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Question Number 224079    Answers: 0   Comments: 0

For the given function f(x),let x_0 =0,x_1 =0.6 and x_2 =0.9. construct the lagrange interpolating polynomials of degree. (1) at most 1 (2)at most 2 to approximate f(0.45) if (a) f(x)=cosx (b) f(x)=(√(1+x)) (c) f(x)=In(1+x) (d) f(x)=tanx

$$\boldsymbol{{For}}\:\boldsymbol{{the}}\:\boldsymbol{{given}}\:\boldsymbol{{function}}\:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right),\boldsymbol{{let}}\:\boldsymbol{{x}}_{\mathrm{0}} =\mathrm{0},\boldsymbol{{x}}_{\mathrm{1}} =\mathrm{0}.\mathrm{6} \\ $$$$\boldsymbol{{and}}\:\boldsymbol{{x}}_{\mathrm{2}} =\mathrm{0}.\mathrm{9}.\:\boldsymbol{{construct}}\:\boldsymbol{{the}}\:\boldsymbol{{lagrange}}\:\boldsymbol{{interpolating}} \\ $$$$\boldsymbol{{polynomials}}\:\boldsymbol{{of}}\:\boldsymbol{{degree}}.\:\left(\mathrm{1}\right)\:\boldsymbol{{at}}\:\boldsymbol{{most}}\:\mathrm{1}\:\left(\mathrm{2}\right)\boldsymbol{{at}}\:\boldsymbol{{most}}\:\mathrm{2} \\ $$$$\boldsymbol{{to}}\:\boldsymbol{{approximate}}\:\boldsymbol{{f}}\left(\mathrm{0}.\mathrm{45}\right)\:\boldsymbol{{if}}\: \\ $$$$\left(\boldsymbol{{a}}\right)\:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\boldsymbol{{cosx}}\:\:\left(\boldsymbol{{b}}\right)\:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\sqrt{\mathrm{1}+\boldsymbol{{x}}}\:\left(\boldsymbol{{c}}\right)\:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\boldsymbol{{In}}\left(\mathrm{1}+\boldsymbol{{x}}\right) \\ $$$$\left(\boldsymbol{{d}}\right)\:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\boldsymbol{{tanx}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 224078    Answers: 0   Comments: 0

Evaluate ∫^(𝛑/2) _0 sinxdx with h=(𝛑/(12)),correct to 5 decimal places,using (1)Trapezoidal rule (2)Newtonβˆ’Cotes formula for n=4 (3)Simpson 3/8 βˆ’rule then find the truncation error in each case.

$$\boldsymbol{{Evaluate}}\:\underset{\mathrm{0}} {\int}^{\frac{\boldsymbol{\pi}}{\mathrm{2}}} \boldsymbol{{sinxdx}}\:\boldsymbol{{with}}\:\boldsymbol{{h}}=\frac{\boldsymbol{\pi}}{\mathrm{12}},\boldsymbol{{correct}}\:\boldsymbol{{to}} \\ $$$$\mathrm{5}\:\boldsymbol{{decimal}}\:\boldsymbol{{places}},\boldsymbol{{using}} \\ $$$$\left(\mathrm{1}\right)\boldsymbol{{Trapezoidal}}\:\boldsymbol{{rule}} \\ $$$$\left(\mathrm{2}\right)\boldsymbol{{Newton}}βˆ’\boldsymbol{{Cotes}}\:\boldsymbol{{formula}}\:\boldsymbol{{for}}\:\boldsymbol{{n}}=\mathrm{4} \\ $$$$\left(\mathrm{3}\right)\boldsymbol{{Simpson}}\:\mathrm{3}/\mathrm{8}\:βˆ’\boldsymbol{{rule}} \\ $$$$\boldsymbol{{then}}\:\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{truncation}}\:\boldsymbol{{error}}\:\boldsymbol{{in}}\:\boldsymbol{{each}}\:\boldsymbol{{case}}. \\ $$

Question Number 224076    Answers: 1   Comments: 0

D=(√((mβˆ’(n^2 /4))^2 +(e^m βˆ’n)^2 ))+(n^2 /4)(m,n∈R),D_(min) =?

