Question

how to solve for the derivative of Position(t) = 0.2274t^2 + 8.5977t - 1.9418 using
f(x+h)-f(x) / h --> NO POWER RULE

Answer 1

if we call the coefficients a,b and c (for easy typing)

then you do
\[ \frac{1}{h}( a (t+h)^2 + b(t+h) + c - (a t^2 + bt + c) ) \]
you should get
\[ \frac{1}{h}( at^2 +2aht +ah^2 + bt +bh+c -at^2 -bt-c) \]
some of the terms cancel, so you get
\[ \frac{1}{h}(2aht+ah^2 +bh) = 2at+ah+b\]
we now take the limit as h->0
\[ \lim_{h \rightarrow 0} 2at+ah+b = 2at+b\]

Answer 2

i'm really confused...

Answer 3

Well, as @phi is saying..you take your function (Lets call it f(x) just to stay consistent with the variables)

\[\large f(x)=0.2274x^2+8.5977x−1.9418\]


Your definition of a derivative is \(\large \frac{f′(x)=f(x+h)−f(x)}{h}\)

So you take your function and apply it

\(\large f(x + h)\) means you take your function and everywhere you see an 'x' you replace it with (x+h)

*Also, for simplicity of typing...making the coefficients just 'a' 'b' and 'c' makes this a little easier to get to the end result

Answer 4

So if I make \[\large a=0.2274 \] \[\large b=8.5977\] and \[\large c=1.9418\]

We can now make this whole process look "neater" in terms of writing

Answer 5

So now: we have our function \[\large f(x)=ax^2 + bx - c\]

Now lets work on the derivative...first...what is \(\large f(x+h)\)?? Remember...anywhere you see an 'x' in your equation...replace it with (x+h)

\[\large f(x+h) = a(x+h)^2 + b(x+h) - c\]

Now we have to subtract \(\large f(x)\) *Which is just our original function right?

So our derivative equation will look like
\[\large f'(x) = \frac{f(x+h) - f(x)}{h} = \frac{(a(x+h)^2 + b(x+h) - c) - (ax^2 + bx - c)}{h}\]

Answer 6

Crap...got cut off...that last equation is:

\[\large f'(x) = \frac{f(x+h) - f(x)}{h} = \] \[\large \frac{(a(x+h)^2 + b(x+h) - c) - (ax^2 + bx - c)}{h}\]

Answer 7

Now it's just a matter of expanding and then simplifying

Answer 8

how would this fit in with the problem? like what would be a,b, and c?

Answer 9

the a,b,and c are the numbers in your problem. It was just easier to type.

as for being confused, maybe we have to start with more basic ideas?
for example

if you have
f(t) = t^2
do you know what f(2) is ?

Answer 10

oh wait. i understand you're first statement and figured it out! thank you so much.