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Question Number 62679 Answers: 1 Comments: 5

Let f be defined in the neighborhood of x and that f ′′(x) exists. Prove that lim_(h→0) ((f(x+h)+f(x−h)−2f(x))/h^2 )=f ′′(x) .

$${Let}\:{f}\:{be}\:{defined}\:{in}\:{the}\:{neighborhood} \\ $$$${of}\:{x}\:{and}\:{that}\:{f}\:''\left({x}\right)\:{exists}. \\ $$$${Prove}\:{that}\:\: \\ $$$$\:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{f}\left({x}+{h}\right)+{f}\left({x}−{h}\right)−\mathrm{2}{f}\left({x}\right)}{{h}^{\mathrm{2}} }={f}\:''\left({x}\right)\:. \\ $$

Question Number 62626 Answers: 1 Comments: 4

Find the limit of ((n!)/4^n ) as n approach infinity

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{of}\:\:\:\:\:\frac{\mathrm{n}!}{\mathrm{4}^{\mathrm{n}} }\:\:\:\mathrm{as}\:\:\mathrm{n}\:\:\mathrm{approach}\:\mathrm{infinity} \\ $$

Question Number 62624 Answers: 2 Comments: 0

Question Number 62623 Answers: 0 Comments: 0

Question Number 62622 Answers: 0 Comments: 0

Question Number 62613 Answers: 3 Comments: 2

Question Number 62612 Answers: 1 Comments: 0

The number of real solutions of the equation ((9/(10)))^x = −3+x−x^2 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\left(\frac{\mathrm{9}}{\mathrm{10}}\right)^{{x}} =\:−\mathrm{3}+{x}−{x}^{\mathrm{2}} \:\:\:\:\:\mathrm{is} \\ $$

Question Number 62596 Answers: 1 Comments: 0

∫sin^(100) (x) cos^(100) (x) dx

$$\int\mathrm{sin}^{\mathrm{100}} \left(\mathrm{x}\right)\:\mathrm{cos}^{\mathrm{100}} \left(\mathrm{x}\right)\:\mathrm{dx} \\ $$

Question Number 62595 Answers: 1 Comments: 1

what is 2sin^2 θcos^2 θ ?

$${what}\:{is}\:\mathrm{2}{sin}^{\mathrm{2}} \theta{cos}^{\mathrm{2}} \theta\:? \\ $$

Question Number 62591 Answers: 2 Comments: 1

If 2^x = 3^y = 6^z , then (1/x) + (1/y) + (1/z) = ____.

$$\mathrm{If}\:\:\mathrm{2}^{{x}} =\:\mathrm{3}^{{y}} =\:\mathrm{6}^{{z}} ,\:\mathrm{then}\:\frac{\mathrm{1}}{{x}}\:+\:\frac{\mathrm{1}}{{y}}\:+\:\frac{\mathrm{1}}{{z}}\:=\:\_\_\_\_. \\ $$

Question Number 62585 Answers: 1 Comments: 0

find the resultant force of a system of three forces O^− P^→ =9N,O^− R^→ =10N and O^− Q^→ 10N acting at point O where angle POR is 135°,angle POQ is 135° and QOR is 90°

$${find}\:{the}\:{resultant}\:{force}\:{of}\:{a}\:{system} \\ $$$${of}\:{three}\:{forces}\:\overset{−} {{O}}\overset{\rightarrow} {{P}}\:=\mathrm{9}{N},\overset{−} {{O}}\overset{\rightarrow} {{R}}\:=\mathrm{10}{N}\:{and}\:\overset{−} {{O}}\overset{\rightarrow} {{Q}}\:\mathrm{10}{N}\: \\ $$$${acting}\:{at}\:{point}\:{O}\:{where}\:{angle}\:{POR}\:{is} \\ $$$$\mathrm{135}°,{angle}\:{POQ}\:{is}\:\mathrm{135}°\:\:{and}\:{QOR}\:{is}\: \\ $$$$\mathrm{90}° \\ $$

Question Number 62583 Answers: 1 Comments: 0

find the vector sum of two vectors of magnitude of 7 and 8 making an angle of 120° to each other

$${find}\:{the}\:{vector}\:{sum}\:{of}\:{two}\:{vectors} \\ $$$${of}\:{magnitude}\:{of}\:\mathrm{7}\:{and}\:\mathrm{8}\:{making}\:{an} \\ $$$${angle}\:{of}\:\mathrm{120}°\:{to}\:{each}\:{other} \\ $$

