Solve using Implicit Differentiation showing all steps:

a.) xy^2-y^2=6

b.) x^2 − 2xcosy + y^2 = 3

c.) x = x + y/2y − x

d.) x^2 + 4cos(xy) − y^2 = 8

Thank you!

Consider the function. f(x) = x^2 − 1, x ≥ 1

(a) Find the inverse function of f.

f ^−1(x) =

(b) Graph f and f ^−1 on the same set of coordinate axes.

(c) Describe the relationship between the graphs. The graphs of f and f^−1 are reflections of each other across the line ____answer here___________.

(d) State the domain and range of f and f^−1. (Enter your answers using interval notation.)

Domain of f

Range of f

Domain of f ^−1

Range of f ^−1

Use Newton's method to approximate the indicated root of the equation correct to six decimal places.

The root of x⁴ − 2x³ + 4x² − 8 = 0 in the interval [1, 2]

x = ?

Question 1

f is a function whose domain is (0,∞) and whose first derivative is f ' (x) = (4 − x)x ^-3 for x > 0

a) Find the critical number(s) of f. Does f have a local minimum, a local maximum, or neither at its critical number(s)? Explain.

b) On which interval(s) if f concave up, concave down?

c) At which value(s) of x does f have a point of inflection, if any?

Question 2

A rectangular box with a square base has a surface area of 150 cm² , obtained by adding the areas all six sides of the box. Find the dimensions of the box with the largest possible volume.​​​​​​​

Please give a brief description about the Remote data like REST, JSON, XML, AJAX and SOAP API’s ?

Suppose f is a twice differentiable function such that f′(x)>0 and f′′(x)<0 everywhere, and consider the following data table.

x      0       1       2

f(x)   3       A       B

For each part below, determine whether the given values of A and B are possible (i.e., consistent with the information about f′and f′′ given above) or impossible, and explain your answer.

a)A= 5, B= 6

(b)A= 5, B= 8

(c)A= 6, B= 6

(d)A= 6, B= 6.1

(e)A= 6, B= 9

The plane

x + y + 2z = 12

intersects the paraboloid

z = x² + y²

in an ellipse. Find the points on the ellipse that are nearest to and farthest from the origin.

1. You are given the function f(x) = 3x^4 + 4x^3

a) Find the x and y- intercepts.

b) Find the critical number(s) of f.

c) Find the interval(s) of increase and decrease of f.

d) Find the relative maximum and minimum value(s) of f.

e) Find the hypercritical number(s) of f.

f) Find the interval(s) of upward and downward concavity of f.

g) Find the point(s) of inflection of f.

h) Sketch the graph of f.

2. You are given the function f(x) = x/(1−x)

a) Find the x and y- intercepts.

b) Find the horizontal asymptote(s).

c) Find the vertical asymptote(s) and do a limit analysis of the behavior of f on either
side of each vertical asymptote.

d) Find the critical number(s) of f.

e) Find the interval(s) of increase and decrease of f.

f) Find the relative maximum and minimum value(s) of f.

g) Find the hypercritical number(s) of f.

h) Find the interval(s) of upward and downward concavity of f.

I) Find the point(s) of inflection of f.

j) Sketch the graph of f.

3. Find the absolute maximum and absolute minimum of f(x) = x^3 − 3x on [0, 3].

4. Why doesn’t the Extreme Value Theorem apply to f(x) =1/x on the interval [-3, 3]?

Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 900 cm3 , the pressure is 160 kPa, and the pressure is increasing at a rate of 40 kPa/min. At what rate is the volume decreasing at this instant?

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Step 1: Write all the rates in the problem (including the one you are trying to find) as
derivatives.

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Step 2: Identify the dependent variables in the derivatives. Find an equation relating
the dependent variables in the derivatives. You will often need a formula from
geometry for this.

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Step 3: Differentiate both sides of the equation from Step 2 with respect to the
independent variable time. The result will be an equation relating the rates (relating the
derivatives).

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Step 4: Substitute the values that you are given in the problem for the derivatives and
the variables and solve for the derivative for which the question is asking.

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Thanks for any help!!!

In an RL series circuit with L = 1/100 H, R = 20 Ω, and E = 60 V. Determine the limit of the maximum current reached (At the function you found to determine the current as a function of time apply the limit when t goes to infinity) and determine the time in which it reaches half of that value. Take i (0) = 0 A.

Create your own scenario in which you can compare the cost for two choices for the same service.

You will need to create two equations, each of which should include::

1. fixed cost

2. variable cost

Then write an inequality to compare the two choices and solve it. Interpret your solution to further explain your comparison. Below is an example that you may use as a model to help you develop your own scenario.

I am trying to decide whether to install a solar energy system or replace my existing gas system. The solar system will cost $20,000 to install but only $25 per month. The equation for the total cost is Solar Cost = 25x + 20000 where x is the number of months. The gas system will only cost $5,000 to install but will cost $125 per month. The equation is Gas Cost = 125x + 5000. To find out how long I need to live in my home for the solar to be more cost effective, I need to solve the following inequality: 25x + 20000 < 125x + 5000. When I solve the inequality, x > 150. That means I need to live in the house longer than 150 months, or 12.5 years, for the solar system to be a better deal.

A store sells about 360 calculators peryear. It would cost the store $8 to store one calculator for the whole year. On the other hand, it costs $10 each time the store reorders the calculators, plus $8for each calculator. How many times per year should the store reorderthe calculators and how many at a time to minimize inventory costs?

1. A 20 foot ladder leans against a vertical wall. The bottom of the ladder slides away from the wall at 2 ft/sec.

a. How fast is the top of the ladder sliding down the wall when the top of the ladder is 12 feet from the ground?

b. Interpret the meaning of the sign.