User:PseudonympH

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Just to clarify, it's actually Pseudonym-pH. I tacked the pH on when regular "Pseudonym" was taken on Battle.net because it's, well, a science term. I also thought it was cool that it flowed well like that, so I kept the suffix. And no, as some people might think, I am not a female. :P

[edit] Math Class

If you're here because of the link posted on Nicenet, here's where you can find the nifty formulas and stuff. I did it here instead of there because i can do pretty math formatting like this:

[edit] Midyear review, number 11.

(Because it was the only one left :) )

For the function

f(x) = 2x² − lnx

a. Find the equation of the tangent line to the graph of f(x) at (1,2).

First, we start off by finding the derivative of f(x)

Taking this when x = 1 will give us m for our y=mx+b equation. We can therefore conclude that m=3, and we can them solve for b:

y = mx + b

2 = (3)(1) + b

b = − 1

The equation of the tangent line is then y=3x-1.

b. Prove that f(x) is equal to 3 somewhere in the interval [1,2].

We can prove this easily using the Intermediate Value Theorem. First evaluate when x=1:

f(1) = 2(1)² − ln1 = 2

Then evalute when x=2:

f(2) = 2(2)² − ln2 = 8 − ln2

Here we hit a bit of a snag, as we can't (easily) evaluate ln(2). However, since 2 is less than e, ln(2) must be less than 1. Therefore,

f(2) > 7

By the Intermediate Value Theorem, we can conclude that there is a number k on [1,2] such that f(k)=3.