Question Number 62577 Answers: 1 Comments: 1

Question Number 62576 Answers: 0 Comments: 0

Question Number 62571 Answers: 0 Comments: 6

Find lim_(n→∞) (n((1/((2n+1)^2 )) + (1/((2n+3)^2 )) + ... + (1/((4n−1)^2 ))))

$$\mathrm{Find} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left({n}\left(\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{3}\right)^{\mathrm{2}} }\:+\:...\:+\:\frac{\mathrm{1}}{\left(\mathrm{4}{n}−\mathrm{1}\right)^{\mathrm{2}} }\right)\right) \\ $$

Question Number 62570 Answers: 0 Comments: 1

Question Number 62556 Answers: 1 Comments: 2

((1+3)/3) + ((1+3+5)/3^2 ) + ((1+3+5+7)/3^3 ) + ... = (a/b) , a, b ∈ Z^+

$$\frac{\mathrm{1}+\mathrm{3}}{\mathrm{3}}\:+\:\frac{\mathrm{1}+\mathrm{3}+\mathrm{5}}{\mathrm{3}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}+\mathrm{3}+\mathrm{5}+\mathrm{7}}{\mathrm{3}^{\mathrm{3}} }\:+\:...\:\:=\:\:\:\frac{{a}}{{b}}\:\:,\:\:{a},\:{b}\:\in\:\:\mathbb{Z}^{+} \: \\ $$

Question Number 62554 Answers: 0 Comments: 0

If a & b are two natural numbers then the following relation holds always. a×b=gcd(a,b)×lcm(a,b) ^• If a,b & c are three natural numbers what relation(anologous to the above) holds between numbers and their gcd & lcm?

$${If}\:{a}\:\&\:{b}\:{are}\:{two}\:{natural}\:{numbers} \\ $$$${then}\:{the}\:{following}\:{relation}\:{holds} \\ $$$${always}. \\ $$$$\:\:\:\:\:{a}×{b}=\mathrm{gcd}\left({a},{b}\right)×\mathrm{lcm}\left({a},{b}\right) \\ $$$$\:^{\bullet} {If}\:{a},{b}\:\&\:{c}\:\:{are}\:{three}\:{natural}\:{numbers} \\ $$$$\:\:\:{what}\:{relation}\left({anologous}\:{to}\:{the}\:{above}\right) \\ $$$$\:\:\:\:{holds}\:{between}\:{numbers}\:{and}\:{their} \\ $$$$\:\:\:\mathrm{gcd}\:\&\:\mathrm{lcm}? \\ $$

Question Number 62542 Answers: 1 Comments: 1

Question Number 62539 Answers: 1 Comments: 0

lim_(x→∞) ((x^3 +sin(2x)−2sin(x))/(arctan(x^3 )−(arctan(x))^3 ))

$$\mathrm{li}\underset{\mathrm{x}\rightarrow\infty} {\mathrm{m}}\frac{\mathrm{x}^{\mathrm{3}} +\mathrm{sin}\left(\mathrm{2x}\right)−\mathrm{2sin}\left(\mathrm{x}\right)}{\mathrm{arctan}\left(\mathrm{x}^{\mathrm{3}} \right)−\left(\mathrm{arctan}\left(\mathrm{x}\right)\right)^{\mathrm{3}} } \\ $$

Question Number 62538 Answers: 0 Comments: 0

Question Number 62587 Answers: 1 Comments: 1

three forces having equal magnitude s of 10N,20N and 30N make angles of 30°,120° and 210° respectively with the positive direction of the x axis. By scale drawing find the magnitude and the direction of the resultant force

$${three}\:{forces}\:{having}\:{equal}\:{magnitude} \\ $$$${s}\:{of}\:\mathrm{10}{N},\mathrm{20}{N}\:{and}\:\mathrm{30}{N}\:{make}\:{angles}\: \\ $$$${of}\:\mathrm{30}°,\mathrm{120}°\:{and}\:\mathrm{210}°\:{respectively}\:{with} \\ $$$${the}\:{positive}\:{direction}\:{of}\:{the}\:{x}\:{axis}. \\ $$$${By}\:{scale}\:{drawing}\:{find}\:{the}\:{magnitude} \\ $$$${and}\:{the}\:{direction}\:{of}\:{the}\:{resultant}\: \\ $$$${force} \\ $$

Question Number 62534 Answers: 1 Comments: 0

Question Number 62519 Answers: 1 Comments: 0

Question Number 62523 Answers: 1 Comments: 0

Question Number 62517 Answers: 4 Comments: 2

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