Which curve bends more sharply?
Two curves, y = f(x) and y = g(x), run through the point P(x0,y0).
You know that f'(x0) = 0, f''(x0) = 3, g'(x0) = 3/4, and g''(x0) = 5.
- Which curve bends more sharply in P?
- Calculate the rate of change of their tangent angles θf(x0) and θg(x0) at P. Does this confirm your answer to question 1? If not, explain.
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1 Expert Answer
Reginald J. answered • 08/21/23
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10+ Year Experienced Calculus Whiz (1st session free)
We can apply the curvature formula to both and compare: |f''(x)|/(1+(f'(x)^2))^3/2.
For f: 3/(1+0^2)^3/2=3
For g: 5/(1(3/4)^2)^3/2=5/((25/16)^3/2)=5/(125/64)=320/125=2.56
So f bends more sharply at P because 3>2.56
As for the tangential angles, compute tan^-1(f'(xo)) for both: For f, 71.56 deg For g, 36.87 deg
f is also larger here, so it confirms question 1.
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Reginald J.
We can apply the curvature formula to both and compare: |f''(x)|/(1+(f'(x)^2))^3/2. For f: 3/(1+0^2)^3/2=3 For g: 5/(1+(3/4)^2)^3/2=5/((25/16)^3/2)=5/(125/64)=320/125=2.56 So f bends more sharply at P because 3>2.56 As for the tangential angle, compute tan^-1(f'(xo)) for both: For f, 71.56 deg For g, 36.87 deg f is also larger here, so it confirms question 1.08/21/23