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Given that the function f(x) = x^3 is differentiable in the interval (−2,2) us the mean value theorem to find the value of x for which the tangent to the curve is parrallel to the chord through the points (−2,8) and (2,8).

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:\mathrm{is}\: \\ $$$$\mathrm{differentiable}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval}\:\left(−\mathrm{2},\mathrm{2}\right)\:\mathrm{us}\:\mathrm{the}\:\mathrm{mean} \\ $$$$\mathrm{value}\:\mathrm{theorem}\:\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\: \\ $$$$\mathrm{tangent}\:\mathrm{to}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{is}\:\mathrm{parrallel}\:\mathrm{to}\:\mathrm{the}\:\mathrm{chord}\: \\ $$$$\mathrm{through}\:\mathrm{the}\:\mathrm{points}\:\left(−\mathrm{2},\mathrm{8}\right)\:\mathrm{and}\:\left(\mathrm{2},\mathrm{8}\right). \\ $$

Question Number 83378 Answers: 0 Comments: 2

x+y+z=1 x^2 +y^2 +z^2 =2 x^3 +y^3 +z^3 =3 find x^4 +y^4 +z^4 =?

$${x}+{y}+{z}=\mathrm{1} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{2} \\ $$$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} =\mathrm{3} \\ $$$${find} \\ $$$${x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} =? \\ $$

Question Number 83373 Answers: 0 Comments: 4

Question Number 83372 Answers: 0 Comments: 1

Question Number 83370 Answers: 0 Comments: 3

Question Number 83369 Answers: 1 Comments: 2

Question Number 83367 Answers: 0 Comments: 1

∫_(−∞) ^∞ ((sin^7 (x))/x^7 )dx

$$\int_{−\infty} ^{\infty} \frac{{sin}^{\mathrm{7}} \left({x}\right)}{{x}^{\mathrm{7}} }{dx} \\ $$

Question Number 83361 Answers: 1 Comments: 1

lim_(x→0^+ ) ((1−cos (x))/(√x)) =

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\left(\mathrm{x}\right)}{\sqrt{\mathrm{x}}}\:=\: \\ $$

Question Number 83354 Answers: 0 Comments: 2

Question Number 83352 Answers: 1 Comments: 6

(√(2sin^2 ((x/2))(1−cos (x)))) = −sin (−x)−5cos (x)

$$\sqrt{\mathrm{2sin}\:^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left(\mathrm{1}−\mathrm{cos}\:\left(\mathrm{x}\right)\right)}\:=\:−\mathrm{sin}\:\left(−\mathrm{x}\right)−\mathrm{5cos}\:\left(\mathrm{x}\right) \\ $$

Question Number 83342 Answers: 1 Comments: 0

∫_0 ^1 ∫_0 ^( z) ∫_y ^1 ((z^(n+1) Li_1 (y))/((zx)^2 ))dx dy dz , ∀ n∈z

$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\:{z}} \int_{{y}} ^{\mathrm{1}} \frac{{z}^{{n}+\mathrm{1}} {Li}_{\mathrm{1}} \left({y}\right)}{\left({zx}\right)^{\mathrm{2}} }{dx}\:{dy}\:{dz}\:,\:\forall\:{n}\in{z} \\ $$

Question Number 83341 Answers: 1 Comments: 1

Find the maximum and minimum of the expression 𝚺_(i=1) ^n a_i x_i with 𝚺_(i=1) ^n (x_i −b_i )^2 =c^2 , where a_i , b_i and c are constants. (extracted and modified from Q83331)

$${Find}\:{the}\:{maximum}\:{and}\:{minimum} \\ $$$${of}\:{the}\:{expression}\:\underset{\boldsymbol{{i}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\boldsymbol{\sum}}{a}}_{\boldsymbol{{i}}} \boldsymbol{{x}}_{\boldsymbol{{i}}} \:{with} \\ $$$$\underset{\boldsymbol{{i}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\boldsymbol{\sum}}}\left(\boldsymbol{{x}}_{\boldsymbol{{i}}} −\boldsymbol{{b}}_{\boldsymbol{{i}}} \right)^{\mathrm{2}} =\boldsymbol{{c}}^{\mathrm{2}} ,\:{where}\:\boldsymbol{{a}}_{\boldsymbol{{i}}} ,\:\boldsymbol{{b}}_{\boldsymbol{{i}}} \:{and}\:\boldsymbol{{c}}\:{are} \\ $$$${constants}. \\ $$$$ \\ $$$$\left({extracted}\:{and}\:{modified}\:{from}\:{Q}\mathrm{83331}\right) \\ $$

Question Number 83340 Answers: 0 Comments: 0

by using the lagrange method solve the partial equatio (p−q=(z/(x+y)))

$${by}\:{using}\:{the}\:{lagrange}\:{method}\:{solve}\:{the}\:{partial}\:{equatio}\:\left({p}−{q}=\frac{{z}}{{x}+{y}}\right) \\ $$

Question Number 83338 Answers: 1 Comments: 0

∫ sin (3x) tan (2x) dx ?

$$\int\:\mathrm{sin}\:\left(\mathrm{3x}\right)\:\mathrm{tan}\:\left(\mathrm{2x}\right)\:\mathrm{dx}\:? \\ $$

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