CRL 4.31 Exercises MrG 2011.1219

3616 days ago by calcpage123

#1) var('a') f(x)=arctan(a*x^2) show(f(x)) show(diff(f(x),x)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\arctan\left(a x^{2}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, a x}{a^{2} x^{4} + 1}
\newcommand{\Bold}[1]{\mathbf{#1}}\arctan\left(a x^{2}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, a x}{a^{2} x^{4} + 1}
#3) f(x)=arcsec((x^2+1)/(x^2-1)) show(f(x)) show(diff(f(x),x).simplify_radical()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm arcsec}\left(\frac{x^{2} + 1}{x^{2} - 1}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, x}{{\left(x^{2} + 1\right)} {\left| x \right|}}
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm arcsec}\left(\frac{x^{2} + 1}{x^{2} - 1}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, x}{{\left(x^{2} + 1\right)} {\left| x \right|}}
#5) f(x)=arccot(x^2-5) show(f(x)) show(diff(f(x),x)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm arccot}\left(x^{2} - 5\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, x}{{\left(x^{2} - 5\right)}^{2} + 1}
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm arccot}\left(x^{2} - 5\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, x}{{\left(x^{2} - 5\right)}^{2} + 1}
#7) f(x)=arccsc(1/(2*x^2-1)) show(f(x)) show(diff(f(x),x).simplify_rational()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm arccsc}\left(\frac{1}{2 \, x^{2} - 1}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{4 \, x}{\sqrt{-4 \, x^{4} + 4 \, x^{2}}}
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm arccsc}\left(\frac{1}{2 \, x^{2} - 1}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{4 \, x}{\sqrt{-4 \, x^{4} + 4 \, x^{2}}}
#9) f(x)=arctan(sqrt(1-x)) show(f(x)) show(diff(f(x),x)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\arctan\left(\sqrt{-x + 1}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2 \, \sqrt{-x + 1} {\left(x - 2\right)}}
\newcommand{\Bold}[1]{\mathbf{#1}}\arctan\left(\sqrt{-x + 1}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2 \, \sqrt{-x + 1} {\left(x - 2\right)}}