Question and Answers Forum

All Questions      Topic List

Relation and Functions Questions

Previous in All Question      Next in All Question      

Previous in Relation and Functions      Next in Relation and Functions      

Question Number 116365 by soumyasaha last updated on 03/Oct/20

Answered by Olaf last updated on 03/Oct/20

$$f\left(x\right)=\underset{k=0}{\sum ^3}\frac{f^{\left(k\right)}\left(a\right)}{k!}{(x-a)}^k+R_3\left(x\right)$$$$f^{\left(0\right)}\left(a\right)=f\left(\frac{\pi }{2}\right)=\cos \left(\frac{\pi }{2}\right)=0$$$$f^{\left(1\right)}\left(a\right)=f^{{}^{\prime }}\left(\frac{\pi }{2}\right)=-\sin \left(\frac{\pi }{2}\right)=-1$$$$\text{Prime causes double exponent: use braces to clarify}$$$$\text{Prime causes double exponent: use braces to clarify}$$$$f\left(x\right)=-(x-\frac{\pi }{2})+\frac{{(x-\frac{\pi }{2})}^3}{6}+R_3\left(x\right)$$$$\mathrm{\exists }\xi \in [0;2\pi ]╲\frac{\pi }{2},R_3\left(x\right)=\frac{f^{\left(4\right)}\left(\xi \right)}{4!}{(x-\frac{\pi }{2})}^4$$$$\mid R_3\left(x\right)\mid ⩽\frac{\mid f^{\left(4\right)}\left(\xi \right)\mid }{4!}{(x-\frac{\pi }{2})}^4⩽\frac{1}{24}{(x-90°)}^4$$$$\mathrm{For}x=80°:$$$$\mid R_3\left(80°\right)\mid ⩽\frac{1}{24}{\left(10°\right)}^4=\frac{1}{24}{\left(\frac{\pi }{18}\right)}^4$$$$\cos 80°\approx \frac{\pi }{18}-\frac{1}{6}{\left(\frac{\pi }{18}\right)}^3+\frac{1}{24}{\left(\frac{\pi }{18}\right)}^4$$$$\cos 80°\approx 0,17369$$

Commented by soumyasaha last updated on 04/Oct/20

$$\mathrm{Thanks}\mathrm{Sir}.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com