calculus and analysis: limit on infinity of arbitrary functions

Here is the code that takes the limit of an expression.

Limit((-I E^(I x) f1(y))/(g2^(Prime)(Prime))(y), x -> (Infinity) )

and the output returned is INDETERMINATE while the desired output is $ infty $. Or if I had to do this instead

Limit((g2^(Prime)(Prime))(y)/(-I E^(I x) f1(y)), x -> (Infinity) )

I would like to get 0 and not INDETERMINATE.

How would you tell Mathematica that the $ f $ Y $ g $ What functions are irrelevant when evaluating the limit?

Thanks for any help.

calculus and analysis: how to get the step-by-step derivative of this?

I'm very new to Mathematica, and I wanted to use it to help me solve some questions that I don't have the mathematical backing to help me solve on paper. I would like to take this equation and make it a partial derivative. However, I would also like a step by step to help me hold on and know what is going on. Instead, what I get is this error message. Could anyone help me?

WolframAlpha("D(-T0/ ProductLog(-E^(-1+dS/dC), dS)")

WolframAlpha::timeout: The call to WolframAlpha(D(-T0/ ProductLog(-E^(-1+dS/dC), dS)) has exceeded 30. seconds. Increasing the value of the TimeConstraint option may improve the result.

Thank you!

calculus and analysis: triple integration numerical solution of a region with variable limits?

They ask me to calculate the mass of the region delimited by the plane (2x + 3y - z = 2) and down the triangle in the xy plane with vertices (0,2),(1,0),(4,0). Density is proportional to the distance from the xy plane, so the function is d(x,y,z) = kz.

My limits are:

0 ≤ z ≤ (2x + 3y - 2)

(1 - 0.5y) ≤ x ≤ (4 - 2y)

Y 0 ≤ y ≤ 2.

My code is:

Integrate(d(x,y,z),{z,0,2x+3y-2},{x,1-0.5y,4-2y},{y,0,2})

but Mathematica doesn't return a number (if I'm correct it should be 19k).

Instead I get: k(3-1.5y)(-2+2x+3y)^2.

However, if I integrate the function step by step:

Integrate(Integrate(Integrate(d(x,y,z),{z,0,2x+3y-2}),{x,1-0.5y,4-2y}),{y,0,2}),

I get a numerical value.

Is there a syntax that Mathematica requires that I am unaware of?

calculus and analysis: multiple integral resolution with symbolic limits

I am trying to solve a double integral, although I am not sure how it could handle symbolic limits.
I have tried the following code:

i2 = (xprime^2*yprime^2)/2
xminp = x - 1/2 (y - yprime);
xmaxp = x + 1/2 (y - yprime);
yminp = 0 ;
ymaxp = y ;
I2 = NIntegrate(i2, {xprime, xminp, xmaxp}, {yprime, yminp, ymaxp})

But this gives the following error:

NIntegrate: xprime = x +1/2(-y +yprime) is not a valid limit of integration)

Does anyone know how I could perform this symbolic integration in Mathematica?
Thank you.

calculus and analysis – Derived with prime for vector function

Consider first the following vector function of a real variable:

   s(t_) := {Sin(t), Cos(t)}

So this works as expected:

   s'(t)
(* {Cos(t), -Sin(t)} *)

Why use the following prime to take derivatives do not Also work?

   soln(t) := {x(t), y(t)} /. 
  First@ DSolve({Derivative(1)(x)(t) == y(t), 
     Derivative(1)(y)(t) == -x(t), x(0) == 0, y(0) == 1}, {x(t), 
     y(t)}, t)

   soln(t)
(* {Sin(t), Cos(t)} *)

   soln'(t)
(* soln'(t) *)

Please note that the following make job:

   D(soln(t), t)
(* {Cos(t), -Sin(t)} *)

calculus – Newton's method for the root of the cube

I have a text that states that the following is Newton's method for cube roots, where $ and $ is an approximation to the cube root of $ x $:

a

I understand that the form can be derived from:
yes

where each $ x_n $ is a better approximation of the root with each iteration and $ x_0 $ being a rough initial approximation.

