Approximations

Question 1

Error

Answer 1

Approx - Actual

Question 2

Absolute error

Answer 2

modulus of error (approx - actual)

Question 3

Relative error

Answer 3

Ratio of error to exact value - (approx - actual) / actual if x ≠ 0
Can be rewritten as approx/actual - 1

Question 4

Absolute relative error

Answer 4

Modulus of relative error

Question 5

Percentage error

Answer 5

Relative error x 100

Question 6

What sign used for approximations

Answer 6

≈

Question 7

What is a radian

Answer 7

the angle of a sector where all sides are equal (to radius r)
also equal to 360/2n

Question 8

maximum error when a number is rounded to n decimal places

Answer 8

5 x 10^-n-1
or 0.00…005

Question 9

max error when number chopped to n decimal places

Answer 9

1 x 10^-n
or 0.00…99…

Question 10

expected error (of rounding/ chopping)
and values for 500 numbers to 2dp

Answer 10

average error (max error / 2) x number of numbers being rounded/chopped
e.g for 500 numbers expected error of rounding to 1dp is 500 x 0.005/2 = 1.25
for chopping is 500 x 0.01/2 = 2.5

Question 11

if X is approx of x with relative error m
and Y is approx of y with relative error n
whats the relative error of XY as approx of xy and X/Y as an approx of x/y

Answer 11

XY - m + n
X/Y - m - n

Question 12

what minimum sf should you round to for approximations

Answer 12

3sf

Question 13

Ill-conditioned problems

Answer 13

Where tiny changes have a large impact

Question 14

Examples of ill conditioned problems

Answer 14

Polynomials, exponentials, trigonometry

Question 15

How to ‘handle’ (seriously) ill conditioned problems

Answer 15

Avoid rounding until the very end
Dont restart calculator so previous answers are stored and can be reused