## 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 $$x$$:

I understand that the form can be derived from:

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:
$$A = frac {1} {2} r ^ 2 theta$$
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 == 1`and 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.