# Why is this another way of writing the formula for the tangent line?

• Mar 18th 2013, 08:12 PM
Paze
Why is this another way of writing the formula for the tangent line?
We’re starting a new unit: applications of diﬀerentiation.
We’re going to do two applicationstoday. The ﬁrstislinear approximations,
which are encompassed by the single formula:
f(x) ≈ f(x0)+f'(x0)(x−x0),
but it will take at least half an hour to explain how this formula is used.

The teacher explains that the right hand side is the equation for the tangent line? Or am I misunderstanding?

If I am not, how come this is the equation for the tangent line? I thought there was only one definition for that.
• Mar 18th 2013, 08:16 PM
Gusbob
Re: Why is this another way of writing the formula for the tangent line?
Quote:

Originally Posted by Paze
We’re starting a new unit: applications of diﬀerentiation.
We’re going to do two applicationstoday. The ﬁrstislinear approximations,
which are encompassed by the single formula:
f(x) ≈ f(x0)+f'(x0)(x−x0),
but it will take at least half an hour to explain how this formula is used.

The teacher explains that this is the right hand side is the equation for the tangent line? Or am I misunderstanding?

If I am not, how come this is the equation for the tangent line? I thought there was only one definition for that.

I hope you can see that the right hand side is actually an equation of a line, and yes it is an equation for the tangent line. What is your usual definition for the tangent line?
• Mar 18th 2013, 08:19 PM
MINOANMAN
Re: Why is this another way of writing the formula for the tangent line?
Paze

Study little the tangent lines
the equation of the tangent TO A CURVE AT THE POINT (X0,Y0) is y-y0=(gradient)(x-x0)
the value of the derivative f'(x0) at the point (x0,y0) is the gradient .therefore the formula is :
y=y+f"(x0)(x-x0) it is simple.....
MINOAS
• Mar 18th 2013, 08:33 PM
Paze
Re: Why is this another way of writing the formula for the tangent line?
Quote:

Originally Posted by Gusbob
I hope you can see that the right hand side is actually an equation of a line, and yes it is an equation for the tangent line. What is your usual definition for the tangent line?

Thank you guys, but I am being very obtuse unfortunately. I have proven to myself in geogebra that you are in fact right. However, I am having problems visualizing this.

My usual definition for the tangent line of a function at any given point 'x' is the limit as delta x approaches 0 of (f(x+deltax)-f(x))/deltax if that helps you understand my stubborn mindset.

For general derivatives I simply use the power rule, chain rule and all that which I have already proven before...This one must have slipped me.
• Mar 18th 2013, 08:44 PM
Gusbob
Re: Why is this another way of writing the formula for the tangent line?
Quote:

Originally Posted by Paze
Thank you guys, but I am being very obtuse unfortunately. I have proven to myself in geogebra that you are in fact right. However, I am having problems visualizing this.

My usual definition for the tangent line of a function is the limit as delta x approaches 0 of (f(x+deltax)-f(x))/deltax if that helps you understand my stubborn mindset.

If I recall correctly, this is how high schoolers are taught: to find a tangent line at the point $\displaystyle (a,f(a))$, you take $\displaystyle L=\displaystyle{\lim_{\Delta x\to 0}} \frac{f(a+\Delta x)-f(a)}{\Delta x}$. This gives you the slope of the tangent line. Since the tangent line passes through the point $\displaystyle (a,f(a))$, the point-slope formula gives $\displaystyle y-f(a)=L(x-a)$, which can be rearranged as $\displaystyle y=f(a)+L(x-a)$. Here's the kicker: $\displaystyle f'(a)=L=\displaystyle{\lim_{\Delta x\to 0}} \frac{f(a+\Delta x)-f(a)}{\Delta x}$ by definition, so you actually do get $\displaystyle y=f(a)+f'(a)(x-a)$.
• Mar 18th 2013, 08:47 PM
Paze
Re: Why is this another way of writing the formula for the tangent line?
I am understanding this better now. I am finding the EQUATION of the tangent line, duh. Not the slope.

However I am still trying to understand how this formula does that (I need to prove it). What I usually do is just take the derivative at a point and then do the good ol' y=kx+m routine to find the equation.
• Mar 18th 2013, 08:50 PM
Paze
Re: Why is this another way of writing the formula for the tangent line?
Quote:

Originally Posted by Gusbob
If I recall correctly, this is how high schoolers are taught: to find a tangent line at the point $\displaystyle (a,f(a))$, you take $\displaystyle L=\displaystyle{\lim_{\Delta x\to 0}} \frac{f(a+\Delta x)-f(a)}{\Delta x}$. This gives you the slope of the tangent line. Since the tangent line passes through the point $\displaystyle (a,f(a))$, the point-slope formula gives $\displaystyle y-f(a)=L(x-a)$, which can be rearranged as $\displaystyle y=f(a)+L(x-a)$. Here's the kicker: $\displaystyle f'(a)=L=\displaystyle{\lim_{\Delta x\to 0}} \frac{f(a+\Delta x)-f(a)}{\Delta x}$ by definition, so you actually do get $\displaystyle y=f(a)+f'(a)(x-a)$.

Yes, thank you.

The only thing I now need is to understand this 'point-slope' formula. I usually chug through these things with deductive reasoning so I am not used to seeing formulas that do things for me :/

I found a proof for the point-slope formula. Thank you very much.
• Mar 18th 2013, 08:57 PM
Gusbob
Re: Why is this another way of writing the formula for the tangent line?
I'll demonstrate that the two are the same thing:

I'm going to set $\displaystyle L=f'(a)$ so the following is easier to read. When we take the derivative of $\displaystyle f(x)$ at $\displaystyle a$, we're finding the slope of the tangent line at $\displaystyle a$, which we have established to be $\displaystyle L=f'(a)$. Using your method, you have $\displaystyle y=Lx+b$ where $\displaystyle b$ is the $\displaystyle y$ intercept. To find $\displaystyle b$, we substitute the only point we know which satisfies this equation: $\displaystyle (a,f(a))$. This gives $\displaystyle f(a)=La+b \implies b=f(a)-La$. Therefore the tangent line is $\displaystyle y=Lx+b=Lx+f(a)-L(a)=f(a)+L(x-a)$.
• Mar 19th 2013, 06:37 AM
HallsofIvy
Re: Why is this another way of writing the formula for the tangent line?
Quote:

Originally Posted by Paze
We’re starting a new unit: applications of diﬀerentiation.
We’re going to do two applicationstoday. The ﬁrstislinear approximations,
which are encompassed by the single formula:
f(x) ≈ f(x0)+f'(x0)(x−x0),
but it will take at least half an hour to explain how this formula is used.

The teacher explains that the right hand side is the equation for the tangent line? Or am I misunderstanding?

If I am not, how come this is the equation for the tangent line? I thought there was only one definition for that.

It would have helped if, when you first posted this you told us what you thought the "only definition" for a tangent line was! What you give here is, I think, what most people first see as "the tangent line to f(x) at (x0, f(x0))".