Academic Detention Reflection #2

The second blog of this detention I will start by explaining linearization. If the word approximate is used then you should know to use linearization. The steps for working linearization problems are:
1. Identify the equation
2. Use the formula f(x)+f ' (x)dx
3. Determine your dx in the problem
4. Then determine your x in the problem
5. Plug in everything you get
6. Solve the equation

Example problem:(sr=square root) Use differentiability to approximate sr(4.5)
f(x)=sr(x) sr(4)+(1/2 sr(4) )(.5)=1.125
sr(x)+(1/2 sr(x) ) dx error=.005
dx=.5
x=4

The next topic I will talk about is integration. Integration finds the area under a curve. The Riemann sum approximates the area using the rectangles or trapezoids. The Riemanns Sums are:
LRAM-Left hand approximation=delta x[f(a)+f(a+delta x)+...f(b-delta x)]
RRAM-Right hand approximation=delta x[f(a+delta x)+...f(b)]
MRAM-Middle approximation=delta x[f(mid)+f(mid)+...]
Trapezoidal-delta x/2[f(a)+2f(a+delta x)+2f(a+2 delta x)+...f(b)]
delta x=b-a/number of subintervals

Example problem: Find the area of f(x)x-3 on the interval [0,2] with 4 subintervals.
delta x=2-0/4=1/2
LRAM=1/2[f(0)+f(1/2)+f(1)+f(3/2)]
1/2[-3+-5/2+-2+-3/2]
1/2[-9]= -9/2
RRAM=1/2[f(1/2)+f(1)+(3/2)+f(2)]
1/2[-5/2+-2+-3/2+-1]
1/2[-7]= -7/2
MRAM=1/2[f(1/4)+f(3/4)+f(5/4)+f(7/4)]
1/2[-11/4+-9/4+-7/4+-5/4]
1/2[-8]= 4
Trapezoidal=1/4[f(0)+2f(1/2)+2f(1)+2f(7/2)+f(2)]
1/4[-3+-10/2+-4+-6/2+-1]
1/4[-16]= 4

For things I do not know I forgot how to do average speed and seem to have lost my notes on this so if someone can remind me on how to do these problems I would greatly appreciate this.