Calculus Archive: Questions from September 19, 2022
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Given \( y=4 \cos (\theta)+10 \sin (\theta) \), find \( \left.\frac{d^{2} y}{d \theta^{2}}\right|_{\theta=\pi}=? \)1 answer -
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\( 2.1 \quad \int_{0}^{\frac{\pi}{2}} x^{2} \sin x d x \) \( 2.2 \int \frac{14 x^{2}-7 x-3}{\left(x^{2}-1\right)(2 x-1)} d x \) \( 2.3 \int \frac{3 x-6}{x^{2}+10 x+28} d x \) \( 2.4 \int \frac{\pi \co1 answer -
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For Problems 1-7, evaluate the limit, if it exists. 1. (See 2.3.12) \( \lim _{x \rightarrow-3} \frac{x^{2}+3 x}{x^{2}-x-12} \) 2. (See 2.3.14) \( \lim _{x \rightarrow 4} \frac{x^{2}+3 x}{x^{2}-x-12} \2 answers -
Sea \( y_{1}(x)=x^{2} \cos (\ln x) \) una solución de la ecuación diferencial \( x^{2} y^{\prime \prime}-3 x y^{\prime}+5 y=0 \). Encuentre una segunda solución. \[ y_{2}(x)=x^{2} \tan (\ln x) \] \2 answers -
Encuentre la solución general de la ecuación diferencial \( \frac{d^{3} t}{d s^{3}}+5 \frac{d^{4} t}{d s^{4}}-2 \frac{d^{3} t}{d s^{3}}-10 \frac{d^{2} t}{d s^{2}}+\frac{d t}{d s}+5 t=0 \) \[ t=c_{1}1 answer -
51) \( \int \frac{d x}{2 \sqrt{x}(1+x)} \) 51) A) \( \frac{1}{2} \ln |\times|+C \) B) \( \frac{1}{2} \tan ^{-1} \sqrt{x}+C \quad O \frac{1}{2} \sin ^{-1} \sqrt{x}+C \) D) \( \tan ^{-1} \sqrt{x}+C \) 51 answer -
How can I get the highlighted part from (1-2cos(theta)+cos^2(theta)???????
\( =2 \int_{0}^{\pi / 2}(1-\cos \theta)^{2} d \theta=2 \int_{0}^{\pi / 2}\left(1-2 \cos \theta+\cos ^{2} \theta\right) d \theta \) \( =2 \int_{0}^{\pi / 2}\left[1-2 \cos \theta+\frac{1}{2}(1+\cos 2 \t2 answers -
Given \( y=4 \cos (\theta)+10 \sin (\theta) \), find \( \left.\frac{d^{2} y}{d \theta^{2}}\right|_{\theta=\pi}= \) ?1 answer -
Find the horizontal asymptote(s) of \( f(x)=\frac{5 e^{x}+3}{1+e^{x}} \) - A. \( y=0 \) only - B. \( y=1 \) only - C. \( y=\ln (5) \) and \( y=\ln (3) \) - D. \( y=5 \) and \( y=3 \) - E. \( y=\frac{31 answer -
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Solve using an appropriate substitution \[ \frac{d y}{d x}=\sin (x-y) \] \( \arctan (x-y)-\sec ^{2}(x-y)=y+C \) \( \tan (x-y)+\sec (x-y)=x+C \) \( \tan (x-y)+\sec (x-y)=y+C \) \( \tan (x-y)-\sec ^{2}(1 answer -
Find all the second partial derivatives. \[ f(x, y)=x^{9} y^{4}+2 x^{7} y \] \[ f_{x x}(x, y)= \] \[ f_{x y}(x, y)= \] \[ f_{y x}(x, y)= \] \[ f_{y y}(x, y)= \]1 answer -
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#13 with all steps please
7.2 EXERCISES 1-49 Evaluate the integral. 15. \( \int \cot x \cos ^{2} x d x \) 16. \( \int \tan ^{2} x \cos ^{3} x d x \) 1. \( \int \sin ^{2} x \cos ^{3} x d x \) 2. \( \int \sin ^{3} \theta \cos ^{1 answer -
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I need 3d graphs for each function
(a) \( f(x, y)=\frac{1}{1+x^{2}+y^{2}} \) (b) \( f(x, y)=\frac{1}{1+x^{2} y^{2}} \) (c) \( f(x, y)=\ln \left(x^{2}+y^{2}\right) \) (d) \( f(x, y)=\cos \left(\sqrt{x^{2}+y^{2}}\right) \) (e) \( f(x, y)1 answer -
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1. \( y^{\prime \prime \prime}-y=0 \) 2. \( 4 y^{\prime \prime}-20 y^{\prime}+5 y=0 \) given \( y(0)=2 ; y^{\prime}(0)=-4 \) 3. \( y^{\prime \prime}+4 y^{\prime}-2 y=2 x^{2}-3 x+6 \) 4. \( y^{\prime \1 answer -
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Find \( y^{\prime} \) and \( y^{\prime \prime} \). \[ y=e^{\alpha x} \sin (\beta x) \] \( y^{\prime} \) \[ y^{\prime \prime}= \]1 answer -
Use the Mean Value Theorem to find upper and lower bounds for the following
(a) \( \iiint_{B} e^{-x^{2}-y^{2}-z^{2}} d V \), donde \( B=\{(x, y, z): 1 \leq x \leq 2,-1 \leq y \leq 2,-3 \leq z \leq-2\} \) (b) \( \iiint_{B} \frac{1}{\ln (2+x+y+z)} d V \), donde \( B=\{(x, y, z)1 answer -
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Differentiate the following to \( x \) and simplify where possible: 2.1 \( y=x^{4} \tan 3 x \) \( 2.2 \quad y=\ln \left(2 x^{2}+1\right) \) 2.3 \( y=\frac{e^{x}+1}{e^{x}-1} \) \( 2.4 \quad y=\frac{2 \1 answer