Calculations with Uncertainties
Uncertainties in Functions
Recap
Uncertainty Calculations - Functions
Wilfrid Laurier University
Terry Sturtevant
Wilfrid Laurier University
May 13, 2013
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Calculations with uncertainties
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Calculations with uncertainties
When quantities with uncertainties are combined, the results
have uncertainties as well.
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Calculations with uncertainties
When quantities with uncertainties are combined, the results
have uncertainties as well.
Following is a discussion of uncertainty in functions of one
or more variables.
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Calculations with uncertainties
When quantities with uncertainties are combined, the results
have uncertainties as well.
Following is a discussion of uncertainty in functions of one
or more variables.
For the following examples, the values of x = 2 ± 1 and
y = 32.0 ± 0.2 will be used.
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainties in Functions
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainties in Functions
Functions with uncertainties
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Two methods of calculating
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Two methods of calculating
For functions, there are two ways of calculating uncertainties:
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Two methods of calculating
For functions, there are two ways of calculating uncertainties:
1. by inspection
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Two methods of calculating
For functions, there are two ways of calculating uncertainties:
1. by inspection
2. by algebra
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Two methods of calculating
For functions, there are two ways of calculating uncertainties:
1. by inspection
2. by algebra
Both ways will be discussed.
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainty in a function - by inspection
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainty in a function - by inspection
The uncertainty in a quantity essentially means the difference
between its maximum value and its nominal value.
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainty in a function - by inspection
The uncertainty in a quantity essentially means the difference
between its maximum value and its nominal value.
x =2±1
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainty in a function - by inspection
The uncertainty in a quantity essentially means the difference
between its maximum value and its nominal value.
x =2±1
→ x can be as big as 3
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainty in a function - by inspection
The uncertainty in a quantity essentially means the difference
between its maximum value and its nominal value.
x =2±1
→ x can be as big as 3
√
Therefore, if z = x
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainty in a function - by inspection
The uncertainty in a quantity essentially means the difference
between its maximum value and its nominal value.
x =2±1
→ x can be as big as 3
√
Therefore, if z = x
Then the nominal value of z =
Terry Sturtevant
√
2 ≈ 1.4
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainty in a function - by inspection
The uncertainty in a quantity essentially means the difference
between its maximum value and its nominal value.
x =2±1
→ x can be as big as 3
√
Therefore, if z = x
√
2 ≈ 1.4
√
Then the maximum value of z = 3 ≈ 1.7
Then the nominal value of z =
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainty in a function - by inspection
The uncertainty in a quantity essentially means the difference
between its maximum value and its nominal value.
x =2±1
→ x can be as big as 3
√
Therefore, if z = x
√
2 ≈ 1.4
√
Then the maximum value of z = 3 ≈ 1.7
Then the nominal value of z =
→ ∆z ≈ 1.7 − 1.4 = 0.3
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainty in a function - by inspection
The uncertainty in a quantity essentially means the difference
between its maximum value and its nominal value.
x =2±1
→ x can be as big as 3
√
Therefore, if z = x
√
2 ≈ 1.4
√
Then the maximum value of z = 3 ≈ 1.7
Then the nominal value of z =
→ ∆z ≈ 1.7 − 1.4 = 0.3
In general,
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainty in a function - by inspection
The uncertainty in a quantity essentially means the difference
between its maximum value and its nominal value.
x =2±1
→ x can be as big as 3
√
Therefore, if z = x
√
2 ≈ 1.4
√
Then the maximum value of z = 3 ≈ 1.7
Then the nominal value of z =
→ ∆z ≈ 1.7 − 1.4 = 0.3
In general,
∆ (f (x)) ≈ fmax (x) − f (x)
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Alternatively,
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Alternatively,
The uncertainty in a quantity essentially means the difference
between its nominal value and its minimum value.
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Alternatively,
The uncertainty in a quantity essentially means the difference
between its nominal value and its minimum value.
x =2±1
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Alternatively,
The uncertainty in a quantity essentially means the difference
between its nominal value and its minimum value.
x =2±1
→ x can be as small as 1
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Alternatively,
The uncertainty in a quantity essentially means the difference
between its nominal value and its minimum value.
x =2±1
→ x can be as small as 1
√
Therefore, if z = x
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Alternatively,
The uncertainty in a quantity essentially means the difference
between its nominal value and its minimum value.
x =2±1
→ x can be as small as 1
√
Therefore, if z = x
Then the nominal value of z =
Terry Sturtevant
√
2 ≈ 1.4
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Alternatively,
The uncertainty in a quantity essentially means the difference
between its nominal value and its minimum value.
x =2±1
→ x can be as small as 1
√
Therefore, if z = x
√
2 ≈ 1.4
√
Then the minimum value of z = 1 = 1
Then the nominal value of z =
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Alternatively,
The uncertainty in a quantity essentially means the difference
between its nominal value and its minimum value.
x =2±1
→ x can be as small as 1
√
Therefore, if z = x
√
2 ≈ 1.4
√
Then the minimum value of z = 1 = 1
Then the nominal value of z =
→ ∆z ≈ 1.4 − 1.0 = 0.4
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Alternatively,
The uncertainty in a quantity essentially means the difference
between its nominal value and its minimum value.
