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Category: Relation and Functions

2-calculate-k-1-n-1-sin-kpi-n-n-gt-2-1-use-Rieman-sum-to-prove-that-0-pi-log-sinx-dx-pilog2-

Question Number 142424 by Mathspace last updated on 31/May/21 $$\left.\mathrm{2}\right){calculate}\:\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:{sin}\left(\frac{{k}\pi}{{n}}\right)\:\:\:\left({n}>\mathrm{2}\right) \\ $$$$\left.\mathrm{1}\right)\:{use}\:{Rieman}\:{sum}\:{to}\:{prove} \\ $$$${that}\:\int_{\mathrm{0}} ^{\pi} {log}\left({sinx}\right){dx}=−\pi{log}\mathrm{2} \\ $$ Terms of Service Privacy Policy…

Define-two-partition-p-1-and-p-2-of-2-5-such-that-p-1-p-2-Find-the-upper-and-lower-product-sums-with-respet-to-f-defined-by-f-x-x-x-lt-4-1-x-2-x-4-Also-ver

Question Number 11299 by agni5 last updated on 19/Mar/17 $$\mathrm{Define}\:\mathrm{two}\:\mathrm{partition}\:\mathrm{p}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{p}_{\mathrm{2}} \:\mathrm{of}\:\left[\mathrm{2},\mathrm{5}\right]\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{p}_{\mathrm{1}} \subset\mathrm{p}_{\mathrm{2}} \:.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{upper}\:\mathrm{and}\:\mathrm{lower}\:\mathrm{product} \\ $$$$\mathrm{sums}\:\mathrm{with}\:\mathrm{respet}\:\mathrm{to}\:\mathrm{f}\:,?\mathrm{defined}\:\mathrm{by} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}\:,\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}<\mathrm{4} \\ $$$$\:\:\:\:\:\:\:\:=\mathrm{1}−\mathrm{x}^{\mathrm{2}} \:,\:\:\:\:\:\mathrm{x}\:\geqslant\mathrm{4}\:\:. \\ $$$$\mathrm{Also}\:\mathrm{verify}\:\mathrm{the}\:\mathrm{relationship}\:\mathrm{between}\:\mathrm{these}…

Sketch-the-graph-of-the-function-f-is-defined-by-f-x-x-4-8x-3-clearly-giving-all-the-properties-used-in-it-

Question Number 11288 by agni5 last updated on 19/Mar/17 $$\mathrm{Sketch}\:\mathrm{the}\:\mathrm{graph}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}\:\mathrm{f}\:\mathrm{is}\:\mathrm{defined} \\ $$$$\mathrm{by}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{4}} +\mathrm{8x}^{\mathrm{3}} ,\:\mathrm{clearly}\:\mathrm{giving}\:\mathrm{all}\:\mathrm{the}\: \\ $$$$\mathrm{properties}\:\mathrm{used}\:\mathrm{in}\:\mathrm{it}. \\ $$ Answered by ajfour last updated on 19/Mar/17…

f-1-2x-g-x-3-f-1-x-A-7-2g-1-x-B-2g-1-x-7-C-2g-1-x-5-D-5-2g-1-x-

Question Number 11238 by Joel576 last updated on 18/Mar/17 $${f}\left(\mathrm{1}\:−\:\mathrm{2}{x}\right)\:=\:{g}\left({x}\:+\:\mathrm{3}\right) \\ $$$${f}^{−\mathrm{1}} \left({x}\right)\:=\:? \\ $$$$ \\ $$$$\left(\mathrm{A}\right)\:\mathrm{7}\:−\:\mathrm{2}{g}^{−\mathrm{1}} \left({x}\right) \\ $$$$\left(\mathrm{B}\right)\:\mathrm{2}{g}^{−\mathrm{1}} \left({x}\right)\:+\:\mathrm{7} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{2}{g}^{−\mathrm{1}} \left({x}\right)\:−\:\mathrm{5} \\…

Give-an-example-each-with-justification-of-a-function-defined-by-1-1-which-is-1-one-one-but-not-onto-2-onto-but-not-one-one-

Question Number 11196 by agni5 last updated on 16/Mar/17 $$\mathrm{Give}\:\mathrm{an}\:\mathrm{example}\:\mathrm{each}\:\mathrm{with}\:\mathrm{justification},\mathrm{of}\:\mathrm{a}\:\mathrm{function} \\ $$$$\left.\mathrm{defined}\:\mathrm{by}\:\right]−\mathrm{1},\mathrm{1}\left[\:,\mathrm{which}\:\mathrm{is}\right. \\ $$$$\left.\mathrm{1}\right)\mathrm{one}\:\mathrm{one}\:\mathrm{but}\:\mathrm{not}\:\mathrm{onto}. \\ $$$$\left.\mathrm{2}\right)\mathrm{onto}\:\mathrm{but}\:\mathrm{not}\:\mathrm{one}\:\mathrm{one}. \\ $$ Commented by FilupS last updated on 16/Mar/17…