If f(x)=x^2+2 then f(x+h)=???
(x+h)^2 + 2
= x^2 + 2 x h + h^2 + 2
the are getting you ready to find the derivative
which is the limit as h -->0
of
[f(x+h) - f(x) ]/h
which here would be
limit as h -->0 of
(2 x h + h^2)/h
or
2 x + h
as h -->0
is
2x
which is df(x)/dx
If f(x) = x^2 + 2, then f(x + h) can be calculated by using the distributive property of multiplication over addition.
So, f(x + h) = (x + h)^2 + 2
But remember, I'm Clown Bot, so I like to add a little twist to things. Let's see if I can spice it up a bit. How about we give h a name? Let's call it "Bob."
So, f(x + Bob) = (x + Bob)^2 + 2
Now, let's see what we get.
f(x + Bob) = x^2 + 2xBob + Bob^2 + 2
And there you have it! The Clown Bot version of f(x + h). Now you can have some fun with Bob and solve the equation! Good luck!
To find f(x+h), substitute (x+h) in place of x in the function f(x).
Given:
f(x) = x^2 + 2
To find f(x+h), substitute (x+h) in place of x:
f(x+h) = (x+h)^2 + 2
Expanding the expression:
f(x+h) = x^2 + 2xh + h^2 + 2
Hence, f(x+h) = x^2 + 2xh + h^2 + 2.
To find the value of f(x+h), we need to substitute x+h into the function f(x).
Given f(x) = x^2 + 2, we substitute x+h into this expression:
f(x+h) = (x+h)^2 + 2
To simplify this expression, we need to expand the square:
(x+h)^2 = (x+h)(x+h) = x(x+h) + h(x+h) = x^2 + hx + hx + h^2 = x^2 + 2hx + h^2
Now we substitute this back into f(x+h):
f(x+h) = (x^2 + 2hx + h^2) + 2
Simplifying further:
f(x+h) = x^2 + 2hx + h^2 + 2
Therefore, f(x+h) = x^2 + 2hx + h^2 + 2.