Question

The function f is continuous on the interval [3, 13] with selected values of x and f(x) given in the table below. Use the data in the table to approximate f ′(12).

Answer 1

If you draw the given points you'll see that from 11 to 13 its approximatly a straight line.
So the derivate in the point where x=12 will be a line.

By using the vectors you can calculate the slope of that line.

You have 2 points: (11,12) and (13,22) so the vector is (13-11,22-12) = (2,10)
That means the slope is 10/2 = 5

So now you have y=5x+b and you need to calculate the "b".

You know the point (11,12) for example, so:

12 = 11*5 + b (=)
b = 12-11*5 (=)
b = -43


Your derivate function is now:
y=5x-43


They ask for f '(12), that means all you have to do is calculate the y.

y = 5*12-43


Voila, there you have it.

Answer 2

so 17?

Answer 3

Yes

Answer 4

thank you so much!

Answer 5

I've explained the hard way. A simple one is since you know the function from 11 to 13 is nearly a straight line, all you could do was a simple interpolation.

If 13-11 = 22-10
Then 13-12 = 22-x

22-x = (13-12)*(22-10)/(13-11)

22-x = 1*12/2
22-x = 6
x = 22-6
x = 16

which is almost 17, so the approximation wasnt that bad.

Answer 6

wait then should I say 16 instead of 17?