Solving Arc Length Problem: 45 Degrees & x-Axis

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SUMMARY

The discussion focuses on solving the arc length problem for the curve defined by the equation y² = x³, specifically from the origin to the point where the tangent line forms a 45-degree angle with the x-axis. The key steps include determining the gradient of the tangent line at the point of interest, which is 1 for a 45-degree angle, and finding the corresponding x-coordinate on the curve. The arc length formula, s = ∫_a^b √(1 + (dy/dx)²) dx, is introduced as the method to calculate the length of the curve between specified limits.

PREREQUISITES
  • Understanding of calculus, specifically differentiation and integration.
  • Familiarity with the concept of arc length in curves.
  • Knowledge of implicit differentiation to find dy/dx for the given curve.
  • Basic trigonometry to relate angles and slopes of lines.
NEXT STEPS
  • Calculate dy/dx for the curve y² = x³ using implicit differentiation.
  • Determine the x-coordinate where the tangent line has a slope of 1.
  • Apply the arc length formula to compute the length of the curve from the origin to the identified point.
  • Explore additional examples of arc length problems for practice.
USEFUL FOR

Students and educators in calculus, mathematicians interested in curve analysis, and anyone looking to deepen their understanding of arc length calculations in relation to curves.

CACain
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..Or I think this is considered that...

Here's the problem as written then I'll get to it:
Find the length of the curve y^2=x^3 from the orign to the point where the tangent makes an angle of 45 degrees with the x-axis.

Okay, by me posting this, I don't want anyone (nor am I looking for someone) to give me the answer. ...that being said, I feel completely comfortable saying I don't have the FIRST clue of how to start this :)

So if someone could just help give me a kick-start on this, I'll do my best to take it from there!

Thanks guys.
 
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OK, first of all, what is the gradient of a line which makes an angle of 45 degrees with the x-axis ? So what is [tex]\frac{dy}{dx}[/tex] at this point ? Find the x-coordinate of the point on the curve where this condition for the tangent is met.

The arc length s of a curve between x = a and x = b is given by

[tex]s = \int_a^b{\sqrt{1 + {(\frac{dy}{dx})}^2}dx}[/tex]

Can you proceed with that ?
 
Last edited:

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