C13 (Numerical Methods)

Question 1

locating a root

Answer 1

look for a sign change

a root is where f(x) = 0

Question 2

improving the precision of a root (2 methods)

Answer 2

interval bisection: cutting the interval in half after every iteration (i.e if root is between 5 and 5.8, then test 5.4, then 5.2 then 5.3 or 5.1)

decimal search: systematically increasing no. sig figs (i.e if root between 1-2, then test 1.1, 1.2, 1.3 … then 1.31, 1.32, 1.33.

if it is asking for the root to 2dp, you need to calculate the interval to 3dp, to see if it rounds up

Question 3

iterative formula

Answer 3

Question 4

rearrange into iterative formula (and solve if u want)

Answer 4

so -1.487...

Question 5

how to draw cobweb/staircase diagram

Answer 5

sketch curve and line y = x

draw line vertically to curve, then horizonatally to y = x, then vertically to curve, then horizontally to y = x …

Question 6

staircase diagram

Answer 6

Successive iterations get closer to the root,
approaching the root from the same direction. This
is graphically represented by a “staircase” diagram.

Question 7

cobweb diagram

Answer 7

Successive iterations alternate between being
above and below the root. This is graphically represented
by a “cobweb” diagram.

Question 8

central estimate/difference concept

Answer 8

estimating the gradient of a point by drawing a line parallelish to the tangent and finding its gradient

Question 9

central difference formula

Answer 9

the difference between the function’s (y) values at points (x+h) and (x-h), divided by the distance (2h) between them:

Question 10

foward/backward distance

Answer 10

going from point x to either (x + h) or (x - h) and dividing by h

Question 11

Answer 11

Question 12

finding area under a curve using rectangles

Answer 12

remember, add up y’s, then multiply by h (width of each rectangle)

Question 13

trapezium rule

Answer 13

Question 14

3 applications of numerical methods

Answer 14

gradient of d/t graph = speed

gradient of s/t graph = acceleration
area under s/t graph = distance travelled