Question

find the tangent line approximation to f(x)=√(4+3x) near 0, and use it to approximate √(4.3). compute the percentage error.

Answer 1

The tangent line of \(f(x)=\sqrt{4+3x}\)...Is this for calculus or pre-calc? My point is have you guys discussed the derivative?

Answer 2

it is for calculus. i know about derivatives!

Answer 3

Okay because I saw the work "approximation" and I though that this was for pre-calc approximate rates of change haha

Anyways, we need to the find the derivative of \(f(x)\) so we have:
\[f'(x)=\frac{3}{2\sqrt{4+3x}}\]
So this derivative at zero is:
\[f'(0)=\frac{3}{2\sqrt{4}}=\frac{3}{4}\]

Answer 4

okay i understand so far!

Answer 5

Now, to approximate \(\sqrt{4.3}\), In general, if you have a number \(\sqrt{x}\). The approximation can be found by looking for the square root nearest to it, a, and calculate:
\[\sqrt{x}\approx \sqrt{a}+\frac{|x-a|}{2\sqrt{a}}\]
So for \(\sqrt{4.3}\), notice that it is real close to the \(\sqrt{4}\) So then our x value is 4.3 and our a value is 4 so:
\[\sqrt{4.3}\approx\sqrt{4}+\frac{|4.3-4|}{2\sqrt{4}}\approx2+\frac{0.3}{4}\approx2+\frac{3}{40}\approx2.075\]
This is pretty darn close!
The actual value is 2.0736

Answer 6

i am a little confused. it said to use 3/4 to approximate √(4.3)

Answer 7

I'm not sure... the only relation i see is that \(\frac{3}{40}=\frac{f'(0)}{10}\)The percentage error is defined as:
\[\%_{ERROR}=\frac{|answer_{ACTUAL}-answer_{ESTIMATE}|}{answer_{ACTUAL}}\]
This is:
\[\%_{ERROR}=\frac{|\sqrt{4.3}-2.075|}{\sqrt{4.3}}\approx0.0654\%\]

Answer 8

thanks!