How do you use the tangent line approximation to approximate the value of #ln(1003)# ?
1 Answer
The answer is
Another term for tangent line approximation is linear approximation. The linear approximation function is:
#L(x)~~f(a)+f'(a)(x-a)#
So we need to find the derivative:
#f(x)=ln(x)#
#f'(x)=1/x#
Now, we need to pick an
#f(a)=f(10^3)=ln(10^3)=3ln(10)#
#f'(a)=f'(1000)=1/(1000)#
So our linear approximation is:
#L(x)~~3ln(10)+1/(1000)(x-1000)#
#L(1003)~~3ln(10)+1/(1000)(1003-1000)#
#~~3ln(10)+3/(1000)#
#~~3ln(10)+.003#
We should leave this as the answer since it's supposed to be mental math. But let's look at how accurate this is: