Question

If f(-1)=5 and f(-0.9)=5.2 estimate f '(-1)?

Answer 1

@SithsAndGiggles

Answer 2

You can estimate the instantaneous rate of change by using the average rate of change:
\[f'(c)\approx\frac{f(b)-f(a)}{b-a}\]
if \(c=b\) or \(c=a\).

In this case you can let \(a=-1\) and \(b=-0.9\).

Answer 3

Of course, this is assuming the function is continuous over the interval \([-1,-0.9]\), but I think we can suppose that to be the case.

Answer 4

i got 2??

Answer 5

Yep

Answer 6

how about if the question is like this:
If f′(x)=3x and f(2)=−1, find the approximation to f(2.1)?

is it same process? i will just sub. f′(x)=3x and solve for f(2.1)?

Answer 7

No, this is a slightly different problem. Here you're told what the slope of the tangent line is at any point \(x\).
As a general example,

Basically, you want to find the tangent line to the given \(x\), then approximate \(f(x+\delta)\) by using the tangent line to \(x\). This is called the linear approximation, and is given by (according the the sketch)
\[f(x+\delta)\approx f'(x+\delta)((x+\delta)-x)+f(x)\]