Analyzing

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describe everything

Completely analyze $T(r) = \tan (\sin (\pi r))$

$\blacktriangleright $ Domain:

We know that $-1 \leq \sin (\pi r) \leq 1$ and $[-1,1] \subset \left ( -\frac {\pi }{2}, \frac {\pi }{2} \right )$ and $\tan (x)$ is continuous on $\left ( -\frac {\pi }{2}, \frac {\pi }{2} \right )$. Therefore $T(r)$ is defined for all $r$.

$\blacktriangleright $ Continuity:

Since both $\tan (x)$ and $\sin (x)$ are continuous, $T$ is the composition of two continuous functions. This also tells us that $T(r)$ is a continuous function.

$T(r)$ will be using $[-1,1]$ out of tangent’s $\left ( -\frac {\pi }{2}, \frac {\pi }{2} \right )$ waveform.

Graph of $y = T(r)$.

$[-1,1]$ is the output of $\sin (\pi r)$, which then becomes the input to tangent.

For this to happen, the input to sine must be $\left [-\frac {\pi }{2}, \frac {\pi }{2}\right ]$, which means that $r \in \left [-\frac {1}{2}, \frac {1}{2}\right ]$.

But that is only half the story.

As $r$ moves from $-\frac {1}{2}$ to $\frac {1}{2}$, $\sin (\pi r)$ moves from $-1$ to $1$. As $r$ continues, it moves from $\frac {1}{2}$ to $\frac {3}{2}$ and $\sin (\pi r)$ moves from $1$ back down to $-1$. Then the whole thing repeats.

Now we have a full period.

That gives a full wave over the interval $\left [-\frac {1}{2}, \frac {3}{2}\right ]$ and a period of $2$.

As $r$ increases, the sine wave will oscillate between $-1$ and $1$ and this piece of the tangent graph will also oscillate.

Since our period is $\left [-\frac {1}{2}, \frac {3}{2}\right ]$, let’s regraph with integer tick marks.

$\blacktriangleright $ Maximums:

The maximum of sine is $1$ and it occurs at $\frac {\pi }{2}$.

Tangent is an increasing function, so the maximum of $T(r)$ is also going to occur at $\frac {\pi }{2}$. However, in $T(r)$, the inside of sine is $\pi r$.

The maximum of the sine waveform occurs at $r = \frac {1}{2}$.

\[ \tan \left (\sin \left (\pi \cdot \frac {1}{2}\right )\right ) = \tan \left (\sin \left (\frac {\pi }{2}\right )\right ) = \tan (1) \approx 1.557 \]

The minimum of $-\tan (1) \approx -1.557$ occurs at $r = -\frac {1}{2}$.

That is just for one period. These maximums and minimums are periodic with a period of $2$.

$\blacktriangleright $ Minimums:

The minimum of sine is $-1$ and it occurs at $-\frac {\pi }{2}$.

Tangent is an increasing function, so the minimum of $T(r)$ is also going to occur at $-\frac {\pi }{2}$. However, in $T(r)$, the inside of sine is $\pi r$.

The minimum of the sine waveform occurs at $r = -\frac {1}{2}$.

\[ \tan \left (\sin \left (-\pi \cdot \frac {1}{2}\right )\right ) = \tan \left (\sin \left (-\frac {\pi }{2}\right )\right ) = \tan (-1) \approx -1.557 \]

That is just for one period. These maximums and minimums are periodic with a period of $2$.

We have also discovered that the critical numbers for $T$ are $\left \{ \frac {1}{2} + 2k \, | \, k \in \mathbb {Z} \right \}$ and $\left \{ -\frac {1}{2} + 2k \, | \, k \in \mathbb {Z} \right \}$

$\blacktriangleright $ Rate-of-Change:

$\tan (t)$ increases on any interval in its domain. Therefore, $T(r) = \tan (\sin (\pi r))$ will increase and decrease with $\sin (\pi r)$, which we know everything about.

$T(r)$ increases on $\left [ -\frac {1}{2}, \frac {1}{2} \right ]$ and decreases on $\left [\frac {1}{2}, \frac {3}{2} \right ]$.

And, this agrees with our graph.

Increasing

We have two versions of increasing and we are beginning to notice.

First, we have a Algebra version of increasing. The Algebra version occurs over an interval.

$f(x)$ increases on the interval $[a,b]$, provided whenever $c < d$, then $f(c) < f(d)$.

We often attach the work average to the Algebra version.

Separate from that is a Calculus version of increasing. The Calculus version occurs at a domain number.

$f(x)$ increases at $x=d$, provided $f'(d) > 0$.

We often attach the work instantaneous to the Calculus version.

These are different. A function can be increasing at every domain number and yet not be an increasing function over the whole domain. We’ll want to keep these ideas separate.

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2025-05-17 23:12:03