slope of a tangent to a curve

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April 7th 2008, 02:59 PM
OzzMan

slope of a tangent to a curve

I could use some help on this one. Calculate the slope of a line tangent to the curve of each of the functions $y = f(x)$ for the given point $P$ .

$Y = x^2$ ; $P$ is $(2,4)$
April 7th 2008, 03:05 PM
Mathstud28

Quote:

Originally Posted by OzzMan

I could use some help on this one. Calculate the slope of a line tangent to the curve of each of the functions $y = f(x)$ for the given point $P$ .

$Y = x^2$ ; $P$ is $(2,4)$

First find the derivative $f'(x)=2x$ ...then you want to find the slope of the tangent line at the point $(2,4)$ and the derivative evaluated at a point x has the same slope as the tangent line at that point so the slope of the tangent is $f'(2)=2\cdot{2}=4$
April 7th 2008, 03:17 PM
OzzMan

Uh I'm new to calculus so if you could kind of walk me through it a little bit more that be great.
April 7th 2008, 03:28 PM
Mathstud28

ok

Quote:

Originally Posted by OzzMan

Uh I'm new to calculus so if you could kind of walk me through it a little bit more that be great.

If you haven't learned differntiation rules then we will go with the quotient difference...ok the slope of a curve is given by $f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$ ..so to get the devative of $f(x)=x^2$ before you know differentiation rules is to go through this thus the derivative of $x^2$ is $\lim_{h \to 0}\frac{(x+h)^2-x^2}{h}=\lim_{h \to 0}\frac{x^2+2xh+h^2-x^2}{h}=\lim_{h \to 0}\frac{2xh+h^2}{h}$ $=\lim_{h \to 0}\frac{2xh}{h}+\lim_{h \to 0}\frac{h^2}{h}=2x+\lim_{h \to 0}h=2x$ ...so the slope at the point x is $2\cdot{x}$ ...so the slope of the curve of $x^2$ at x=2 is $2\cdot{2}=4$ ...so the slope of the tangent line at the point x=2 has the same slope as the curve so its slope is 4
April 7th 2008, 03:31 PM
OzzMan

wow lol. and im only in chapter 2. i got 7 more chapters to go tonight and tomorrow.
April 7th 2008, 03:36 PM
Mathstud28

here let me make it a little easier

Quote:

Originally Posted by OzzMan

wow lol. and im only in chapter 2. i got 7 more chapters to go tonight and tomorrow.

all that work just to get $x^2$ ...but I'll tell you a little trick if $f(x)=x^{n}$ ...then the derivative of f(x) or f'(x) is $f'(x)=nx^{n-1}$ ...so to get the derivative of x²...just multiply the coefficient by the exponent and reduce the exponent by one..so it would be $f'(x)=2*x^{2-1}=2x^1=2x$
April 7th 2008, 03:39 PM
OzzMan

I'm a little confused though where you got those equations from. In my book they seem to be a little different. Damn I'm lost.
April 7th 2008, 03:49 PM
Mathstud28

?

Quote:

Originally Posted by OzzMan

I'm a little confused though where you got those equations from. In my book they seem to be a little different. Damn I'm lost.

WHich equations?
April 7th 2008, 04:03 PM
OzzMan

The 4th post in this thread
April 7th 2008, 04:04 PM
Mathstud28

If you mean

Quote:

Originally Posted by OzzMan

The 4th post in this thread

the limit stuff...hmm...thats something you just have to learn...here I think this will really help you its a great site..good luck! if you have any more questions just ask

Visual Calculus
April 7th 2008, 04:15 PM
OzzMan

Well what I mean is that it just appeared a little differently in my book. It's all good though.
April 7th 2008, 04:16 PM
Mathstud28

Ok then

Quote:

Originally Posted by OzzMan

Well what I mean is that it just appeared a little differently in my book. It's all good though.

Well good luck!