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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Continuity and Differentiability

NDA 2017 · Mathematics · Continuity and Differentiability

If F(x)= 9-x^2 , then what is _ x arrow 1 (F(x)-F(1))/(x-1) equal...

If⁡F(x)=9−x2\operatorname{If} F(x)=\sqrt{9-x^2}, then what is lim⁡x→1F(x)−F(1)x−1\lim _{x \rightarrow 1} \frac{F(x)-F(1)}{x-1} equal to?

Options

  1. A−142-\frac{1}{4 \sqrt{2}}
  2. B18\frac{1}{8}
  3. C−122-\frac{1}{2 \sqrt{2}}
  4. D122\frac{1}{2 \sqrt{2}}

Correct answer

C. −122-\frac{1}{2 \sqrt{2}}

Step-by-step solution

lim⁡x→1F(x)−F(1)x−1\lim _{x \rightarrow 1} \frac{F(x)-F(1)}{x-1} We know, lim⁡x→aF(n)−F(a)x−a=F′(a)\lim _{x \rightarrow a} \frac{F(n)-F(a)}{x-a}=F^{\prime}(a)∴F′(x)=ddx(9−x2)=1(0−2x)29−x2=−x9−x2F′(1)=−19−1=−18=−122\begin{aligned} & \therefore F^{\prime}(x)=\frac{d}{d x}\left(\sqrt{9-x^2}\right) \\ & =\frac{1(0-2 x)}{2 \sqrt{9-x^2}}=\frac{-x}{\sqrt{9-x^2}} \\ & F^{\prime}(1)=\frac{-1}{\sqrt{9-1}}=\frac{-1}{\sqrt{8}}=\frac{-1}{2 \sqrt{2}} \end{aligned}$$
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