Linear Approximation
- Thread starter Woodstalk
- Start date
- Joined
- Nov 24, 2012
- Messages
- 3,021
Hello, and welcome to FMH! 
We are being asked to estimate the cube of a quantity, so let's let:
[MATH]f(x)=x^3[/MATH]
Now, we can use the approximation:
[MATH]\frac{\Delta f}{\Delta x}\approx\frac{df}{dx}[/MATH]
Hence:
[MATH]f(x+\Delta x)\approx\frac{df}{dx}\Delta x+f(x)[/MATH]
Now, using the function we defined, with \(x=1\) and \(\Delta x=0.02\), we may state:
[MATH](1+0.02)^3\approx3(1)^2(0.02)+1^3[/MATH]
Or
[MATH](1.02)^3\approx1.06[/MATH]
Does that make sense?
We are being asked to estimate the cube of a quantity, so let's let:
[MATH]f(x)=x^3[/MATH]
Now, we can use the approximation:
[MATH]\frac{\Delta f}{\Delta x}\approx\frac{df}{dx}[/MATH]
Hence:
[MATH]f(x+\Delta x)\approx\frac{df}{dx}\Delta x+f(x)[/MATH]
Now, using the function we defined, with \(x=1\) and \(\Delta x=0.02\), we may state:
[MATH](1+0.02)^3\approx3(1)^2(0.02)+1^3[/MATH]
Or
[MATH](1.02)^3\approx1.06[/MATH]
Does that make sense?