x =2±1
→ x can be as small as 1
√
Therefore, if z = x
√
2 ≈ 1.4
√
Then the minimum value of z = 1 = 1
Then the nominal value of z =
→ ∆z ≈ 1.4 − 1.0 = 0.4
In general,
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Alternatively,
The uncertainty in a quantity essentially means the difference
between its nominal value and its minimum value.
x =2±1
→ x can be as small as 1
√
Therefore, if z = x
√
2 ≈ 1.4
√
Then the minimum value of z = 1 = 1
Then the nominal value of z =
→ ∆z ≈ 1.4 − 1.0 = 0.4
In general,
∆ (f (x)) ≈ f (x) − fmin (x)
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Alternatively,
The uncertainty in a quantity essentially means the difference
between its nominal value and its minimum value.
x =2±1
→ x can be as small as 1
√
Therefore, if z = x
√
2 ≈ 1.4
√
Then the minimum value of z = 1 = 1
Then the nominal value of z =
→ ∆z ≈ 1.4 − 1.0 = 0.4
In general,
∆ (f (x)) ≈ f (x) − fmin (x)
Notice that these two different methods may give
slightly different results, so that’s why there’s an
“approximately equal” sign.
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainty in a function - algebra
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
Uncertainty in a function - algebra
Determining uncertainties algebraically is easiest illustrated
graphically.
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
y = f (x)
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
y = f (x)
x
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
y = f (x)
f (x)
x
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
y = f (x)
f (x)
x
Terry Sturtevant
x + ∆x
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
y = f (x)
f (x + ∆x)
f (x)
x
Terry Sturtevant
x + ∆x
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
y = f (x)
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
y = f (x)
x
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
y = f (x)
f (x)
x
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
y = f (x)
f (x)
x
Terry Sturtevant
x + ∆x
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
f (x + ∆x)
y = f (x)
f (x)
x
Terry Sturtevant
x + ∆x
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
f (x) + f 0 (x) × ∆x
f (x + ∆x)
y = f (x)
f (x)
x
Terry Sturtevant
x + ∆x
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
So for the previous example.
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
So for the previous example.
x =2±1
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
So for the previous example.
x =2±1
Therefore, if z =
√
x
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
So for the previous example.
x =2±1
Therefore, if z =
∆z ≈
√
x
|f 0 (x)| ∆x
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
So for the previous example.
x =2±1
Therefore, if z =
√
x
|f 0 (x)| ∆x
∆z ≈
≈ 2√1 x ∆x
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
So for the previous example.
x =2±1
Therefore, if z =
√
x
|f 0 (x)| ∆x
∆z ≈
≈ 2√1 x ∆x
Then the value of ∆z ≈ 2√1 x 1
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
So for the previous example.
x =2±1
Therefore, if z =
√
x
|f 0 (x)| ∆x
∆z ≈
≈ 2√1 x ∆x
Then the value of ∆z ≈ 2√1 x 1
→ ∆z ≈
1
2×1.4
× 1 ≈ 0.4
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
To summarize,
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
To summarize,
By inspection,
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
To summarize,
By inspection,
∆ (f (x)) ≈ fmax (x) − f (x)
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
To summarize,
By inspection,
∆ (f (x)) ≈ fmax (x) − f (x)
or
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
To summarize,
By inspection,
∆ (f (x)) ≈ fmax (x) − f (x)
or
∆ (f (x)) ≈ f (x) − fmin (x)
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
To summarize,
By inspection,
∆ (f (x)) ≈ fmax (x) − f (x)
or
∆ (f (x)) ≈ f (x) − fmin (x)
By algebra,
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Two methods of calculating
Uncertainty in a function - by inspection
Uncertainty in a function - algebra
To summarize,
By inspection,
∆ (f (x)) ≈ fmax (x) − f (x)
or
∆ (f (x)) ≈ f (x) − fmin (x)
By algebra,
∆f (x) ≈ |f 0 (x)| ∆x
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Recap
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Recap
1. By inspection, ∆ (f (x)) ≈ fmax (x) − f (x)
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Recap
1. By inspection, ∆ (f (x)) ≈ fmax (x) − f (x)
For example,
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Recap
1. By inspection, ∆ (f (x)) ≈ fmax (x) − f (x)
For example,
√
2±1≈
√
2±
√
2+1−
√ 2
≈ 1.4 ± (1.7 − 1.4)
≈ 1.4 ± 0.3
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Recap - continued
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Recap - continued
2. By algebra, ∆z ≈ |f 0 (x)| ∆x
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Recap - continued
2. By algebra, ∆z ≈ |f 0 (x)| ∆x
For example,
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Recap - continued
2. By algebra, ∆z ≈ |f 0 (x)| ∆x
For example,
√
√
1
√ ×1
2±1≈ 2±
2× 2
≈ 1.4 ± 0.4
Terry Sturtevant
Uncertainty Calculations - Functions Wilfrid Laurier University
Calculations with Uncertainties
Uncertainties in Functions
Recap
Recap - continued
2. By algebra, ∆z ≈ |f 0 (x)| ∆x
For example,
√
√
1
√ ×1
2±1≈ 2±
2× 2
≈ 1.4 ± 0.4
since the derivative of
√
x is
Terry Sturtevant
1
√
2 x
Uncertainty Calculations - Functions Wilfrid Laurier University