SUPER simple calculus question (derivative): I'll pick a Best Answer TODAY! THANK YOU!?

Question

Hi! The question is the find the slope of the tangent line of f at P.

f(x) = x³ + x P = (2, 10)

I know that the formula is:

mtan = lim [f(x) -f(c)]/(x-c)
............x->c

But what I get (and I'm sure I'm doing something wrong) is:

lim (x³ + x - 10)/(x-2)
x->2

Did I do something wrong? Please explain how to solve this! THANK YOU SO MUCH! You have no idea how important this is!

Answer 1

So far so good. Now divide x^3+x-10 by x - 2 to get
x^2+2x+5 so the limit as x->2 is 2^2+2(2)+5 = 13.
This is the slope.

Answer 2

To find the slope, just like you were doing, find the derivative of the function and plug in x:
f(x) = x^3 + x
f'(x) = 3x^2 + 1
f'(2) = 3(2)^2 +1
= 3(4) +1
= 12 + 1 = 13 // Doing the simple version of the derivative

By the definition of the derivative:

f'(x) = lim h-> 0 [f(x+h) - f(x)]/h
= lim h->0 [((x + h)^3 + (x +h) ) - (x^3 +x)] / h
= lim h->0 [(x^3 + 3x^2h + 3xh^2 + h^3 + x + h - x^3 - x)]/ h
= lim h->0 [ 3x^2h + 3xh^2 + h^3 + h]/ h // Factor an h from each term
= lim h->0 h(3x^2 + 3xh + h^2 + 1) / h // h's cancel
= lim h->0 (3x^2 + 3xh +h^2 + 1) // Plug 0 for every h
f'(x) = (3x^2 + 1)
f'( 2) = 3(2)^2 + 1
= 12 + 1
m= 13

Answer 3

Yes you did wrong.
The better way to do this is to find f'(x)
which is 3x^2 +1 and EVALUATE it at x=2, which is 13. [the 10 value is if you wanted to write the tangent line equation, but that is not necessary]

Answer 4

the slope of the tangent line of a function at a point is equal to its derivative at that point so for ur function :

f'(x) = 3x^2 +1 evaluating this at p=(2,10)
f'(2) = 3*(2)^2 +1 = 13

I hope this is helpfull

Answer 5

in case you simplify the equation you get R(x) = a million/(x+a million). in case you graph this you've a discontinuity at x = -a million. in case you attitude -a million from the left, R(x) --> - infinity. in case you attitude -a million from the right R(x) --> +infinity. So R(x) is easily discontinuous at x= -a million considering the fact that drawing close from the left provides a diverse reduce than at the same time as drawing close from the right. If we use the simplified equation, the equation is non-stop at x=a million, because an same reduce (.5) is gained in case you attitude from the left or the right. in case you take advantage of L'medical institution's rule you locate that (x-a million)/(x^2-a million) lim x --> a million = a million/2x lim x --> a million = a million/2 = .5