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Calculus and Analysis

Parallel planes

Which of the following planes are parallel? Are any of them identical? P1: 4x - 2y - 6z = 3 P2: 4x - 2y - 2z = 6 P3: -6x + 3y -9z = 5 P4: z = 2x - y - 3 please explain each step in detail

Cylindrical and spherical coordinates.

Write the equations i) x^2 - y^2 - 2z^2 = 4 and ii) z = x^2 - y^2 in a) cylindrical coordinates b) spherical coordinates give detailed explanation for each step of the solutions.

Derivative

Find f'(0) where f(x)=(x+1)(x+2)...(x+1000)

Moivre-Laplace Formula

Moivre-Laplace formula exp(ix) = cos(x) + i sin(x), where i = (-1)^(1/2) , and which is widely used in different items of mathematics is usually deduced from the Maclaurin expansions of the functions involved. But the theory of Taylor (Maclaurin) expansions is a part of more general theory developed in the course of the fun

Integration

Intergrate the follow function f(x)=(1+x^2)/(1+x^4)

Sine Cancellation Laws

Using Cancellation Laws and other methods solve the following problems: 1) arcsin(1) 2) sin(arcsin(1/3))

Evaluation of a Function

A certain rational function f(x) contains quadratic functions in both its numerator and denominator. Aside from that, we also know the folliwing things about f: f has a vertical asymptote at x=5 f has a single x-intercept of x=2 f is removably discontinous at x=1, lim as (x)approaches 1 of f(x)= -1/9 evaluate lim of f(

Application of antiderivatives

A rocket lifts off the surface of the earth with a constant acceleration of 20 m/sec.sq. How fast will the rocket be going 1 minute later? What I did: a=20 m/sec.sq. v=20t+C m/sec, 1 min. = 60 sec. Initial conditions: v=0 when t=0 At t=0, C=0 Speed = |v|=20(60)+0, or 1200 m/sec. Question: Is this correct, or am I leav

Calculating the radius of a hemisphere

Given that the volume of a hemisphere (half a sphere) of radius r is 2pir^3/3, choose the one option closest to the radius of a hemisphere whose volume is 100cm^3. Options A. 0.28cm B. 3.63 cm C. 4.64cm D. 5.94cm E. 7.78cm F. 47.74cm

Interval of convergence

Find the interval of convergence of (a) f(x), (b) f'(x), (c) f''(x), (d) {f(x)dx En=1 [(-1)^n+1 (x-2)^n ] / 2

Power series centered at 0

Use the power series 1 / 1+x = En=0 (-1)^n x^n to determine the power series, centered at 0, for the function h(x) = x / x^2-1 = 1 / 2(1+x) - 1 / 2(1-x)

Taylor Polynomial

Find the nth Taylor polynomial centered at c f(x)= (x)^1/3 n = 3 c = 8

Theorem 8.11

Use theorem 8.11 to determine the convergence or divergence of the p-series En=1 3 / (n5/3)

Differentiation: Word problem - rate of change

A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30cm wide at the bottom, 80 cm wide at the top, and has a height of 50 cm. If the trough is being filled with water at a rate of 0.2m3/min, how fast is the water level rising when the water is 30cm deep?

Rate of Change Word Problem

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1m higher than the bow of the boat. If the rope is pulled in at a rate of 1m/s, how fast is the boat approaching the dock when it is 8m from the dock?

Length of curve

Length of curve interval [0,4] y= x (to the three halves) minus 1 y=x^3/2 - 1