2007 AB FRQs - AP Review -- Sage

#4a) f(t)=e^(-t)*sin(t) g(t)=diff(f(t),t) show(f(t)) show(factor(g(t))) show(solve(g(t)==0,t))

 $\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(-t\right)} \sin\left(t\right)$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}{\left(\cos\left(t\right) - \sin\left(t\right)\right)} e^{\left(-t\right)}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left[\sin\left(t\right) = \cos\left(t\right)\right]$

#4b) h(t)=diff(g(t),t) show(h(t)) var('A') equ=A*h(t)+g(t)+f(t)==0 show(equ) show(solve(equ,A))

 $\newcommand{\Bold}[1]{\mathbf{#1}}-2 \, \cos\left(t\right) e^{\left(-t\right)}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-2 \, A \cos\left(t\right) e^{\left(-t\right)} + \cos\left(t\right) e^{\left(-t\right)} = 0$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left[A = \left(\frac{1}{2}\right)\right]$

#5a) tangent at t=5: y-y1=m*(x-x1) --> r-r(5)=r'(5)*(x-5) r(x)=2*(x-5)+30 show(r(x)) show(r(5.4))

 $\newcommand{\Bold}[1]{\mathbf{#1}}2 \, x + 20$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}30.8000000000000$

#5b) v=(4/3)*pi*r^2 v=(4/3)*pi*r^3 show(v) show(diff(v,r)) show(4*pi*30^2*2)

 $\newcommand{\Bold}[1]{\mathbf{#1}}\frac{4}{3} \, \pi r^{3}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}4 \, \pi r^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}7200 \, \pi$

#5c) rsum show(2*4+3*2+2*1.2+4*.6+.5)

 $\newcommand{\Bold}[1]{\mathbf{#1}}19.3000000000000$

#5d) rsum underestimates as function is decreasing

#6a) var('k') f(x)=k*sqrt(x)-ln(x) g(x)=diff(f(x),x) h(x)=diff(g(x),x) show(f(x)) show(g(x)) show(h(x))

 $\newcommand{\Bold}[1]{\mathbf{#1}}k \sqrt{x} - \log\left(x\right)$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}\frac{k}{2 \, \sqrt{x}} - \frac{1}{x}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{k}{4 \, x^{\frac{3}{2}}} + \frac{1}{x^{2}}$

#6b) use second derivative test! show(g(1)) show(solve(g(1)==0,k))

 $\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, k - 1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left[k = 2\right]$

#6c) show(solve(f(x)==0,k)) show(solve(h(x)==0,k)) show(solve(log(x)/sqrt(x)==4/sqrt(x),x))

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left[k = \frac{\log\left(x\right)}{\sqrt{x}}\right]$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left[k = \frac{4}{\sqrt{x}}\right]$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = e^{4}\right]$

2007 AB FRQs - AP Review

3321 days ago by MATH4R2013