9 Numerical Methods

Question 1

show x^3/5 =x(x-1)

has a root between 3 and 4

Answer 1

x^3 =5x^2 -5x
x^3 -5x^2 +5x =0

3^3 -5(3)^2 +5(3) =-3
4^3-5(4)^2 +5(4) =4

sign change so there is a root between

Question 2

x^3 -12x +12=0

Answer 2

x= x^3/12 +1

y=x
and y=x^3/12 +1

3 points of intersection, finding a root will tell you
“confirm there is a root between x=1 and x=2”
put x=2 into y=x^3/12 +1

take that value and put into equation until both Y and X equal the same number

where the numbers converge the graphs intersect there
put that value into x^3-12x +12=0

do the same but with x=1

Question 3

what to remember with cobwebs

Answer 3

always draw to the curve first

Question 4

positive gradient curves

Answer 4

staircase

Question 5

curves with negative gradient

Answer 5

cobweb

Question 6

continuous and discontinuous graphs

Answer 6

discontinuous are ones with a break, certain fractions usually

Question 7

how to find the limit

Answer 7

L replaces both the xn+1 and the xn

Question 8

show 2 x^3-x^2 -5x-1

x= sqrrt(5x+1/x-1)

write a recurrence relation

Answer 8

x^3-x^2 =5x +1
x^2(x-1) =5x+1
…

xn+1= sqrrt(5xn+1/xn-1)

Question 9

mid ordinate rule

Answer 9

area= width of strip x sum of mid ordinates

Question 10

lnx 2<5 (or the same as)

6 strips

Answer 10

5-2 /6= 0.5

x= 2.25, 2.75……4.75
y=0.811

0.5x(y1+y2+y3…)= 3.66

Question 11

numerical interrogation

Answer 11

assessing whether you have made an over or under estimate

Question 12

simpson’s rule

Answer 12

1/3h[(y0+yn)+(4x odd ordinates) +(2xeven ordinates)]