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Chapter 10: Problem 872

If \(\mathrm{m}=\tan \theta\) is the slope of the tangent to the curve \(\mathrm{e}\) \(\mathrm{y}=1+\mathrm{x}^{2}\) than (a) \(|\tan \theta| \geq 1\) (b) \(|\tan \theta|<1\) (c) \(\tan \theta<1\) (d) \(|\tan \theta| \leq 1\)

Short Answer

Expert verified
The correct option is (a) \(|\tan \theta| \geq 1\).

Step by step solution

01

Find the first derivative of the given equation

Given equation is: \[y = 1 + x^2\] To find the first derivative, differentiate with respect to x: \[\frac{dy}{dx} = \frac{d}{dx}(1+x^2)\] Using the power rule, we get: \[\frac{dy}{dx} = 0 + 2x\] So, the first derivative is: \[\frac{dy}{dx} = 2x\]
02

Compare the first derivative with the given slope

We have the slope `m` is equal to the tangent of the angle `θ`: \[m = \tan \theta\] Also, we found the first derivative as: \[\frac{dy}{dx} = 2x\] As the first derivative represents the slope of the tangent to the curve, therefore: \[ m = 2x \Rightarrow \tan\theta = 2x \] Now, we need to find the range of `|\tan θ|`.
03

Find the range of |\tan θ|

Recalling the inequality: \[ \tan\theta = 2x \] Therefore, \[ |\tan\theta| = |2x| \] As the function \(x^2\) is always non-negative, we can say that \[ 1 + x^2 \geq 1 \] So the minimum value of \(|2x|\) when \(x^2 \geq 1\), which is true when \(x \geq 1\) or \(x \leq -1\). Therefore, \[ |\tan\theta| \geq |\tan\theta| \] Comparing with the given options, we can conclude that the correct option is: (a) \(|\tan \theta| \geq 1\)

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Most popular questions from this chapter

If the curves \(2 \mathrm{x}^{2}+3 \mathrm{y}^{2}=6\) and \(\mathrm{ax}^{2}+4 \mathrm{y}^{2}=4\) intersect orthogonally than a = (a) 2 (b) 1 (c) 3 (d) \(-3\)

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If \(\mathrm{y}=\tan ^{-1}\left[1 /\left(\mathrm{x}^{2}+\mathrm{x}+1\right)\right] \tan ^{-1}\left[1 /\left(\mathrm{x}^{2}+3 \mathrm{x}+3\right)\right]\) \(+\tan ^{-1}\left[1 /\left(\mathrm{x}^{2}+5 \mathrm{x}+7\right)\right] \ldots \ldots\) to \(\mathrm{n}\) terms then \((\mathrm{dy} / \mathrm{d} \mathrm{x})\) \(=\) (a) \(\left[1 /\left\\{1+(\mathrm{x}+\mathrm{n})^{2}\right\\}\right]+\left[1 /\left(1+\mathrm{x}^{2}\right)\right]\) (b) \(\left[1 /\left\\{1+(\mathrm{x}+\mathrm{n})^{2}\right\\}\right]-\left[1 /\left(1+\mathrm{x}^{2}\right)\right]\) (c) \(\left[2 /\left\\{1+(\mathrm{x}+\mathrm{n})^{2}\right\\}\right]+\left[1 /\left(1+\mathrm{x}^{2}\right)\right]\) (d) \(\sum \mathrm{n}\)
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