A Wild Function: Derivative + Integral + Series + Infinite Product

We are building and visualizing a very complicated function

$$F(x)= \frac{d}{d x}\left[e^{x^{2}} \int_{0}^{x} \frac{\sin (t^{3})+\ln (1+t^{2})}{1+t^{4}} dt\right] - \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}} \cos (n x^{2}+\sqrt{n}) - \lim_{N\to\infty} \prod_{k=1}^{N}\left(1+\frac{x^{2}}{k^{2}} e^{-x / k}\right).$$

This panel explains each piece conceptually, and the canvas shows how these pieces combine and behave as you vary parameters.

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1. First term: derivative of a product with an integral

Consider

$$G(x) = e^{x^{2}} \int_{0}^{x} \frac{\sin (t^{3})+\ln (1+t^{2})}{1+t^{4}} \, dt.$$

The first term of $F(x)$ is $G'(x)$. This is a product of:

• an outer factor $e^{x^{2}}$, and
• an integral that depends on the upper limit $x$.

To differentiate $G(x)$, we use:

1. Product Rule

$$(u v)' = u'v + u v',$$

with

• $u(x)=e^{x^{2}}$, so $u'(x) = 2x e^{x^{2}}$,
• $v(x) = \displaystyle \int_{0}^{x} f(t)\,dt$, where

$$f(t) = \frac{\sin(t^{3}) + \ln(1+t^{2})}{1+t^{4}}.$$

2. Fundamental Theorem of Calculus (FTC):

$$\frac{d}{dx} \int_{0}^{x} f(t)\,dt = f(x).$$

Putting these together:

$$G'(x) = 2x e^{x^{2}} \int_{0}^{x} f(t)\,dt + e^{x^{2}} f(x).$$

So the first term of $F(x)$ is a combination of:

• the accumulated area under the curve $f(t)$ from 0 to $x$, and
• the instantaneous value of $f$ at $t=x$, all weighted by the growing factor $e^{x^{2}}$.

In the visualization, you can:

• slide $x$ to see how the integral $\int_0^x f(t) dt$ builds up,
• see how $e^{x^{2}}$ amplifies this accumulation, and
• see the derivative $G'(x)$ as a moving point on a curve.

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2. Second term: oscillating infinite series

$$S(x)=\sum_{n=1}^{\infty} \frac{(-1)^n}{n^{2}}\cos(n x^{2} + \sqrt{n}).$$

Features:

• Oscillations: each term is a cosine with phase $n x^{2} + \sqrt{n}$, so as $n$ increases, the oscillation frequency in $x$ becomes more and more rapid.
• Alternating sign: the factor $(-1)^n$ makes successive terms flip sign.
• Decay: $1/n^{2}$ makes each term smaller as $n$ grows.

The decay $1/n^{2}$ is strong enough that the series converges for every real $x$. Intuitively, the more we add high-index terms, the more refined “ripples” we add, but they are smaller and smaller.

In the visualization, you can:

• adjust the number of terms $N$ used in the partial sum $S_N(x)=\sum_{n=1}^N \dots$,
• see the graph of this finite approximation and how adding more terms refines the wavy structure but does not blow up.

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3. Third term: infinite product

$$P(x) = \lim_{N\to\infty} \prod_{k=1}^{N}\left(1+\frac{x^{2}}{k^{2}} e^{-x / k}\right).$$

For each fixed $x$:

• We multiply many small factors $1 + \frac{x^{2}}{k^{2}} e^{-x/k}$.
• As $k$ grows, $\frac{x^{2}}{k^{2}}$ tends to 0 and $e^{-x/k}\to 1$, so each factor tends to 1.
• The infinite product converges if these small deviations from 1 are “summably small”.

Intuitively, $P(x)$ behaves like:

• 1 when $x$ is very small (all factors are close to 1),
• something more structured for larger $x$, depending on the whole infinite ladder of multiplicative corrections.

In the visualization, we approximate

$$P_N(x) = \prod_{k=1}^{N}\left(1+\frac{x^{2}}{k^{2}} e^{-x / k}\right)$$

with a finite $N$ and show how the curve stabilizes as you increase $N$.

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4. The full function $F(x)$

Putting everything together,

$$F(x) = G'(x) - S(x) - P(x).$$

Each part contributes a different "personality":

• First term $G'(x)$: smooth growth shaped by the integral and boosted by the exponential.
• Second term $S(x)$: oscillatory “ripples” at many scales due to the cosine series.
• Third term $P(x)$: more subtle multiplicative shaping through the infinite product.

Around special points (like $x=0$) you can analyze:

• The integral near 0: the integrand is continuous and bounded, so $\int_0^x f(t) dt \approx f(0)x$ for small $x$.
• The exponential: $e^{x^{2}} \approx 1 + x^{2}$.
• The series and the product can be studied term-by-term at $x=0$:
  • $S(0) = \sum_{n=1}^{\infty} \frac{(-1)^n}{n^{2}} \cos(\sqrt{n})$ — a convergent numerical constant.
  • $P(0) = \prod_{k=1}^{\infty} (1+0) = 1$.

The visualization does not attempt to compute exact symbolic values but lets you see how:

• the derivative of the integral term behaves as you move $x$,
• the partial sums of the series add finer oscillations,
• the partial products shape the overall magnitude.

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How to use the visualization

• Use the x-position slider to explore $F(x)$ along the horizontal axis.
• Use term / product depth sliders to see how the series and product parts approach their infinite limits.
• Use component toggles to isolate each building block: integral-derivative, series, and product.
• Watch the animated construction along the curve as time passes (the tracer moves with a parameter along the graph).

Mathematically, this kind of function is an example used to discuss:

• convergence (of the series and product),
• smoothness (differentiability, continuity),
• and how powerful tools (FTC, product rule, infinite sums/products) combine to create highly nontrivial functions.

The canvas turns these abstract constructions into a live, layered picture of how $F(x)$ is built.