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Area enclosed between two graphs

  • #1
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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.
 

Answers and Replies

  • #2
andrewkirk
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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}##?
 
  • #3
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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.

upload_2016-6-15_9-27-57.png
 
  • #4
andrewkirk
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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.
 
  • #5
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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.
 
  • #6
Ray Vickson
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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|.
 

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