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Magazine Problem 76
Find the antiderivative. \(\int \frac{e^{2 x}}{1+e^{4 x}} d x\)
Problem 77
Find the derivative. \(\frac{d}{d t} \int_{0}^{t} \frac{\sin x}{\sqrt{1+x^{2}}} d x\)
Problem 78
\(x=2 y^{2}-y^{3}, x=0\), and \(y=0\) rotated around the \(x\) -axis using cylindrical shells
Problem 78
Find the derivative. \(\frac{d}{d x} \int_{1}^{x^{3}} \sqrt{4-t^{2}} d t\)
Problem 79
Find the derivative. \(\frac{d}{d x} \int_{1}^{\ln (x)}\left(4 t+e^{t}\right) d t\)
Problem 80
\(y=x^{2}-x\) and \(x=0\)
Problem 81
The following problems consider the historic average cost per gigabyte of RAM on a computer. $$\begin{array}{|l|l|} \hline \text { Year } & \text { 5-Year Change (\$) } \\ \hline 1980 & 0 \\ \hline 1985 & -5,468,750 \\ \hline 1990 & -755,495 \\ \hline 1995 & -73,005 \\ \hline 2000 & -29,768 \\ \hline 2005 & -918 \\ \hline 2010 & -177 \\ \hline \end{array}$$ If the average cost per gigabyte of RAM in 2010 is \(\$ 12\), find the average cost per gigabyte of RAM in 1980 .
Problem 82
The following problems consider the historic average cost per gigabyte of RAM on a computer. $$\begin{array}{|l|l|} \hline \text { Year } & \text { 5-Year Change (\$) } \\ \hline 1980 & 0 \\ \hline 1985 & -5,468,750 \\ \hline 1990 & -755,495 \\ \hline 1995 & -73,005 \\ \hline 2000 & -29,768 \\ \hline 2005 & -918 \\ \hline 2010 & -177 \\ \hline \end{array}$$ The average cost per gigabyte of RAM can be approximated by the function \(C(t)=8,500,000(0.65)^{t}\), where \(t\) is measured in years since 1980, and \(C\) is cost in USS. Find the average cost per gigabyte of RAM for 1980 to 2010 .
Problem 83
\(y=5+x, y=x^{2}, x=0\), and \(x=1\)
Problem 84
The following problems consider the historic average cost per gigabyte of RAM on a computer. $$\begin{array}{|l|l|} \hline \text { Year } & \text { 5-Year Change (\$) } \\ \hline 1980 & 0 \\ \hline 1985 & -5,468,750 \\ \hline 1990 & -755,495 \\ \hline 1995 & -73,005 \\ \hline 2000 & -29,768 \\ \hline 2005 & -918 \\ \hline 2010 & -177 \\ \hline \end{array}$$ The velocity of a bullet from a rifle can be approximated by \(v(t)=6400 t^{2}-6505 t+2686\), where \(t\) is seconds after the shot and \(v\) is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: \(0 \leq t \leq 0.5\). What is the total distance the bullet travels in \(0.5 \mathrm{sec} ?\)
