Solve Tangent of Line Homework | y=f(x) Slope 4√2x+7

[menco]

Homework Statement

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)

Homework Equations

The Attempt at a Solution

I am very lost with this as I can't find much info in my textbook. 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.

 

[gabbagabbahey]

Homework Statement

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)

Homework Equations

The Attempt at a Solution

I am very lost with this as I can't find much info in my textbook. Any help where to start?

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)

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.

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.

 

[menco]

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

 

[gabbagabbahey]

I know how to find the slope and equation of a tangent line fairly easy

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)

 

[menco]

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

 

[gabbagabbahey]

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

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 $4\sqrt{2x + 7}$?

 

[menco]

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

 

[gabbagabbahey]

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

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

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

 

[HallsofIvy]

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

 

[menco]

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?

 

[vela]

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?

 

[menco]

f(9/2) = 256/3

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

 

[vela]

$\frac{4}{3}(2x+7)^{3/2}$ 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.

 

[Accretion]

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

 

[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