Let f(x) = √(x^2 + 1) - 1 (taking the positive real square root, as usual).
When x = 10^-3, compute f(x), working to 5 significant figures at every stage of the calculation.
Show (algebraically) that f(x) =(x^2)/[√(x^2 + 1) +1].
Use this expression for f to compute f(x), again working to 5 significant figures at every stage.
To compute f(x) when x = 10^-3, we will first substitute x = 10^-3 into the expression f(x) = √(x^2 + 1) - 1:
f(x) = √((10^-3)^2 + 1) - 1
Simplifying the expression inside the square root:
f(x) = √(10^-6 + 1) - 1
= √(1.000001) - 1
Now, in order to compute f(x) working to 5 significant figures at every stage, we will evaluate √(1.000001) to 5 significant figures:
√(1.000001) ≈ 1.000000
= 1
Substituting this result back into the expression for f(x):
f(x) = 1 - 1
= 0
So, when x = 10^-3, f(x) = 0.
To show algebraically that f(x) = (x^2)/[√(x^2 + 1) + 1], we start with the expression for f(x) = √(x^2 + 1) - 1:
f(x) = √(x^2 + 1) - 1
To simplify this expression, we can multiply the numerator and denominator by (√(x^2 + 1) + 1):
f(x) = (√(x^2 + 1) - 1) * (√(x^2 + 1) + 1) / (√(x^2 + 1) + 1)
Using the difference of squares formula (a^2 - b^2 = (a - b)(a + b)), we simplify the numerator:
f(x) = [(√(x^2 + 1))^2 - 1^2] / (√(x^2 + 1) + 1)
f(x) = (x^2 + 1 - 1) / (√(x^2 + 1) + 1)
Canceling out the 1s in the numerator:
f(x) = x^2 / (√(x^2 + 1) + 1)
So, we have shown algebraically that f(x) = (x^2)/[√(x^2 + 1) + 1].
Now, to compute f(x) using this expression when x = 10^-3, we substitute x = 10^-3 into the expression:
f(x) = (10^-3)^2 / [√((10^-3)^2 + 1) + 1]
Simplifying the numerator:
f(x) = 10^-6 / [√(10^-6 + 1) + 1]
= 10^-6 / (√1.000001 + 1)
To compute this expression to 5 significant figures at every stage, we will evaluate √1.000001:
√1.000001 ≈ 1.000000
= 1
Substituting this result back into the expression for f(x):
f(x) = 10^-6 / (1 + 1)
= 10^-6 / 2
To get the answer to 5 significant figures, we can convert 10^-6 to scientific notation:
f(x) ≈ 1.00 x 10^-6 / 2
≈ 0.50 x 10^-6
≈ 5.0 x 10^-7
So, when x = 10^-3, f(x) ≈ 5.0 x 10^-7.