Chapter 3: Applications of Differentiation
Section 3.5: Curvature of a Plane Curve
Use the appropriate formula from Table 3.5.1 to determine the curvature of yx=x3/2,x≥0, then obtain the curvature from first principles, that is, by calculating the rate at which the tangent turns as arc length increases.
The curvature is given by
=34x 4+9 x3/243/2
=6x 4+9 x3/2
To obtain this result from first principles, begin by obtaining the arc-length function
sx=∫0x1+y′t2ⅆt = ∫0x1+94t dt = 4+9 x3/2−8/27
and its inverse, x=xs=8+27 s2/3−4/9. The angle made by the tangent line and the x-axis is θ=arctany′x. The curvature is the rate at which this angle varies as s changes. Hence, the derivative of θ must be taken with respect to s. Either the chain rule or the substitution x=xs must be used. Making the substitution leads to θ=arctan328+27 s2/3−4/9, so the derivative with respect to s becomes
Replacing s with sx then gives
κ=θ′sx=a|f(x)s=sx = 619⁢4+9⁢x3/22/3−49⁢4+9⁢x3/2 = 6x 4+9 x3/2
Calculate the curvature as per Table 3.5.1.
Define the function yx
Context Panel: Assign Function
yx=x3/2→assign as functiony
Compute the curvature for x>0
Write the expression for the curvature.
Context Panel: Evaluate and Display Inline
Context Panel: Simplify≻Assuming Positive
y″x1+y′x23/2 = 34⁢x⁢1+94⁢x3/2→assuming positive6x⁢4+9⁢x3/2
Calculate the curvature from first principles.
Obtain sx, the arc-length function on the interval 0,x
Write the appropriate integral.
Context Panel: Assign to a Name≻S
∫0x1+y′t2ⅆt = −827+127⁢4+9⁢x3/2→assign to a nameS
Obtain xs, the inverse of the arc-length function
Apply the solve command and select the first (of three) solutions, the last two of which are complex. Assign this solution to the name X.
Obtain θs=arctany′s, where the derivative is taken with respect to x
Assign to the name θ, the arctangent of y′x, which is then evaluated at x=xs=X.
Compute θ′s and replace s with sx
Calculus palette: Differentiation template
Differentiate θ with respect to s and press the Enter key.
(This is the rate of change of θ taken with respect to the arc length.)
Context Panel: Evaluate at a Point
Replace s with sx=S
ⅆⅆ s θ
→evaluate at point
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