Questions tagged [calculus-and-analysis]

Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.

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7
votes
4answers
992 views

How to transform the result of this integral into powers of $\sin\,x$?

I want to calculate the integral $I = \displaystyle\int{\cos ^5 x\, \mathrm dx}$. Using the variable $t = \sin x$, I get $$I = \sin x - \frac{2}{3}\sin^3 x + \frac{1}{5}\sin^5 x + C.$$ With ...
0
votes
2answers
921 views

Integral with the derivative of Dirac delta does not evaluate correctly?

Integrate[DiracDelta'[y - z] DiracDelta[x - z], {z, -Infinity, Infinity}, Assumptions -> x \[Element] Reals && x \[Element] Reals]
8
votes
2answers
229 views

Differentiate the product of some terms

How can I compute the following derivative, $$ \frac{\partial}{\partial \lambda_j} \prod_{i=1}^k (1+\lambda_i)e^{\lambda_i} \quad \text{for }\; 1\le j \le k $$ for some positive integer $k$ which is ...
2
votes
1answer
434 views

Asymptotic Rate of Growth

How can I calculate the Asymptotic Rate of growth of a function, for instance like: $X^3 - X^2 - X -1$ EDIT: For instance, as you can see in this graph, after the 1200 the function approximates to ...
4
votes
2answers
1k views

Sum of series with log in each term [closed]

I was solving recurrence relation of Introduction to Algorithms by CLRS, 3rd. edition. Problem 4-3 (i) $$ T(n) = T(n-2) + \frac{1}{lg \; n} $$ I tried few ways, like expending with iteration method. ...
1
vote
1answer
1k views

How to define a derivative/Variation operation in Mathematica from scratch

I am looking for a method in Mathematica 8 to define my own derivative/variation (independent of built-in operations). I do not want to re-define built-in objects since I fear that this could cause ...
1
vote
1answer
572 views

How to solve this integral

I'm trying to deal with this article, but I don't understand how equations 5.9 and 5.10 were obtained. I tried to use Mathematica 8 to solve it, but the answer was different. I have a problem with ...
8
votes
2answers
8k views

How to find (numerical) value of a derivative at point?

I have the following function: f[0, 0] = 0 f[x_, y_] := Exp[-(x^2 + y^2)^(-1)] How do I find its partial derivatives at any given point, including $(0,0)$? This ...
8
votes
1answer
459 views

ReplaceAll[] and Limit[] don't give correct results for this expression under extreme variables [duplicate]

Possible Duplicate: Funny behaviour when plotting a polynomial of high degree and large coefficients 1/x^2 + (3 + x)/(6 (1 - Exp[x] + x)) ——This is a ...
17
votes
5answers
2k views

How to make a Line[] with no end?

I'm trying to do this: In this graph, the secant points are aproximated in order to become the tangent, it seems I need some kind of function which plots a line based on two points and it's points ...
3
votes
1answer
479 views

How to move differentation before the integral?

I have quite a lot expressions that need to be integrated after switching the integral sign with differentiation operator. The additional problem is that the bounds are dependent to one of the ...
2
votes
3answers
361 views

Need help evaluating definite integral to a function of Y

Suppose $Y = \sqrt{2T}\cos(U)$, $ 0 \le u \le \pi $, and $ 0 \le \cos^{-1}(\frac{y}{\sqrt {2t}}) \le \pi ) $, so $ -1 \le \frac{y}{\sqrt{2t}} \le 1 $, with all $ \mathbb{R}$. The iterated integral ...
5
votes
3answers
611 views

ComplexInfinity/Indeterminate error when evaluating derivative

Motivation: I want to find the coefficients for the polynomial that is obtained when one adds together the first $n$ natural numbers to the power of $a$; that is, when you consider $1^{a} + 2^{a} + \...
5
votes
3answers
459 views

How to force Dt[ ] to recognize dependencies

I want to differentiate a function f[] for which I don't have a specific expression. f[] depends on ...
13
votes
1answer
490 views

What strategies can I use to evaluate a limit when Limit[] returns unevaluated

I'm trying to find the following limit using Mathematica: $$\lim_{N\to\infty}\sum_{k=1}^N\left(\frac{k-1}{N}\right)^N$$ The problem is taken from here and is known to converge to $\displaystyle\frac{...
1
vote
1answer
186 views

How to collect q[t] from the following integration

As shown in the following program, the q[t] in a can be collected from the integration by defining the integration of ...
1
vote
1answer
130 views

How to explain the results of the following integrals?

