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Question Number 47532 by 143jesus last updated on 11/Nov/18

$$\sqrt{{400}^2+{100}^{2=}}$$

Answered by Joel578 last updated on 11/Nov/18

$$=\sqrt{160000+10000}=\sqrt{170000}=100\sqrt{17}\approx 412.31$$

Answered by arcana last updated on 11/Nov/18

$$\frac{f(x+h)-f\left(x\right)}{h}\approx f^{\prime }\left(x\right)$$$$f(x+h)-f\left(x\right)\approx h\cdot f{}^{\prime }\left(x\right)$$$$f(x+h)\approx f\left(x\right)+h\cdot f{}^{\prime }\left(x\right)$$$$$$$$f\left(x\right)=\sqrt{x},a={400}^{2}h={100}^2$$$$f{}^{\prime }\left(x\right)=\frac{1}{2\sqrt{x}}$$$$f(a+h)\approx f\left(a\right)+h\cdot f{}^{\prime }\left(a\right)$$$$$$$$\sqrt{{400}^2+{100}^2}=f({400}^2+{100}^2)\approx \sqrt{{400}^2}+{100}^2\left(\frac{1}{2\sqrt{{400}^2}}\right)$$$$\approx 400+\frac{100\times 100}{800}=400+12.5=412.5$$$$\sqrt{{400}^2+{100}^2}\approx 412.5$$

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