The lengths of tangent,subtangent,normal,and | vedclass

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(D) Given the curve $y=x^2+x-1$ and the point $(x_1, y_1) = (1, 1)$.First,find the derivative $\frac{dy}{dx} = 2x + 1$.At the point $(1, 1)$,the slope

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$B, A, D, C$

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(D) Given the curve $y=x^2+x-1$ and the point $(x_1, y_1) = (1, 1)$.First,find the derivative $\frac{dy}{dx} = 2x + 1$.At the point $(1, 1)$,the slope $m = \frac{dy}{dx} = 2(1) + 1 = 3$.Now,calculate the lengths:$1$. Length of tangent $A = \left|\frac{y_1 \sqrt{1+m^2}}{m}\right| = \left|\frac{1 \cdot \sqrt{1+3^2}}{3}\right| = \frac{\sqrt{10}}{3} \approx \frac{3.16}{3} \approx 1.05$.$2$. Length of subtangent $B = \left|\frac{y_1}{m}\right| = \left|\frac{1}{3}\right| = \frac{1}{3} \approx 0.33$.$3$. Length of normal $C = \left|y_1 \sqrt{1+m^2}\right| = |1 \cdot \sqrt{1+3^2}| = \sqrt{10} \approx 3.16$.$4$. Length of subnormal $D = |y_1 m| = |1 \cdot 3| = 3$.Comparing the values: $B = 0.33$,$A \approx 1.05$,$D = 3$,$C \approx 3.16$.Thus,the increasing order is $B, A, D, C$.

The lengths of tangent,subtangent,normal,and subnormal for the curve $y=x^2+x-1$ at $(1,1)$ are $A, B, C$,and $D$ respectively. Then their increasing order is:

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