/(√n4^n) (Upper limit of sigma is ∞ and the lower limit of sigma is n=1)

e.) ∑ n (1)/(In(In(n))) (Upper limit of sigma is ∞ and the lower limit of the sigma is n=3)

f.) ∑ (n+2^n)/(n^2*2^n) (Upper limit of sigma is ∞ and the lower limit of the sigma is n=1)

g.) ∑ (1)/(1^2+2^2+ 3^2+...+n^2) (Upper limit of sigma is ∞ and the lower limit of the sigma is n=1)

evaluate the indefinite integral

sqrt(4-x^2)

A cheetah can run at 105 ft/s, but only for 7 seconds. The cheetah is at (0,0) when it sees an antelope which is moving accoording to the parametric equation (x,y)= (-39+40t , 228+30t), where t is in seconds and (x,y) are measured in feet. If the cheetah started to run at t=0 it could catch the antelope. How many seconds can the cheetah afford to wait before starting? Assume that the cheetah does not change direction when it runs.

Which of the following is the volume of the region bounded by x * 2 + y * 2 = 4 cylinders, z = 0 and z = 4 - x planes?

Find the absolute maximum value and absolute minimum value, if any, of the function g(x) = x √(4 − x2) on the interval [−2, 2].