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Home Question Bank Math Calculus Archive 2024 December 31

Calculus Questions from Dec 31,2024

Browse the Calculus Q&A Archive for Dec 31,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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4. Let \( f(x)=\ln \left(x^{2}+1\right) \), then \( f^{\prime}(x)= \) Let \( f(x)=e^{x} \), then \( f^{n}(x)= \) The function \( f(x)=-2 x^{3}+a x^{2}+b x+c \) the turnig point of \( p(8,-9) \) and \( (5,18) \), then the value of \( a= \) and \( b= \). Which function satisfies the differential equation \( y^{\prime \prime}=-y \) What is the solution to the differential equation \( \frac{d y}{d x}+x y=x, y(0)=-6 ? \) What is the solution to the following differential equation, subject to the initial condition \( y(0)=2 \) ? \( \frac{d y}{d x}=6 x+6 \) What is the general solution to the following: \( \int 2 x \sin (x) d x \) ? Integrate the following: \( \int x e^{2 x} d x \) 20. \( \int x \tan ^{2} x d x \) Un objeto se mueve a lo largo de una línea y su aceleración está dada por \( a(t) = 12t \). Si en el tiempo inicial \( t=0 \) la velocidad es \( v(0)=4 \), encuentra la expresión para la velocidad. 4) If the function \( f: f(x)=\left\{\begin{array}{ll}\frac{\sin 2 x+\tan x}{1+\sin 4 x} & , x \in] 0, \frac{\pi}{3}\left[-\left\{\frac{\pi}{4}\right\}\right. \\ k & , x=\frac{\pi}{4}\end{array}\right. \) is continuous at \( x=\frac{\pi}{4} \), then \( k=\cdots \) \( \begin{array}{ll}\text { (a) } \frac{1}{2} & \text { (c) } 3\end{array} \) DEVELOPPEMENTS LIM EXERCICE I Calculer les développements limités de la fonctio 1) \( f(x)=\frac{1}{\sqrt{3-x}} \) à l'ordre 3 , au voisinage \( \left.2^{\circ}\right) f(x)=\sqrt{x} \), à l'ordre 3 , au voisinage de 2 \( \left.3^{\circ}\right) f(x)=\frac{\sqrt{x+2}}{\sqrt{x}} \), à l'ordre 3 , au voisinage 4) \( f(x)=\sqrt{1-x}+\sqrt{1+x} \), à l'ordre 4 , au \( \left.5^{\circ}\right) f(x)=(\ln (1+x))^{2} \), a lordre 4, au voisin 6) \( f(x)=\frac{1}{(x+1)(x-2)} \), à l'ordre 3 , au vo 7) \( f(x)=\frac{x^{2}+1}{x^{2}+2 x+2} \), à l'ordre 3 , au voisi 8) \( f(x)=\frac{\ln (1+x)}{1-x} \), à l'ordre 4 , au voisina 9) \( f(x)=\frac{\sin x}{x}-\frac{x}{e^{x}-1} \), à l'ordre.4, au v \( 10^{\circ} \) ) \( f(x)=\ln \left(\frac{1}{\cos x}\right) \), à l'ordre 4 , au vois 11.) \( f(x)=\operatorname{Arccos}\left(\frac{1+x}{2+x}\right) \), à l'ordre 2 , a 12.) \( f(x)=(1+\sin x)^{\cos x} \), à l'ordre 4, aul 13 ) \( f(x)=\operatorname{Artan} \sqrt{\frac{1+x}{2+x}} \), à l'ordre 2, au 14 ) \( f(x)=\operatorname{Argsh}\left(1+2 x+3 x^{2}\right) \), à l'ordr 15 ) \( f(x)=\sqrt{\tan x} \), à l'ordre 3 , au voisina EXERCICE II A l'aide des développements limités, calculer \[ \begin{array}{l} \lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}-\frac{1}{(\operatorname{Arcin} x)^{2}}\right) \\ \lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\ln (1+x)}\right) \\ \lim _{x \rightarrow 0} \frac{\ln \left(\frac{1+x}{1-x}\right)}{\operatorname{Arctan}(1+x)-\operatorname{Arctan}(1-x))} \\ \lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)^{\frac{1}{\sin x}} \quad \lim _{x \rightarrow 0} \frac{\sqrt{1+x}-\cos x}{\ln (1+x)} \end{array} \]
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