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8 problems match “logarithms”. Clear

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1. Per-pass speed loss in aerobraking ●●○○○ 20 ptsA spacecraft approaches a planet at 18000 m/s and must slow to 8000 m/s over exactly 50 orbital passes. On each pass it keeps the same fraction of its speed. What percentage of its speed must it lose on each pass? Round to two decimal places.
2. Simplify expression with imaginary cosine and sine ●●●○○ 50 ptsSimplify $2\cos^2(i\log 2009)+i\sin(i\log 4036081)$, where $\log$ is the natural logarithm. Give your answer as a decimal rounded to eight decimal places.
3. Limit of xth root of factorial over x ●●●○○ 50 ptsFind the value of $\lim_{n\to\infty} \frac{\sqrt[n]{n!}}{n}$, where $n$ is a positive integer. Round to two decimal places.
4. Area of a parallelogram on a log graph ●●●○○ 50 ptsGiven points $A(3,1)$ and $B\left(\frac{5}{3},2\right)$. The four vertices of parallelogram $ABCD$ all lie on the graph of $f(x)=\log_{2}\frac{ax+b}{x-1}$. Find the area of parallelogram $ABCD$. Round to two decimal places.
5. Roots of a symmetric odd function ●●●○○ 50 ptsLet $y=f(x)$ be an odd function defined on $\mathbf{R}$ whose graph is symmetric about the line $x=1$. When $0<x\leq 1$, $f(x)=\log_3 x$. Find the sum of all real roots of the equation $f(x)=-\frac{1}{3}+f(0)$ in the interval $(0,10)$.
6. Limit of nth root of factorial over n ●●●○○ 50 ptsFind $\lim_{n\to\infty} \frac{(n!)^{1/n}}{n}$. Round to three decimal places.
7. Complex line integral of 1/(1+z^2) ●●●○○ 50 ptsEvaluate $\int_1^{1+i}\frac{1}{1+z^2}\,dz$ along the straight line segment from $z=1$ to $z=1+i$. Give the real part and the imaginary part of the result separately. Round each to two decimal places.
8. Intersections of a log and a sine graph ●●●●○ 100 ptsHow many real numbers $x$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$?