# Tangent of a line

1. Aug 19, 2012

### menco

1. The problem statement, all variables and given/known data
The graph of y = f(x) passes through the point (9/2, 100/3). Also the tangent line to the graph at any point (x,y) has the slope 4*sqrt(2x+7). Find f(x)

2. Relevant equations

3. The attempt at a solution
I am very lost with this as I can't find much info in my text book. Any help where to start? I am assuming I am trying to find the equation of the line y=mx+c given the slope of the tangent and current points.

2. Aug 19, 2012

### gabbagabbahey

Re: Tagent of a line

Given a function $y=f(x)$, how does one find the slope of the tangent line? (If you aren't sure, you had better open up your textbook and find out the definition of tangent line)

No, you are asked to find the original function $f(x)$ (which will not be a straight line), not the tangent line at some point.

3. Aug 19, 2012

### menco

Re: Tagent of a line

I know how to find the slope and equation of a tangent line fairly easy but trying to reverse it is confusing me

4. Aug 19, 2012

### gabbagabbahey

Re: Tagent of a line

Again, describe how to find the slope of the tangent line to a function $y=f(x)$. (Don't say that you know how, demonstrate that you know)

5. Aug 19, 2012

### menco

Re: Tagent of a line

To find the slope of a tangent line take the derivative of the function and substitute in the point of contact if known.

6. Aug 19, 2012

### gabbagabbahey

Re: Tagent of a line

Right, and so at a general point $x$, the slope is just $f'(x)$. So, what can you say about $f(x)$ if the slope of the tangent line at a point $x$ is $4\sqrt{2x + 7}$?

Last edited: Aug 19, 2012
7. Aug 19, 2012

### menco

Re: Tagent of a line

the rate of change of the function is 4*sqrt(2x+7)?

8. Aug 19, 2012

### gabbagabbahey

Re: Tagent of a line

Yes, $f'(x)=4\sqrt{2x+7}$.

So, $f(x)=?$...

9. Aug 19, 2012

### HallsofIvy

Staff Emeritus
Re: Tagent of a line

The opposite of the derivative is the anti-derivative, also called the "indefinite integral". Have you studied those?

Last edited by a moderator: Aug 25, 2012
10. Aug 24, 2012

### menco

Re: Tagent of a line

Yes we have just started integrals, by using substitution

I found f(x) = (4(2x+7)^3/2) / 3

Does the point (9/2, 100/3) have anything to do with the problem?

11. Aug 24, 2012

### vela

Staff Emeritus
Re: Tagent of a line

Yes, the graph of y=f(x) passes through that point, so you should have that 100/3 = f(9/2). Is that the case for your f(x)? If not, how do you fix it?

12. Aug 24, 2012

### menco

f(9/2) = 256/3

So it is not the case, I'm a little unsure of what you mean by fix it?

13. Aug 24, 2012

### vela

Staff Emeritus
$\frac{4}{3}(2x+7)^{3/2}$ is not the only function whose derivative is $4\sqrt{2x+7}$. You need to find another one, one where f(9/2)=100/3.

14. Aug 25, 2012

### Accretion

menco, take this equation for example: $\int x^2 = \frac{x^3}{3} + C$, do you remember why we put +C there?

15. Aug 25, 2012

### menco

Ah yes i see I forgot all about +C, which is the constant of integration.

So if I use 100/3 = 256/3 + C, C = (-52)

Therefore the final function will be ((4(2x+7)^3/2) / (3)) - 52

then when f(9/2) = 100/3