Final Will Focus on Material Since the Midterm

Final exam is Thursday Dec. 19 from 7pm-9pm

1210 LeBaron Hall (alternate seating)

Final will focus on material since the midterm

Related rates

Max min

Graphing

Integration

Def. integral (FTC, area, distance)

Sep Diff eqs

Orders of Growth

Limits, L’Hopital’s Rule

Chapter 7

Logarithms and Exponentials (TMI)

There are 2 schools of thought on how logs should be taught in calculus. That’s the reason for this weird book organization.

1.  Students see logs and exponentials in high school, so let’s use them right away in calculus!
Remember chapter 3: What is the derivative of 2x? We said that 2x'=limh→02x+h-2xh=0.69*2x=ln22x
Is there a number X such that Xx=1*Xx=Xx→X=2.71828=e
So we decided ex'= ex
The inverse of ex was decided to be ln⁡(x). Therefore:

ea=b↔lnb= a

2.  Do it the historically accurate way as an integral. No more hand-waving!
We know that ∫xndx=xn+1n+1+c (n≠-1). What if we want the antiderivative of 1/x?
Define lnx to be the antiderivative of 1x.
lnx= 0x1tdt x>0

Section 7.1 develops this and eventually shows that lnx=loge(x)

Section 7.2 Separable Differential Equations

A differential equation is an equation that involves derivatives of a function. Idea: Solve equation, find functions that “work” in that equation.

Example:

d2ydx2-2dydx-3y=0

Verify that y=e3x and y=e-x solve this equation.

Plug in different components.

y=e3x
dydx=3e3x
d2ydx2=9e3x
Plug these in, you’ll find that it equals 0.

Important stuff!

Easiest differential equations are separable equtions (put variables on each side of the equation)

dydxy2=y3+3x2

We can get all of the y variables onto one side, all the x’s on the other.

y2dyy3+3= x2dx

y2y3+3dy=x2dx
u=y3+3
du=3y2dy→13du=y2dy
1u*13du=x33+C

13lnu=13x3+C

13lny3+3=13x3+C

This is our solution! How lame! Most of the time the solution is implicit.

Initial value problems

As we just saw, a differential equation usually has infinitely many solutions. But! If you’re trying to describe some physical phenomena, there is often a set of conditions that will allow for a single solution. It helps you “know C”.

dydxy2=y3+3x2 ;y1= 4