# 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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### Finding unit tangent, normal, and binormal vectors for a given r(t)

For my Calc III class, I need to find $T(t), N(t)$, and $B(t)$ for $t=1, 2$, and $-1$, given $r(t)=\{t,t^2,t^3\}$. I've got Mathematica, but I've never used it before and I'm not sure how to coerce ...
4answers
8k views

### Usage of Assuming for Integration

For some reason, when I enter the following integration in Mathematica Assuming[{k ∈ Integers}, Integrate[ Exp[ I k t], {t, -π, π}]] the result turns out to be 0....
5answers
20k views

3answers
8k views

### Step-by-step definite integration

I am interested in obtaining the intermediate steps for rather straightforward symbolic definite integrations (e.g., from an elementary calculus text). As an example, for $\int_0^1 x\,dx$ I'd like ...
5answers
3k views

### Graphically approximating the area under a curve as a sum of rectangular regions

I am completely new to Mathematica. Basically I was trying to write a code to plot a function and draw the approximate area by rectangles. To be more precise, plot a function ...
2answers
4k views

### How does Mathematica understand branchcuts of the complex logarithm?

Say I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. Then it seems to be me that Mathematica when given such a ...
3answers
4k views

### Can we teach Mathematica about functional differentiation?

The key relation for functional differentiation is $$\frac{\delta}{\delta f(y)}f(x)=\delta(x-y),$$ where $\delta(x-y)$ is the Dirac delta function, and the usual properties of differentiation (e.g. ...
2answers
2k views

### Bug in mathematica analytic integration?

I found Mathematica provides me a wrong answer for a relatively simple analytically solvable integral: ...
3answers
5k views

### how to simplify symbolic integration

The following is the code: ...
2answers
2k views

### Simplifying the derivative of $|x|$

Context In:= D[Abs[x], x] /. x -> 1 Out= Derivative[Abs] In:= D[x, x] /. x -> 1 Out= 1 Question Why is ...
1answer
240 views

### When to use GenerateConditions -> True

Many functions, usuallly those involving integration, take a GenerateConditions option which often defaults to False, or at ...
3answers
1k views

### Inverting a function in a certain region

InverseFunction works well for globally invertible functions, like f = 2*# + 2 &; InverseFunction[f] ...
3answers
6k views

5answers
5k views

### Using implicit differentiation to find a line that is tangent to a curve at a point

A few days ago I asked about using differentiation to find a line that is tangent to a curve at a given point. J.M. provided a very elegant way to solve these kinds of problems in Mathematica. Now I'...
2answers
656 views

3answers
436 views

### Integrating a real function I get a complex value, while after variable transformation the result is real. Bug?

I have the following integral: Integrate[1/Sqrt[0.7 + 0.3*(1 + z)^3], {z, 0, ∞}, Assumptions -> z ∈ Reals] >> -3.36354 - 3.85013 I The output is complex, ...
4answers
3k views

### Plot a 2D vector path onto a surface

In my calculus 3 course, we're studying gradients and have a project that takes a combination of 3D Gaussian radial surfaces and a basic parametric path $r(t) = \{x(t),y(t)\}$ to see how the gradient ...
6answers
9k views

### Finding the centroid of the area between two curves

When I have an area bounded by curves, is there a built-in way to find the center of the area? Or do I have to plot it first and then use ComponentMeasurements on ...
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 ...
5answers
648 views

### Limit of partial sums involving inverse squares

Consider the finite sum rs[x_, n_] := x/n Sum[n^2/(i + (n - i) x)^2, {i, 1, n}] Is there a way to bring Mathematica to calculate the limit for ...
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 $n = 3$ ...
2answers
4k views

### Replace rule with function? Derivatives don't evaluate

Say I have an expression (call it expr) involving a function, f[x]. I'd like to be able to evaluate that for a particular choice of ...
3answers
3k views

### Creating the Nabla operator (also known as Del operator) as an operator

How can I define the nabla operator (also known as Del operator) as a an operator, acting on everything to the right of the operator! Also taking \[Del]^2 would ...
1answer
3k views

### Calculating volume between two surfaces of revolution

Once again, I'm very new to Mathematica and winging it as I go. I probably don't have the math education at this point to even justify buying it in the first place, but that's off topic. I wanted to ...
1answer
540 views

### How to code around known MMa special-case failures?

Fourier Analysis and Signal Processing often require these integrals on functions f[x] that won't be known in advance. ...