Find a function y = y(x) that has a y-intercept at (0, 4), a horizontal tangent line at the y-intercept, and satisfies the differential equation y''-10y'+25y=0.

to calculate the integral from (-infinty) to (+ infinity) of [x^2 / (x^4 + 4)[ dx .   poles, residues.

If

ln(8x) = y²,

find

dy/dx.

Given the following vector force field, F, is conservative: F(x,y)=(2x²y⁴+x)i+(2x⁴y³+y)j, determine the work done subject to the force while traveling along any piecewise smooth curve from (-2,1) to (-1,0)

Let f(x) = -2x³ - 9x² - 12x + 3. Find the following:

a) The domain of f Answer can be an interval or in words

b) The y-intercept Answer must be a point

c) f ^'(x)

d) The critical numbers, if any.

e) The open interval(s) where the function is increasing and the open interval(s) where the function is decreasing. The answer to this question will be accepted in one of two ways:

1) Show the number line with the critical numbers along with the open intervals, the test values (don't need to show plugging test values into function), and state where the number line is + or -.

2) Show the number line with the critical numbers along with the open intervals, evaluate at your test values and state the open intervals where the function is increasing or decreasing.

f) State the relative maximum/relative minimum point, if any. The answer must be an ordered pair. Hence, show the work on obtaining the y-value and state order pair(s) that is are a relative minimum and a relative maximum. If there are no relative extreme points, then state “no relative extreme”.

A toy manufacturer's cost for producing q units of a game is given by C(q) = 1460 + 3.5q + 0.0006q². If the demand for the game is given by p = 8.6 − (1/440)q. How many games should be produced to maximize profit? (Round your answer to the nearest integer.)

part 1)

Let f(x) = x^4 − 2x^2 + 3. Find the intervals of concavity of f and determine its inflection point(s).

part 2)

Find the absolute extrema of f(x) = x^4 + 4x^3 − 8x^2 + 3 on [−1, 2].

1. . Find the limit: lim ?→ ∞ (? + √?2 + 2?)

2. If 1200 ??2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

2. The volume of a right circular cone is ? =1/3 ??^2 ℎ , where ? is the radius of
the base and ℎ is the height.

(a) Find the rate of change of the volume with respect to the height if the radius is constant.

(b) Fine the rate of change of the volume with respect to the radius if the height is constant.

4. A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of 2 ??3/?, how fast is the water level rising when the water is 5 cm deep?

Consider the region R between the x-axis and the curve y = x^3 / 3 , between x = 0 and x = 1.

(a) Calculate the surface area of the solid obtained by revolving R about the x-axis.

(b) Write an integral for the the surface area of the solid obtained by revolving R about the y-axis

Solve the problem

43) Find equations for the horizontal and vertical tangent lines to the curve r = 1 - sinθ, 0 ≤ θ < 2π.

Please check if your answer is correct with the following:

Horizontal: y = 1/4 at (1/2, π/6), y = 1/4 at (1/2, 5π/6), y = -2 at (2, 3π/2)

Vertical: x = 0 at (0, π/2), x = -3sqrt(3)/4 at (3/2, 7π/6), x = 3sqrt(3)/4 at (3/2, 11π/6)

Given the equation:

sin(x+y)=x+cos(y)

a) Differentiate y with respect to x

b) Give the equation of the line tangent to the curve of at the point (0,π/4)

A closed rectangular box of volume 324 cubic inches is to be made with a square base. If the material for the bottom costs twice per square inch as much as the material for the sides and top, find the dimensions of the box that minimize the cost of materials.

Consider the following planes. 5x − 3y + z = 2, 3x + y − 5z = 4 (a) Find parametric equations for the line of intersection of the planes. (Use the parameter t.) (x(t), y(t), z(t)) = (b) Find the angle between the planes. (Round your answer to one decimal place.)