Use linear approximation, i.e. the tangent line, to approximate 1/0.104 as follows: Let f(x)=1/x and find the equation of the tangent line to f(x) at a "nice" point near 0.104 . Then use this to approximate 1/0.104 .
L(x)=f(a)+f'(a)(x-a)...i think.
You're notation is slightly different than I use, but yes, that is how to make a linear approximaton.
You're using
L(x)=f(a)+f'(a)(x-a)
whereas I use
f(x+dx)=f(x)+f'(x)dx which is the differential notation I was taught.
If f(x)=1/x = x-1 , then caluclate f'(x) = -x-2.
The "nice" point near 0.104 'might' be .1 since 1/.1 = 10. Now calculate f'(.1) = -.1-2 = -100
Then dx =.004
Then using
L(x)=f(a)+f'(a)(x-a)
L(.104)=f(.1) + f'(.1)*.004
L(.104) =approx. 10 + -100*.004
The correct answer is 9.61538... so you can see the approximation is good to one decimal place.
To use linear approximation to approximate 1/0.104, you need to find the equation of the tangent line to the function f(x) = 1/x at a "nice" point near 0.104.
First, find the derivative of f(x) = 1/x. The derivative of f(x) with respect to x is f'(x) = -1/x^2.
To find a "nice" point, choose a value close to 0.104, such as 0.1. Calculate f(0.1) = 1/0.1 = 10.
Now, calculate f'(0.1) = -1/(0.1)^2 = -100.
To find the equation of the tangent line at x = 0.1, use the point-slope form of a line: y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line.
Plugging in the values, the equation of the tangent line is y - 10 = -100(x - 0.1).
To approximate 1/0.104, plug in x = 0.104 into the equation of the tangent line:
L(0.104) = 10 + (-100)(0.104 - 0.1)
Simplifying, L(0.104) ≈ 10 - 100(0.004)
L(0.104) ≈ 10 - 0.4
L(0.104) ≈ 9.6
Therefore, using linear approximation, you can approximate 1/0.104 as approximately 9.6.
Where did the .004 come from ?
Well, that was quite a mathematical journey! Approximating 1/0.104 using the tangent line method was quite a ride, but it seems like we made it to our destination, albeit with a slight detour.
And let's not forget that "nice" point near 0.104, which turned out to be a little sneaky. Who knew it would be .1? But hey, sometimes math throws us curveballs, or should I say, tangents?
In the end, our approximation of 9.61538 seemed pretty decent, considering we started with 1/0.104. It just goes to show that even in math, a little approximation can go a long way.