Question

Given that f(s)=4s^2+3s+8
Find the ratio (f(t+h)-f(t))/h

Answer 1

Should that original equation be

f(t) = 4s² + 3s + 8

?

*just making sure it is f(t)*

Answer 2

no its f(s)

Answer 3

Well I'm going to go on assuming that it is...

you have f(t) = 4t² + 3t + 8

And you are finding the derivative using the definition ...

\[\frac{ f(t + h) - f(t) }{ h }\]

What you do...is take your equation....and start off like this

\[\frac{ 4(t + h)^2 + 3(t + h) + 8 - f(t) }{ h }\]

Notice that I replaced all the 'x' in the original equation with the new (x + h)

Now you finish it off with that - f(t) all you do now is write in the original equation again since that IS f(t)..so

\[\frac{ 4(t + h)^2 + 3(t + h) + 8 - (4t² + 3t + 8) }{ h }\]

Now just simplify through...

Answer 4

So we'll have

\[\frac{ 4t^2 + 8th + 4h^2 + 3t + 3h + 8 - 4t^2 - 3t - 8 }{ h }\]

And now we just combine like terms to simplify

\[\frac{4h^2 + 8th + 3h }{ h }\]

Now we can divide through by the 'h' to get

\[4h + 8t + 3\]

We know that 'h' is approaching 0 so

f'(t) = 8t + 3

Answer 5

Thank u
:)