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Question Number 159135 Answers: 1 Comments: 4

Question Number 159131 Answers: 1 Comments: 0

Question Number 159130 Answers: 0 Comments: 0

f ∈ C^0 (R,R) , ((f(x))/x)→_(x→+∞) 0 Does lim_(x→+∞) f(x+1) − f(x) exist?

$${f}\:\in\:\mathcal{C}^{\mathrm{0}} \left(\mathbb{R},\mathbb{R}\right)\:,\:\frac{{f}\left({x}\right)}{{x}}\underset{{x}\rightarrow+\infty} {\rightarrow}\mathrm{0} \\ $$$${Does}\:\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:{f}\left({x}+\mathrm{1}\right)\:−\:{f}\left({x}\right)\:{exist}? \\ $$

Question Number 159123 Answers: 2 Comments: 0

Question Number 159122 Answers: 0 Comments: 0

In order to monitor buses in a travel agency, the manager decides to monitor the number of break downs of the buses using the sequence {x_n } defined by x_(n+1) = 1.05 x_n + 4. Given that x_0 = 40. is the number of break downs by the buses from the 1^(st) of january 2000, and that for every n∈N, we denote x_n the number of breakdowns of the buses as from 1^(st) of january of the year (2000 + n) (a) Calculate x_1 , x_2 , x_3 (b) Consider the sequence {y_n } defined by y_n = x_n + 80 for all n ∈ N (i) express y_(n+1) in terms of y_n and deduce the nature of the sequence {y_n }. (ii) Express y_n in terms of n. deduce x_n in terms of n (iv) find the number of break downs that will be registered by 1^(st) january 2021.

Question Number 159121 Answers: 0 Comments: 0

Consider f(x) = x^3 + 2x −1. Use the intermidiate value theorem and the Rolle theorem to establish that the equation f(x) = 0 has a unique solution denoted a_0 ∈] 0,1[.

$$\mathrm{Consider} \\ $$$${f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{2}{x}\:−\mathrm{1}. \\ $$$$\mathrm{Use}\:\mathrm{the}\:\mathrm{intermidiate}\:\mathrm{value}\:\mathrm{theorem}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{Rolle}\:\mathrm{theorem}\:\mathrm{to}\:\mathrm{establish}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{f}\left({x}\right)\:=\:\mathrm{0}\:\mathrm{has}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution} \\ $$$$\left.\mathrm{denoted}\:{a}_{\mathrm{0}} \in\right]\:\mathrm{0},\mathrm{1}\left[.\:\right. \\ $$

Question Number 159119 Answers: 0 Comments: 0

Assume x;y;z>0 and x^2 +y^2 +z^2 =12 Prove that: Σ_(cycl) (((x/y) + 1 + (y/x))/((1/x) + (1/y))) ≤ 9

$$\mathrm{Assume}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} =\mathrm{12} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\underset{\boldsymbol{\mathrm{cycl}}} {\sum}\:\frac{\frac{\mathrm{x}}{\mathrm{y}}\:+\:\mathrm{1}\:+\:\frac{\mathrm{y}}{\mathrm{x}}}{\frac{\mathrm{1}}{\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{y}}}\:\leqslant\:\mathrm{9} \\ $$

Question Number 159118 Answers: 0 Comments: 0

Let F(r) = { ((mkr, 0 ≤ r < R)),((((mgR^2 )/r^2 ), r≥ R)) :} find D_F

$$\mathrm{Let}\:{F}\left({r}\right)\:=\:\begin{cases}{{mkr},\:\:\mathrm{0}\:\leqslant\:{r}\:<\:{R}}\\{\frac{{m}\mathrm{g}{R}^{\mathrm{2}} }{{r}^{\mathrm{2}} },\:{r}\geqslant\:{R}}\end{cases} \\ $$$$\mathrm{find}\:{D}_{{F}} \\ $$

Question Number 159111 Answers: 1 Comments: 0

Given A^→ =5t^2 i^→ +tj^→ −t^3 k^→ and B^→ =sin(t)i^→ −cos(t)j^→ . Calculate ((d(A^→ .B^→ ))/dx) ; ((d(A^→ ∧B^→ ))/dx) and ((d(A^→ .A^→ ))/dx).

