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If x= 9.1, then what is the value of y?

Of the original function or the derivative?

original function

y = \sqrt{ x } + 3 at x=9.1 --> 6.016620626

so we know the point (9.1, 6.016620626) lies on the function curve

P = (9,6)
Q = (9.1, 6.016620626)
slope of PQ = ??

that's what I don't get, I do not know what to do...

you would use the slope formula
\[\LARGE m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]

it doesn't work because both x are 9 (we cannot divide by 0)

x1 = 9
x2 = 9.1

so x2 - x1 = 9.1 - 9 = 0.1

oh i missed that one, the answer is 0.16620626 ... would that be the answer for the first one?

I entered it and it says correct!! thank you! what about b) ?

repeat the same steps as a) but now use x = 9.01 instead of 9.1

I meant the second part of the question

tell me what slopes you get for part a)

for every one of them of just the first one?

each one

0.16620626 -0.1666203961-0.1671322196-0.1667129887

We just do the average?

it's probably not 100% clear but what do you notice that is common to all of these slopes?

sure

Do you know how to help me on this one?

|dw:1443842056603:dw|

keep following @triciaal steps to find the value of f(3)
if x = 3, then y = ??

y=239

so we know (3,239) lies on the curve

and the slope of 324 was correct?

yes

so you know this line has a slope of 324 and goes through (3,239)

so y-y1=m(x-x1) ?

yes

m = 324
(x1,y1) = (3,239)

How do you rearrange it so that the equation looks like y= mx+b ??

you just need to solve for y and simplify

you're welcome