Announcements! ( See All )
4/28 - Final Assessment Part I has been released, and is due Tue 5/5 at 12PM noon.
4/21 - Week 13 Checkpoint, HW 12, and Lab 9 have been released. Week 13 Checkpoint is due Thu 4/23, HW 12 is due Tue 4/28, and Lab 9 is due Sun 5/3.
4/14 - Week 12 Checkpoint and HW11 have been released. Week 12 Checkpoint is due is due Thu 4/16, and HW 11 is due Tue 4/21
4/7 - Week 11 Checkpoint, HW 10, and Lab 8 have been released. Week 11 Checkpoint is due Thu 4/9, HW 10 is due Tue 4/14, and Lab 8 is due Sun 4/19.
3/31 - Homework 9 and the Week 10 Checkpoint have been released. Week 10 Checkpoint is due 4/2 and Homework 9 is due 4/7.
3/20 - Homework 8 and Lab 7 have been released. Homework 8 is due Tue 3/31 and Lab 7 is due Sun 4/5.
3/10 - Homework 7 and Lab 6 have been released. Homework 7 is due Tue 3/17 and Lab 6 is due Sun 3/15.
3/5 - Homework 6 has been released, and is due Tue 3/10. No Lab this week.
2/25 - Lab 5 has been released, and is due Sun 3/1. No Homework this week so you have time to study for the Midterm on Tue 3/3.
2/18 - Homework 5 has been released, and is due Tue 2/25. Lab 4 is due Sun 2/23.
2/11 - Homework 4 and Lab 4 have been posted! Lab 4 is due Sun 2/23. This is a challenging lab; please start early. Lab Party on Thu 2/13 is being converted to a Homework Party.
1/21 - Homework 1 and Lab 1 have been posted! First section Wed 1/22, please be sure to bring a laptop!

Exponential and Log Functions

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Limits and Approximations

For all $x$,

The expansion as a sum implies that

Here the symbol $\sim$ means that the ratio of the two sides goes to 1 as $x$ goes to 0.

You can see this approximation in the figure. Around $x = 0$, the blue graph of $e^x$ and the red graph of $1+x$ are almost indistinguishable.

Take $\log$ on both sides to see that

To spot this in the figure, look at values near 1 on the horizontal axis. You can see that $\log(x) \sim -1 + x$ for $x$ near 1. Write $x$ as 1 plus a small increment $w$. The approximation becomes $\log(1+w) \sim w$ for small $w$, which is the result we had before. It doesn’t matter whether you refer to the small number as $x$ or $w$. But it does matter that it is small.

For exact values, let $\vert x \vert < 1$. Then the Taylor expansion of $\log(1+x)$ is

Bounds

  • $e^x \ge 1+x$ for all $x$. You can see this in the graphs above.

  • For $x \in (0, 1)$:

    • $\log(1+x) < x$
    • $\log(1+x) > x - \frac{x^2}{2}$
    • and so on. You get alternating upper and lower bounds as you go further into the series.

In probability theory you will often come across logs of values that are near 1. Here is another useful lower bound on $\log(1+x)$. It is true for all positive $x$ but it is only close to $\log(1+x)$ when $x$ is close to 0.

  • For $x > 0$, $\log(1+x) \ge \frac{x}{1+x/2} = \frac{2x}{2+x}$

There are many ways of proving inequalities like this one. One way is to show that the difference between the two sides has the right sign, by noticing that it is 0 at $x=0$ and monotone (in the right direction) on the specified domain.

Here are the graphs of $\log(1+x)$ and $2x/(2+x)$.

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The difference $\log(1+x) - 2x/(2+x)$ is an increasing positive function:

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