$$ \\ $$$${D}=\sqrt{\left({m}βˆ’\frac{{n}^{\mathrm{2}} }{\mathrm{4}}\right)^{\mathrm{2}} +\left({e}^{{m}} βˆ’{n}\right)^{\mathrm{2}} }+\frac{{n}^{\mathrm{2}} }{\mathrm{4}}\left({m},{n}\in{R}\right),{D}_{\mathrm{min}} =? \\ $$

Question Number 224069    Answers: 1   Comments: 0

If x^(32) =2^x then solve for x.

$$\mathrm{If}\:\mathrm{x}^{\mathrm{32}} =\mathrm{2}^{\mathrm{x}} \:\mathrm{then}\:\mathrm{solve}\:\mathrm{for}\:\mathrm{x}. \\ $$

Question Number 224065    Answers: 0   Comments: 1

Find x, (√3)xβˆ’3x(√(1βˆ’x^2 ))=1 .

$$\mathrm{Find}\:\mathrm{x},\:\sqrt{\mathrm{3}}\mathrm{x}βˆ’\mathrm{3x}\sqrt{\mathrm{1}βˆ’\mathrm{x}^{\mathrm{2}} }=\mathrm{1}\:. \\ $$

Question Number 224042    Answers: 1   Comments: 0

Question Number 224041    Answers: 1   Comments: 0

Question Number 224034    Answers: 3   Comments: 0

x^3 +(1/x^3 )=18(√3) .Find the value of x.

$$\mathrm{x}^{\mathrm{3}} +\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} }=\mathrm{18}\sqrt{\mathrm{3}}\:.\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}. \\ $$

Question Number 224030    Answers: 1   Comments: 0

Question Number 224029    Answers: 2   Comments: 0

Question Number 224028    Answers: 1   Comments: 0

Question Number 224018    Answers: 2   Comments: 0

x β‰  y Ξ» β‰₯ 1 { ((x + Ξ»^2 = (y βˆ’ Ξ»)^2 )),((y + Ξ»^2 = (x βˆ’ Ξ»)^2 )) :} Find: (((x^2 + y^2 )/(4Ξ»^2 βˆ’ 1)))^(2025) = ?

$$\mathrm{x}\:\neq\:\mathrm{y} \\ $$$$\lambda\:\geqslant\:\mathrm{1} \\ $$$$\begin{cases}{\mathrm{x}\:+\:\lambda^{\mathrm{2}} \:=\:\left(\mathrm{y}\:βˆ’\:\lambda\right)^{\mathrm{2}} }\\{\mathrm{y}\:+\:\lambda^{\mathrm{2}} \:=\:\left(\mathrm{x}\:βˆ’\:\lambda\right)^{\mathrm{2}} }\end{cases} \\ $$$$\mathrm{Find}:\:\:\:\left(\frac{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} }{\mathrm{4}\lambda^{\mathrm{2}} \:βˆ’\:\mathrm{1}}\right)^{\mathrm{2025}} =\:\:? \\ $$

Question Number 224017    Answers: 0   Comments: 0

x,y,z>0 xy+yz+zx+2xyz=1 prove that: (√(1βˆ’x^2 )) + (√(1βˆ’y^2 )) + (√(1βˆ’z^2 )) ≀ ((3 (√3))/2)

$$\mathrm{x},\mathrm{y},\mathrm{z}>\mathrm{0} \\ $$$$\mathrm{xy}+\mathrm{yz}+\mathrm{zx}+\mathrm{2xyz}=\mathrm{1} \\ $$$$\mathrm{prove}\:\mathrm{that}: \\ $$$$\sqrt{\mathrm{1}βˆ’\mathrm{x}^{\mathrm{2}} }\:+\:\sqrt{\mathrm{1}βˆ’\mathrm{y}^{\mathrm{2}} }\:+\:\sqrt{\mathrm{1}βˆ’\mathrm{z}^{\mathrm{2}} }\:\leqslant\:\frac{\mathrm{3}\:\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$

