Area enclosed between two graphs

[Saracen Rue]

Homework Statement

f(x) = √(x+2), g(x) = d/dx (f(x))^(f(x)). Find the total area enclosed between g(x) and √(x^2) correct to 3 decimal places.

Homework Equations

Knowledge of differentiation and integration - specifically areas between curves.

The Attempt at a Solution

I've attempted to solve g(x) = √(x^2) on my calculator but I keep receiving an error message. I've defined f(x) as equaling √(x+2), and I don't see where I could be going wrong. I know after I've found the points of intersection, all I need to do is set up definite integrals over the relevant domains with the graph on bottom over said domains being subtracted from the graph on top.

 

[andrewkirk]

I've attempted to solve g(x) = √(x^2) on my calculator but I keep receiving an error message.

It would be better to, instead of using a calculator, derive an explicit formula for g(x) by differentiating $f(x)^{f(x)}$. It's a bit messy, but quite manageable. Then you can plot that function and it'll be fairly easy to see approximately where it must intersect $\sqrt{x^2}$.

By the way, what is a simplified way to write $\sqrt{x^2}$?

 

[Saracen Rue]

It would be better to, instead of using a calculator, derive an explicit formula for g(x) by differentiating $f(x)^{f(x)}$. It's a bit messy, but quite manageable. Then you can plot that function and it'll be fairly easy to see approximately where it must intersect $\sqrt{x^2}$.

By the way, what is a simplified way to write $\sqrt{x^2}$?

Ah thank you for the advice. The square root of x squared simplifies to x, but the rule $\sqrt{x^2}$ sketches a distinctive graph due to the fact you must square x before square rooting it - this means y cannot be a negative number. Here's a picture to show you.

 

[andrewkirk]

Good. Now try to differentiate $\sqrt{(x+2)}^{\sqrt{x+2}}$ to get a formula for the function $g$.
It may help to first re-write it as $e^{(x+2)^\frac12\cdot \frac12 \log(x+2)}$
You should get a function that is near zero at x=0 and asymptotically approaches zero from above as it heads left, and which then increases in a concave-up curve to the right. By roughly drawing this against the graph above you should be able to see how many times the two curves cross, and approximately where.

 

[Mark44]

Ah thank you for the advice. The square root of x squared simplifies to x, but the rule $\sqrt{x^2}$ sketches a distinctive graph due to the fact you must square x before square rooting it - this means y cannot be a negative number.

No, that's not why. $\sqrt{x^2} = |x|$, the absolute value of x. If what you said was true -- that $\sqrt{x^2}$ simplifies to x, the graph would be a straight line through the origin, with a slope of 1.

 

[Ray Vickson]

Ah thank you for the advice. The square root of x squared simplifies to x, but the rule $\sqrt{x^2}$ sketches a distinctive graph due to the fact you must square x before square rooting it - this means y cannot be a negative number. Here's a picture to show you.

View attachment 102060

You have just drawn a graph of the absolute-value function |x|.