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Home Question Bank Math Calculus Archive 2025 January 16

Calculus Questions from Jan 16,2025

Browse the Calculus Q&A Archive for Jan 16,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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19 A particle \( P \) is moving along a straight line. The fized point \( O \) lies on this line. At time \( f \) seconds where \( f \geqslant 0 \), the displacement, s metres, of \( P \) from \( O \) is given by \[ s=t^{3}+5 t^{4}-9 t+10 \] Find the displacernent of \( P \) from \( O \) when \( P \) is instantancously at rest. Grve your answer in the form \( \frac{a}{b} \) where \( a \) and \( b \) are integers 2. On admet que \( g(x)=(x-1) e^{-x}+2 \) a) Démontrer que l'équation : \( x \in \mathbb{R}, g(x)=0 \) admet une solution unique noté \( \alpha \) b) Vérifier que \( -0,4<\alpha<-0,3 \) c) Démontrer que \( \left\{\begin{array}{l}\text { pour tout } x \in]-\infty ; a[; g(x)<0 \\ \text { pour tout } x \in] a ;+\infty[; g(x)>0\end{array}\right. \) Partie B On considère la fonction définie sur \( \mathbb{R} \) par \( f(x)=2 x-1-x e^{-x} \) On désigne par ( \( C \) ) sa représentation graphique dans le plan muni d'un repère orthonormé \( (O, I, I) \). Unité graphique: 2 cm 1. Calculer leslimites de \( f \) en \( -\infty \) et en \( +\infty \) 2. On admet que \( f \) est dérivablesur \( \mathbb{R} \) Montrer que pour tout \( x \in \mathbb{R}, f^{\prime}(x)=g(x) \) Email : oualefidele@gmailcom \( \mathrm{Cd}_{1} \quad 58 \geq 20709 \quad 02-58-82-72 \quad 05-65-91-86 \) 87 Manathon de Butterfly "Etude de Fonetions" Terminale 10 3. Etudier les variations de \( f \), puis dresser son tableau de variation 4. Calcules \( \lim _{x \rightarrow-\infty} \frac{f(x)}{x} \), puisinterpréter graphiquement le résultat obtenu 5- a) Démontrer que la droite ( \( D \) ): \( y=2 x-1 \) est une asymptote à la courbe ( \( C \) ) \( \lim _ { x \rightarrow + } ( x - \sqrt { 4 x ^ { 3 } + x + 1 } ) \) 1. Considere o cenário simplificado de uma empresa de extração mineira que pretende iniciar atividade e está em negociação com a autarquia relativamente à dimensão da mina a construir. Seja \( y \) a produção alvo por mês da empresa, em milhares de euros, estima-se que o consumo mensal na região pelos trabalhadores da mina seja, em milhares de euros: \[ C(y)=0.012 y+0.000028 y^{2} \] O tratamento da poluição será da responsabilidade de instituições públicas, assumindo-se um custo mensal, em milhares de euros, de: \[ P(y)=0,00021 \times y+0,0000029 \times y^{2}+0,03 \times C(y) \] Adicionalmente, assumindo que os custos com salários da empresa serão \( S(y)=0,02 y \), em milhares de euros, prevê-se que os impostos auferidos mensalmente, em milhares de euros, sejam: \( \quad I(y)=0,15 \times(y-2 S(y))+0,25 \times(S(y)-C(y))+0,17 \times C(y) \) Doc.pedrocastlo ferrera Graph the functions in Exercises 9 a. What are the domain and range of \( f \) ? b. At what points \( c \), if any, does \( \lim _{x \rightarrow c} f(x) \) exist? c. At what points does only the left-hand limit exist? d. At what points does only the right-hand limit exist? 9. \( f(x)=\left\{\begin{array}{ll}\sqrt{1-x^{2}}, & 0 \leq x<1 \\ 1, & 1 \leq x<2 \\ 2, & x=2\end{array}\right. \) Find the area of the region bounded by the curves \( y = \frac{1}{x} \) from \( x = 1 \) to \( x = 2 \) and the line \( y = 1 \). \( \frac { d y } { d x } = ( x + y - 1 ) ^ { 2 } \) Какой объем занимает тело, созданное вращением области между \( y = x + 1 \) и \( y = 3x - x^2 \) вокруг оси Y? Let \( I=\int_{C}(2 z d x+2 y d y+2 x d z) \) where \( x, y, z \) are real, and let \( C \) be the straight line segment from point \( A:(0,2,1) \) to point \( B:(4,1,-1) \). The value of \( I \) is Si une courbe est donnée par la fonction \( y = e^{x} \), calculez sa longueur d'arc entre \( x = 0 \) et \( x = 1 \).
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