Question Number 159109 Answers: 0 Comments: 0

Show that ∀ V_i ^→ and V_j ^→ : V_i ^→ ∧[V_j ^→ ∧(V_j ^→ ∧V_i ^→ )]=−V_j ^→ ∧[V_i ^→ ∧(V_i ^→ ∧V_j ^→ )]

$${Show}\:{that}\:\forall\:\overset{\rightarrow} {{V}}_{{i}} \:{and}\:\overset{\rightarrow} {{V}}_{{j}} : \\ $$$$\overset{\rightarrow} {{V}}_{{i}} \wedge\left[\overset{\rightarrow} {{V}}_{{j}} \wedge\left(\overset{\rightarrow} {{V}}_{{j}} \wedge\overset{\rightarrow} {{V}}_{{i}} \right)\right]=−\overset{\rightarrow} {{V}}_{{j}} \wedge\left[\overset{\rightarrow} {{V}}_{{i}} \wedge\left(\overset{\rightarrow} {{V}}_{{i}} \wedge\overset{\rightarrow} {{V}}_{{j}} \right)\right] \\ $$

Question Number 160090 Answers: 0 Comments: 2

There are 40 oranges, 20 apples, and 20 lemons in a bag. What is the minimum number of fruits that you have to take out of the bag with your eyes closed before you are sure that one of them is an orange?

$$\mathrm{There}\:\mathrm{are}\:\mathrm{40}\:\mathrm{oranges},\:\mathrm{20}\:\mathrm{apples},\:\mathrm{and}\: \\ $$$$\mathrm{20}\:\mathrm{lemons}\:\mathrm{in}\:\mathrm{a}\:\mathrm{bag}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{minimum}\:\mathrm{number}\:\mathrm{of}\:\mathrm{fruits}\:\mathrm{that}\:\mathrm{you}\: \\ $$$$\mathrm{have}\:\mathrm{to}\:\mathrm{take}\:\mathrm{out}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bag}\:\mathrm{with}\:\mathrm{your}\: \\ $$$$\mathrm{eyes}\:\mathrm{closed}\:\mathrm{before}\:\mathrm{you}\:\mathrm{are}\:\mathrm{sure}\:\mathrm{that}\: \\ $$$$\mathrm{one}\:\mathrm{of}\:\mathrm{them}\:\mathrm{is}\:\mathrm{an}\:\mathrm{orange}? \\ $$

Question Number 159125 Answers: 0 Comments: 3

for a, b, c >0 and a+b+c=2 find min(2ab^2 +b^3 c, 2bc^2 +c^3 a, 2ca^2 +a^3 b) or disprove that such a minimum doesn′t exist.

$${for}\:{a},\:{b},\:{c}\:>\mathrm{0}\:{and}\:{a}+{b}+{c}=\mathrm{2} \\ $$$${find}\:{min}\left(\mathrm{2}{ab}^{\mathrm{2}} +{b}^{\mathrm{3}} {c},\:\mathrm{2}{bc}^{\mathrm{2}} +{c}^{\mathrm{3}} {a},\:\mathrm{2}{ca}^{\mathrm{2}} +{a}^{\mathrm{3}} {b}\right) \\ $$$${or}\:{disprove}\:{that}\:{such}\:{a}\:{minimum} \\ $$$${doesn}'{t}\:{exist}. \\ $$

Question Number 159106 Answers: 0 Comments: 0

∫_0 ^π ((sin(nz))/(z^4 sin(πz)))dz

$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{sin}\left({nz}\right)}{\mathrm{z}^{\mathrm{4}} \mathrm{sin}\left(\pi{z}\right)}{dz} \\ $$

Question Number 159099 Answers: 1 Comments: 0