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Question Number 69784 Answers: 0 Comments: 0

calculate f(a) =∫_0 ^∞ e^(−(x^2 +(a/x^2 ))) dx with a>0

$${calculate}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\left({x}^{\mathrm{2}} \:+\frac{{a}}{{x}^{\mathrm{2}} }\right)} {dx}\:\:{with}\:{a}>\mathrm{0} \\ $$

Question Number 69778 Answers: 2 Comments: 5

prove that the equation (b^2 −4ac)x^2 + 4(a + c)x −4 = 0 is always real.

$${prove}\:{that}\:{the}\:{equation}\: \\ $$$$\:\:\left({b}^{\mathrm{2}} −\mathrm{4}{ac}\right){x}^{\mathrm{2}} \:+\:\mathrm{4}\left({a}\:+\:{c}\right){x}\:−\mathrm{4}\:=\:\mathrm{0}\:{is}\:{always}\:{real}. \\ $$

Question Number 69766 Answers: 0 Comments: 4

find (dy/dx) at the point (0,3) when 2x^2 y + y + 4xy^2 = 2x + 3

$${find}\:\:\frac{{dy}}{{dx}}\:\:{at}\:{the}\:{point}\:\:\left(\mathrm{0},\mathrm{3}\right)\:\:{when}\:\:\mathrm{2}{x}^{\mathrm{2}} {y}\:+\:{y}\:+\:\mathrm{4}{xy}^{\mathrm{2}} \:=\:\mathrm{2}{x}\:+\:\mathrm{3}\: \\ $$

Question Number 69765 Answers: 1 Comments: 0

Given that y = (√(5x^2 + 3)) , show that when x^2 = (6/5) , (d^2 y/dx^(2 ) ) = ((125)/8)

$${Given}\:{that}\:\:{y}\:=\:\sqrt{\mathrm{5}{x}^{\mathrm{2}} \:+\:\mathrm{3}}\:,\:{show}\:{that}\:\:{when}\:{x}^{\mathrm{2}} \:=\:\frac{\mathrm{6}}{\mathrm{5}}\:,\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}\:} }\:=\:\frac{\mathrm{125}}{\mathrm{8}} \\ $$

Question Number 69764 Answers: 1 Comments: 1

find (dy/dx) if x = sin^2 t and y= tan t at t = (π/4)

$${find}\:\:\:\frac{{dy}}{{dx}}\:\:{if}\:\:{x}\:=\:{sin}^{\mathrm{2}} {t}\:\:{and}\:\:{y}=\:{tan}\:{t}\:{at}\:\:{t}\:=\:\frac{\pi}{\mathrm{4}} \\ $$

Question Number 69763 Answers: 1 Comments: 2

find (dy/dx) if y = 3^x e^(2x + 1) , at x =1

$${find}\:\frac{{dy}}{{dx}}\:\:{if}\:\:{y}\:=\:\mathrm{3}^{{x}} {e}^{\mathrm{2}{x}\:+\:\mathrm{1}} ,\:{at}\:{x}\:=\mathrm{1} \\ $$

Question Number 69762 Answers: 1 Comments: 1

prove by mathematical induction, that for all positive integers n, Σ_(r=1) ^n r(r + 1) = (n/3)(n + 1)( n + 2)

$${prove}\:{by}\:{mathematical}\:{induction},\:{that}\:{for}\:{all}\:{positive}\:{integers}\:{n}, \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\left({r}\:+\:\mathrm{1}\right)\:=\:\frac{{n}}{\mathrm{3}}\left({n}\:+\:\mathrm{1}\right)\left(\:{n}\:+\:\mathrm{2}\right) \\ $$

Question Number 69754 Answers: 0 Comments: 2

Question Number 69746 Answers: 1 Comments: 1

Question Number 69741 Answers: 2 Comments: 4

Question Number 69738 Answers: 1 Comments: 0

Sarah dances everyday of the week, including saturdays and sundays. In november 2018, Sarah had to miss a few days. To control her absences she marks the day she missed class with a x on the calendar. She marked the 5th, 21st and 27th of november. What percentage indicates Sarah′s absences in november?

$${Sarah}\:{dances}\:{everyday}\:{of}\:{the}\:{week}, \\ $$$${including}\:{saturdays}\:{and}\:{sundays}. \\ $$$${In}\:{november}\:\mathrm{2018},\:{Sarah}\:{had}\:{to}\:{miss} \\ $$$${a}\:{few}\:{days}.\:{To}\:{control}\:{her}\:{absences} \\ $$$${she}\:{marks}\:{the}\:{day}\:{she}\:{missed}\:{class} \\ $$$${with}\:{a}\:\boldsymbol{{x}}\:{on}\:{the}\:{calendar}. \\ $$$${She}\:{marked}\:{the}\:\mathrm{5}{th},\:\mathrm{21}{st}\:{and}\:\mathrm{27}{th} \\ $$$${of}\:{november}. \\ $$$${What}\:{percentage}\:{indicates}\:{Sarah}'{s} \\ $$$${absences}\:{in}\:{november}? \\ $$

Question Number 69735 Answers: 1 Comments: 1

∫_( 0) ^(π/4) ((sin x+cos x)/(3+sin 2x)) dx =

$$\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\:\frac{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{\mathrm{3}+\mathrm{sin}\:\mathrm{2}{x}}\:{dx}\:= \\ $$

Question Number 69734 Answers: 0 Comments: 1

If I_n = ∫_(0 ) ^(π/4) tan^n x dx, n ∈ N, then I_(n+2) +I_n =

$$\mathrm{If}\:{I}_{{n}} =\:\underset{\mathrm{0}\:} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{tan}^{{n}} {x}\:{dx},\:{n}\:\in\:{N},\:\mathrm{then}\:{I}_{{n}+\mathrm{2}} +{I}_{{n}} = \\ $$

Question Number 69715 Answers: 1 Comments: 0

∫((x^2 +1)/((x+1)^2 ))e^x dx

$$\int\frac{{x}^{\mathrm{2}} +\mathrm{1}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{e}^{{x}} {dx} \\ $$

Question Number 69710 Answers: 1 Comments: 1

Question Number 69709 Answers: 0 Comments: 2

Question Number 69692 Answers: 0 Comments: 0

Question Number 71696 Answers: 1 Comments: 0

The side of a square is measured to be 12cm long cofrect to the nearest cm. Find the maximum absolute error and the maximum percentage error for (a) The length of the square (Answer: 0.5cm, 4.17%) (b) The area of the square. (Answer: 12.25cm, 8.5%)

$$\mathrm{The}\:\mathrm{side}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\:\mathrm{is}\:\mathrm{measured}\:\mathrm{to}\:\mathrm{be}\:\:\mathrm{12cm}\:\mathrm{long}\:\mathrm{cofrect} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{nearest}\:\:\mathrm{cm}.\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{absolute}\:\mathrm{error} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{percentage}\:\mathrm{error}\:\mathrm{for} \\ $$$$\left(\mathrm{a}\right)\:\:\:\:\mathrm{The}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}\:\:\:\:\:\left(\mathrm{Answer}:\:\:\mathrm{0}.\mathrm{5cm},\:\:\mathrm{4}.\mathrm{17\%}\right) \\ $$$$\left(\mathrm{b}\right)\:\:\:\:\mathrm{The}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}.\:\:\:\:\:\:\left(\mathrm{Answer}:\:\:\:\:\mathrm{12}.\mathrm{25cm},\:\:\:\mathrm{8}.\mathrm{5\%}\right) \\ $$

Question Number 69681 Answers: 2 Comments: 2