I am confused when deriving form (a). Can anyone explain how that shape is obtained? My derivation shows:

$ f (x) = x ^ 3 – a $ Y $ f & # 39; (x) = 3x $ where $ to $ is the cube, then we find $ x $ for $ f (x) = 0 $

$ x_1 = x_0 – (x ^ 3 – a) / (3x) $

$ x_1 = (x ^ 3 / 3x) – (a / 3x) $

$ x_1 = (x ^ 2/3) – (a / 3x) $

Again, I'm not sure how the form works $ ((a / x ^ 2) + 2y) / 3 $ you get it and that's the question.

Thank you

calculus – is this the volume of a solid of revolution of a sector about a point in space along phi

Consider this sector S what is the area is as follows:
$$
A = frac {1} {2} r ^ 2 theta
$$

Where theta is in radians.
I would like to create a solid from this sector by turning it $ phi $ with the following:
$$
V = int_ {0} ^ {2 pi} Big ( frac {1} {2} r ^ 2 theta Big) phi : : d phi
$$

which evaluates:
$$
V = frac {r ^ 2 theta phi ^ 2} {4}
$$

and then leave $ phi = 2 pi $ (upper limit of the integral):
$$
V = frac {r ^ 2 pi ^ 2 theta} {2}
$$

Is this correct?

Edit: Add image, red is S, light green is the surface, dark green is inside the cross section.
enter the image description here

calculus and analysis: how to reform an expression to improve numerical stability?

In engineering, we generally have to reform an expression to avoid division by a small number, or to eliminate discontinuities.
For example, we can reform $$ frac { frac {1} {t ^ 2} -1} {(t + frac {1} {t}) ^ 2} $$ to $$ frac {1-t ^ 2} {(1 + t ^ 2) ^ 2} $$ then we have definition when x == 0 and better numerical stability when t it is small.

For example, I have an expression that is difficult to manually reform: $$ g (f) = – frac {i-ie ^ {- 2if pi}} {4f pi-4f ^ 3 pi}, (f geq0) $$ Obviously $$ lim_ {f rightarrow0} g (f) = frac {1} {2} $$ Y $$ lim_ {f rightarrow1} g (f) = – frac {1} {4} $$ How can I reform it with Mathematica to remove discontinuities in f == 0 Y f == 1and avoid dividing small numbers if possible?

calculus – Name for this type of integral that is logarithmically divergent in a parameter?

Suppose $ f (x) $ Y $ g (x) $ they are fast-deteriorating functions that approach smoothly $ 1 $ how $ x rightarrow 1 $. For example,

$$ f (x) = e ^ {- x}, , g (x) = frac {1} {1 + x ^ 2} $$

Now consider the following function defined through an integral:

$$ F ( alpha) = int_0 ^ { infty} frac {f (x)} {x + alpha g (x)} dx $$

Suppose you would like to study the little one$ alpha $ behavior of $ F ( alpha) $. You can clearly see that if $ alpha = 0 $, the integrand is logarithmically singular near $ x = 0 $. Also, if you continue analytically $ alpha $, it is clear that $ F ( alpha) $ it is singular along the entire negative-real axis.

So I hope that the little analytic$ alpha $ behavior to look like a logarithm

$$ F ( alpha) approx F_0 textrm {Log} ( alpha) + mathcal {O} ( alpha) $$

Is there a name for integrals defined as $ F ( alpha) $? Or maybe something similar?

differential calculus: what is the derivative of te ^ t power?

I am trying to differentiate f = (t) = t (e ^ t) cott

I know that the derivative of e ^ x is e ^ x. But, there is a variable t in front.

I know that the derivative of cotx is -csc ^ 2x.