For-this-problem-we-define-the-fractional-part-of-x-R-0-as-x-x-x-where-x-is-the-integer-part-of-x-i-e-the-greatest-integer-less-than-or-equal-to-x-a-Draw-the-function-x-in-a-cord

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Question Number 180926 by Mastermind last updated on 19/Nov/22

$$\mathrm{For}\mathrm{this}\mathrm{problem},\mathrm{we}\mathrm{define}\mathrm{the}$$$$\mathrm{fractional}\mathrm{part}\mathrm{of}\mathrm{x}\in {\mathbb{R}}_{⩾0}\mathrm{as}$$$$\left\{\mathrm{x}\right\}=\mathrm{x}-\lfloor \mathrm{x}\rfloor$$$$\mathrm{where}\lfloor \mathrm{x}\rfloor \mathrm{is}\mathrm{the}\mathrm{integer}\mathrm{part}\mathrm{of}\mathrm{x},\mathrm{i}.\mathrm{e}$$$$\mathrm{the}\mathrm{greatest}\mathrm{integer}\mathrm{less}\mathrm{than}\mathrm{or}\mathrm{equal}$$$$\mathrm{to}\mathrm{x}.$$$$\left(\mathrm{a}\right)\mathrm{Draw}\mathrm{the}\mathrm{function}\left\{\mathrm{x}\right\}\mathrm{in}\mathrm{a}\mathrm{cordinate}$$$$\mathrm{system}\mathrm{for}0⩽\mathrm{x}⩽3$$$$\left(\mathrm{b}\right)\mathrm{Find}\mathrm{the}\mathrm{area}{\mathrm{A}}_{\mathrm{n}},\mathrm{under}\mathrm{the}\mathrm{graph}$$$$\mathrm{of}\left\{\mathrm{x}\right\}\mathrm{between}0\mathrm{and}\mathrm{n}\in \mathbb{N}\mathrm{as}\mathrm{given}\mathrm{by}$$$${\mathrm{A}}_{\mathrm{n}}={\int }_0^{\mathrm{n}}\left\{\mathrm{x}\right\}\mathrm{dx}.$$$$$$$$$$$$\mathrm{M}.\mathrm{m}$$

Commented by Frix last updated on 19/Nov/22

$$\mathrm{draw}\mathrm{it},{\mathrm{you}}^{\prime }\mathrm{ll}\mathrm{immediately}\mathrm{see}\mathrm{the}\mathrm{solution}!$$

Commented by Mastermind last updated on 19/Nov/22

$$\mathrm{Do}\mathrm{it}\mathrm{let}\mathrm{see}\mathrm{boss}$$

Commented by Mastermind last updated on 19/Nov/22

$$\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{understand}\mathrm{you}$$$$$$

Commented by Frix last updated on 19/Nov/22

$$\left(\mathrm{a}\right)\mathrm{Draw}\mathrm{the}\mathrm{function}\left\{\mathrm{x}\right\}\mathrm{in}\mathrm{a}\mathrm{coordinate}$$$$\mathrm{system}\mathrm{for}0⩽\mathrm{x}⩽3$$$$\mathrm{Do}\mathrm{this}\mathrm{and}\mathrm{the}\mathrm{rest}\mathrm{is}\mathrm{obvious}.$$

Commented by mr W last updated on 19/Nov/22

$$thegraphofy=\left\{x\right\}isfollowing.$$$$wecanseethat$$$${\int }_0^n\left\{x\right\}dx=n\times \frac{1}{2}=\frac{n}{2}.$$

Commented by mr W last updated on 19/Nov/22

Commented by Mastermind last updated on 19/Nov/22

$$\mathrm{this}\mathrm{is}\mathrm{the}\mathrm{answer}\mathrm{to}\mathrm{B},\mathrm{right}\dots \mathrm{which}$$$$\mathrm{is}\mathrm{area}?$$

Commented by Mastermind last updated on 19/Nov/22

$$\mathrm{this}\mathrm{is}\mathrm{answer}\mathrm{to}\mathrm{A}$$$$\mathrm{thank}\mathrm{you}$$

Commented by MJS_new last updated on 19/Nov/22

$$\mathrm{some}\mathrm{love}\mathrm{to}\mathrm{think}\mathrm{for}\mathrm{themselves},$$$$\mathrm{some}\mathrm{love}\mathrm{to}\mathrm{let}\mathrm{others}\mathrm{think}\mathrm{for}\mathrm{them}\dots$$$$\mathrm{guess}\mathrm{which}\mathrm{ones}\mathrm{might}\mathrm{once}\mathrm{reach}\mathrm{the}$$$$MasteryofMind$$

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