Derivatives and Integration
1. Derivatives
Example 1:
Traveling from Chapel Hill to Raleigh (Distance(D)=30 miles), leaving Chapel Hill at 8:00am, arriving Raleigh City Hall at 8:45am. Therefore the speed of your travel is
v=D/t = 30miles/0.75hr=40 miles/hr.
This actually the average speed, you may travel at a different speed at any particular moment. If one asks what your speed is at 8:15am, we may have to figure that out in the following way:
We know D=v*t, if we know the distance traveled from Chapel Hill at 8:15am, D(8:15am) and the distance your traveled two minute later (i.e. D(8:15am), Then the distance you traveled from 8:15am to 8:17am is D(8:15am)-D(8:15am+2min). then the speed at 8:15am can be estimated as
(D(8:15am)-D(8:17am))mile /2 min
The actual speed at 8:15 may still be different from the above estimation, but it is a better estimation than 40 miles/hr. In fact, the shorter the time you allow your car to travel after 8:15am, the more accurate the speed you calculate. Let t stand for time, and Δt for the time allowed for travel, the speed at 8:15am can be written as:
Vt=8:15am ≈
Example 2:
How many of you have watched the last launch of space shuttle Endeavour on August 8, 2007? How fast the shuttle is traveling at the time it is just off the launch pat? How fast the shuttle is traveling at the just before it reached orbit (18,000 miles/hour)? In order to get rid of the gravitation of Earth, an object has to travel at an accelerating speed of 7.9km/s2. If you do a plot of time and distance the shuttle is traveling, it would look like
The last Endeavour launch took place at 6:36pm on August 8. If I ask how fast the shuttle is traveling at 6:37pm, how would to figure it out the speed?
The shorter the Δt is, the more accurate the speed. Mathematically,
In general: If a function y=f(x) exists at x0, when x increased Δx at x0, i.e. x = x0+Δx, the function has a corresponding increase Δy=f(x0+Δx)-f(x0), if the limit of the ratio of Δy to Δx exists when Δxà0, the limit is called the derivative of y=f(x) at x=x0.
Examples:
y=f(x)=C (C=constant)
This means regardless of what x value is, y is always x. Thus
f(x)=C,
f(x+Δx)=C
Thus, the derivative of any constant is zero.
y=f(x)=x,
f(x)=x,
f(x+Δx)=x+Δx
y=f(x)=x2
f(x)=x2
f(x+Δx)=(x+Δx)2=x2+2xΔx+Δx2
In general
(xn)’ = nxn-1
For convenience, we can figure out the derivatives for the commonly used functions and put them in a table for later use so that we don’t have to do this again and again. Here they are:
(C)’=0
(x)’=1
(xn)’=nxn-1
(sinx)’=cos(x)
(tagc)’=1/cos2x
(ctagx)’=-1/sin2x
(ex)=ex
(ax)’=axln(a)
(ln(x))’=1/x
The Geometric Meaning of the Derivatives
When Δxà0, the angle φà α, therefore, the derivatives of y=f(x) at x, f’(c), is the slope of the tangent line passing (x, y).
The functions we provided with derivative are very simple functions. We often work with more complex functions that are made from the simple ones. , for example, sin(x2), e2x, etc., we call these functions compound functions as they are functions containing fuctions.
Where sin(x2) can be written as sin(u), where u=x2. Similarly, e2x can be written as eu where u=2x. Here are the rules for taking derivatives for compound functions:
If y=f(g(x)) is the compound function of y=f(u) and u=g(x). if the derivatives for u=g(x) exists at x, and y=f(u) exists at u=g(x). then the derivative of the compound function y=f(g(x)) with respective to x is
; or f’(x)=f’(u)*g’(x)
Examples:
1) y=sin(x2), and y=sin(u), u=x2
y’=(sin(u))’×u’=(sin(x2))’×(x2)’=2xcos(x2)
2) y=sin2(x)
Let u=sin(x), y=sin2(x)=u2,
3) y=e2x+sin(x)
Let u=?,
4) y=e2x+cos(2x)
Integration:
The inverse of derivatives: Let me ask the inverse question in Example 1 of derivatives, if I travel at 40 mph on I-40 east, where am I in 45 minutes, how far away I am from Chapel Hill?
We know we traveled 30 miles in 45 min at that speed, is that sufficient to know where we are? What else do we need to know?
In derivative, we can write dS/dt=40. The inverse of that is intergration, i.e.
S=40t+C
Where C is a constant determined by the initial condition.
e.g. (x2)’=2x, in fact (x2+C)’=2x, where C is a constant.
Similarly we can create a table of integration:
(1)
(2) , where n≠-1
(3)
(4)
(5)
Definite Integration: Given a function f(x) which is bounded on [a, b]. Randomly insert n points within [a, b] so that,
a=x0<x1<…<xn=b,
separate the interval [a,b] into n smaller intervals,
[x0, x1], [x1, x2], …, [xn-1, xn],
The lengths of the invervals are, respectively,
Δx1=x1-x0, Δx2=x2-x1, …, Δxn=xn-xn-1.
Take any number εi from any interval above, calculate the product, f(εi)Δxi, and sum the product,
Let λ be the maximum length of the n intervals, if λà0, regardless of how [a, b] is separated, and how εi is taken from the interval [xi-1, xi], S is always approach a finite limit. The limit is the definite integration of f(x) in the interval [a, b].
Newton-Leibniz Formula:
, where f(x)=F’(x)
is the area under the curve from a to b.
Examples:
1)
2)
3)