Why does the following program lead to different results of a21 and a211? ...
2
votes
0answers
422 views

Simple contour integral with a parameter gone wrong

Bug introduced in 7.0 and fixed in 7.0.1 I run into the following problem, I tried to evaluate a very simple integral: ...
7
votes
2answers
303 views

How to simplify the integration result of the following program?

In the following program, the fec[x] is a function depending on $x$ while q3[t] and q5[t] ...
15
votes
3answers
13k views

How to find the nth derivative?

This question is not the same as my last one. How do you find the $n$-th derivative where $n$ is a variable? For example, you can find the nth derivative for a specific $n = 3$ ...
3
votes
2answers
585 views

How to iteratively display the nth derivative?

I am new to Mathematica, but not so new to programming. What I'd like to do in Mathematica (not using an API) is to display on each line, the nth derivative of a function, e.g. for n = 1..10. Is there ...
2
votes
2answers
3k views

How to draw a tangent to a surface in a given direction?

It seems quite embarrassing for me, but somehow I can't draw a tangent line to a surface in a given direction. Assume that I want to visualize directional derivative of the function, say, $(x,y)\...
1
vote
2answers
4k views

Find the dimensions of a cylindrical can that will minimize the cost of the material

A cylindrical can is to be made to hold 1 L of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can. This is a standard calculus problem. The volume of the ...
5
votes
2answers
2k views

Chain rule while differentiating

I am trying to find the derivative of a function defined in polar coordinates with respect to $x$ and $y$. My function is defined as follows: $ v_x(r, \theta ) = v_r \cos (\theta ) - v_{\theta }\sin (...
4
votes
2answers
5k views

How to convert a system of parametric equations to a normal equation?

For example, I have a system of parametric equations (R is a constant number) : ...
13
votes
1answer
347 views

Extra factors appear when evaluating Euler integrals

Note: this is fixed in version 9. When I perform the double integral in Mathematica, Integrate[(x (1 - x))^z (y (1 - y))^z, {x, 0, 1}, {y, 0, 1}] which should ...
56
votes
6answers
5k views

Generating evenly spaced points on a curve

In the KnotData package a simple command such as points = Table[KnotData[{3, 1}, "SpaceCurve"][t], {t, 0, 2 Pi, 0.1}]; will ...
10
votes
2answers
1k views

Why aren't these additions of integrals and summations equal?

I have the following code: Simplify[Integrate[f[x] + g[x], x] == Integrate[f[x], x] + Integrate[g[x], x]] To test: $$\int{\left(f(x) + g(x)\right)dx}=\int{f(x)...
10
votes
2answers
2k views

Simplifying the derivative of $|x|$

Context In[855]:= D[Abs[x], x] /. x -> 1 Out[855]= Derivative[1][Abs][1] In[856]:= D[x, x] /. x -> 1 Out[856]= 1 Question Why is ...
7
votes
4answers
2k views

Mathematica gives wrong answer for an integral

When I execute the following, Integrate[ Exp[-w^2 + I w^3], {w, -∞, ∞}] I get (2 E^(2/27) BesselK[1/3, 2/27])/(3 Sqrt[3]) ...
21
votes
5answers
13k views

How to find the domain and range of a function with Mathematica?

