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Calculus and Analysis

Find all differentiable functions.

Find all differentiable functions f : R ---> R such that (f composed f) = f. R = All Real Numbers (f composed f) = the function f composed with itself

Differential equations to describe infection rates

Use models to describe the population dynamics of disease agents. Total population is a Constant (T). A small group of infected individuals are introduced into a large population. Describe spread of infection within population as a function of time. This disease which, after recovery, confers immunity. The population can be

Taylor Polynomial

Find the Taylor polynomial of degree 4 at c=1 for the equation f(x)=ln x and determine the accuracy of this polynomial at x=2.

Orthogonality

Find a unit vector that is orthogonal to both i + j and i + k give detailed explanation for each step

Angle between a cube's diagonal

Find the angle between a cube's diagonal and one of its sides. (use the vector calculus to get your answer) give detailed response. explain each step.

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?