Question Number 224016    Answers: 0   Comments: 0

a,b,c>0 a+b+c+2=abc prove that: (1/( (√(7+a)))) + (1/( (√(7+b)))) + (1/( (√(7+c)))) ≀ 1

$$\mathrm{a},\mathrm{b},\mathrm{c}>\mathrm{0} \\ $$$$\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{2}=\mathrm{abc} \\ $$$$\mathrm{prove}\:\mathrm{that}:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{7}+\mathrm{a}}}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{7}+\mathrm{b}}}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{7}+\mathrm{c}}}\:\leqslant\:\mathrm{1} \\ $$

Question Number 224015    Answers: 0   Comments: 0

a,b,c>0 a+b+c+2=abc prove that: (√a) + (√b) + (√c) ≀ (3/2) (√(abc))

$$\mathrm{a},\mathrm{b},\mathrm{c}>\mathrm{0} \\ $$$$\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{2}=\mathrm{abc} \\ $$$$\mathrm{prove}\:\mathrm{that}:\:\:\:\sqrt{\mathrm{a}}\:+\:\sqrt{\mathrm{b}}\:+\:\sqrt{\mathrm{c}}\:\leqslant\:\frac{\mathrm{3}}{\mathrm{2}}\:\sqrt{\mathrm{abc}} \\ $$

Question Number 223995    Answers: 1   Comments: 0

Question Number 223965    Answers: 2   Comments: 0

Question Number 223964    Answers: 3   Comments: 0

Question Number 223858    Answers: 1   Comments: 0

Question Number 223823    Answers: 3   Comments: 0

(√(4x+1))+(√(3xβˆ’2))=1 x=?

$$\sqrt{\mathrm{4}{x}+\mathrm{1}}+\sqrt{\mathrm{3}{x}βˆ’\mathrm{2}}=\mathrm{1} \\ $$$${x}=? \\ $$

Question Number 223822    Answers: 2   Comments: 0

(((4/3))^(4/3) ) Rewrite in simplest radical form

$$\left(\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\frac{\mathrm{4}}{\mathrm{3}}} \right) \\ $$$$\:{Rewrite}\:{in}\:{simplest}\:{radical}\:{form} \\ $$

Question Number 223804    Answers: 0   Comments: 3

Question Number 223800    Answers: 1   Comments: 0

Given f(x)= ((x^2 +14x+40)/(g(x)))βˆ’43 h(x)= ((g(x)+51)/(x+4)) m(x)= ((h(x)βˆ’9)/(xβˆ’2)) , xβ‰ 2 m(2)= 2043. If f(x) divided by x^2 +8xβˆ’20 gives remainder is M(x)=ax+b then the value of M(98)=?

$$\:\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)=\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{14x}+\mathrm{40}}{\mathrm{g}\left(\mathrm{x}\right)}βˆ’\mathrm{43} \\ $$$$\:\mathrm{h}\left(\mathrm{x}\right)=\:\frac{\mathrm{g}\left(\mathrm{x}\right)+\mathrm{51}}{\mathrm{x}+\mathrm{4}} \\ $$$$\:\mathrm{m}\left(\mathrm{x}\right)=\:\frac{\mathrm{h}\left(\mathrm{x}\right)βˆ’\mathrm{9}}{\mathrm{x}βˆ’\mathrm{2}}\:,\:\mathrm{x}\neq\mathrm{2} \\ $$$$\:\mathrm{m}\left(\mathrm{2}\right)=\:\mathrm{2043}.\: \\ $$$$\:\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{divided}\:\mathrm{by}\:\mathrm{x}^{\mathrm{2}} +\mathrm{8x}βˆ’\mathrm{20}\: \\ $$$$\:\mathrm{gives}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{M}\left(\mathrm{x}\right)=\mathrm{ax}+\mathrm{b} \\ $$$$\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{M}\left(\mathrm{98}\right)=?\: \\ $$

Question Number 223734    Answers: 1   Comments: 0

Question Number 223703    Answers: 2   Comments: 0

Question Number 223700    Answers: 1   Comments: 0

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