I'm studying calculus and in some exercises I am asked to find the domain and range of a function. Does Mathematica have already a built-in function for this? I can imagine some ways of doing so, ...
3
votes
1answer
1k views

Assumptions with D

I want to give assumptions for the D function. Say I want to calculate $\frac{d|x|}{dx}$ for $x>0$, which is 1. I write ...
3
votes
2answers
412 views

Strange result when integrating BesselJ functions

Consider the following integral: ...
17
votes
2answers
2k views

Getting an InterpolatingFunction from a ContourPlot

I have a function, say minimizeme[Ω_][β_][ϵ_] = ϵ^2 Ω - Log[2 (Cosh[2 β] + Cosh[2 β ϵ])]/(2 β); I want to find its critical points in $\epsilon$ for a given ...
7
votes
4answers
2k views

Can't compute definite integral

Consider a scalar field (in polar co-ordinates), $f(r) = l-r$. Now I want to evaluate the field integral over a circular region of radius $b$, centered at a distance of $x$ from the origin. According ...
5
votes
0answers
214 views

Calculating a limit with a result that is discontinuous in the parameters

The following limit is left unevaluated (Edit: added the assumption that $\epsilon$ is real thanks to the comment below): ...
28
votes
3answers
4k views

How to deal with complicated Gaussian integrals in Mathematica?

As we know, for most Gaussian integrals, we can get the analytical result. Now I have many Gaussian integrals to treat, which have the following general form, ...
3
votes
2answers
2k views

How to substitute numeric values in a symbolic Jacobian matrix?

I have a multi-variate function from $\mathbb{R}^n\to\mathbb{R}^n$. Choosing any desired initial vector, we can produce the corresponding function value, which is a vector as follows. The main problem ...
19
votes
1answer
5k views

How does Mathematica integrate?

Basically, this question can be considered to be an extenstion to my other question. What I wanted to do was this integral as homework (it is indefinite BTW so no approximations using Simpson's Rule ...
2
votes
2answers
931 views

Quickly differentiate and evaluate a function of several variables

How can I differentiate a function with respect to several variables, and evaluate it at the same time? I want to specify also the variable index that I want to differentiate, and the number of times ...
9
votes
2answers
568 views

Check homework integration in Mathematica

Given in my homework I have to compute (by hand) $$\iint\limits_{x^2+y^2\leq 1}(x^2+y^2)\,\mathrm dx\,\,\mathrm dy.$$ My solution so far: Let $f(x,y)=x^2+y^2$ and $K=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq ...
10
votes
2answers
1k views

Derivative from the left and right

I am currently trying to write a script that takes a curve $C$, a starting point $p$ and a direction vector $\vec{v}$ and calculates the path of a beam starting at $p$ in direction $\vec{v}$ when it ...
37
votes
1answer
924 views

How to augment the realm of functions Mathematica thinks it knows how to integrate symbolically

My question involves extending the functionality of Integrate over specific integrals in the most generic manner. Specifically, is it possible to "hack into" ...
6
votes
1answer
1k views

Finding ranges of a parameter for which a function is always positive

I have a complex function of a single variable expressed in analytical form, which also depends on a parameter. I would like to have Mathematica show me for which values of the parameter the real part ...
30
votes
4answers
6k views

How to find the period of an arbitrary mathematical function?

Is there a function to find the period of an arbitrary (possibly complex) function in Mathematica?
8
votes
1answer
1k views

Limit not evaluating

Why is the input Limit[IntegerPart[Sin[x]/x], x -> 0] not being evaluated?
14
votes
3answers
5k views

how to simplify symbolic integration

The following is the code: ...
10
votes
3answers
5k views

Using D to find a symbolic derivative

I need to do the following: Define a function Take the derivative of this function and have a look at the symoblic representation Substitute in some values With the bonus that I want to use the ...
7
votes
1answer
3k views

How to find the smallest root

I have a continuous, differentiable, monotonic, bounded function called F[t]. If t -> Infinity then ...
19
votes
2answers
1k views

Incorrect results for elementary integrals when using Integrate

There is a rather simple integral ($K_0$ is the 0-th order MacDonald function) $$\int_0^\infty e^{-x \cosh\xi}\, d\xi = K_0(x)$$ which mathematica cannot solve. This even though the documentation ...