91影视

By the method used to obtain (12.5)[which is the series(13.1)below], verify each of the other series (13.2)to (13.5)below.

In a water purification process, one-nth of the impurity is removed in the first stage. In each succeeding stage, the amount of impurity removed is one-nth of that removed in the preceding stage. Show that if$\mathit{n}\mathbf{=}\mathbf{2}$, the water can be made as pure as you like, but that$\mathit{n}\mathbf{=}\mathbf{2}$ if, at least one-half of the impurity will remain no matter how many stages are used.

Find the Lagrangian and Lagrange's equations for a simple pendulum (Problem $\mathbf{4}$) if the cord is replaced by a spring with spring constant $\mathbf{k}$. Hint: If the unstretched spring length is ${\mathbf{r}}_{\mathbf{8}}$, and the polar coordinates of the mass $\mathbf{m}$are $\left(r,胃\right)$, the potential energy of the spring is $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{k}{\left(r-r_0\right)}^{\mathbf{2}}$.

Connect the midpoints of the sides of an equilateral triangle to form 4 smaller equilateral triangles. Leave the middle small triangle blank, but for each of the other 3 small triangles, draw lines connecting the midpoints of the sides to create 4 tiny triangles. Again leave each middle tiny triangle blank and draw the lines to divide the others into 4 parts. Find the infinite series for the total area left blank if this process is continued indefinitely. (Suggestion: Let the area of the original triangle be 1; then the area of the first blank triangle is 1/4.) Sum the series to find the total area left blank. Is the answer what you expect? Hint: What is the 鈥渁rea鈥� of a straight line? (Comment: You have constructed a fractal called the Sierpinski gasket. A fractal has the property that a magnified view of a small part of it looks very much like the original.)

In the bouncing ball example above, find the height of the tenth rebound, and the distance traveled by the ball after it touches the ground the tenth time. Compare this distance with the total distance traveled.

Prove theorem $\mathbf{14}\mathbf{.}\mathbf{3}$. Hint: Group the terms in the error as role="math" localid="1657423688910" $\mathbf{(}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{2}}\mathbf{)}\mathbf{+}\mathbf{(}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{3}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{4}}\mathbf{)}\mathbf{+}\mathbf{鈰�}$to show that the error has the same sign as role="math" localid="1657423950271" ${\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{.}$Then group them asrole="math" localid="1657423791335" ${\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{}\mathbf{+}\mathbf{(}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{2}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{3}}\mathbf{)}\mathbf{+}\mathbf{(}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{4}}\mathbf{+}{\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{5}}\mathbf{)}\mathbf{+}\mathbf{鈰�}}$to show that the error has magnitude less than$\mathbf{\left|}{\mathit{a}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{\right|}\mathbf{.}$

Find the following limits using Maclaurin series and check your results by computer. Hint: First combine the fractions. Then find the first term of the denominator series and the first term of the numerator series.

$\mathit{a}\mathbf{)}\mathbf{}\underset{\mathbf{x}\mathbf{鈫�}\mathbf{0}}{\mathbf{l}\mathbf{i}\mathbf{m}}\left(\frac{1}{x}-\frac{1}{e^x-1}\right)$

role="math" localid="1662640763230" $\mathit{b}\mathbf{)}\mathbf{鈥�}\underset{\mathbf{x}\mathbf{鈫�}\mathbf{0}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathbf{(}\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{-}\frac{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{x}}{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{x}}\mathbf{)}$

$\mathit{c}\mathbf{)}\mathbf{}\underset{\mathbf{x}\mathbf{鈫�}\mathbf{0}}{\mathbf{lim}}\left(cs{c}^{2}x-\frac{1}{x^2}\right)$

The velocity Vof electrons from a high energy accelerator is very near the velocity Cof light. Given the voltage Vof the accelerator, we often want to calculate the ratio v/c. The relativistic formula for this calculation is (approximately, for V>>1)

$\frac{\mathbf{v}}{\mathbf{c}}\mathbf{=}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{\left(}\frac{\mathbf{0}\mathbf{.}\mathbf{511}}{\mathbf{V}}\mathbf{\right)}}^{\mathbf{2}}}$, V=Number of million volts

Use two terms of the binomial series (13.5) to find 1-v/cin terms of V. Use your result to find 1-v/cfor the following values of V. Caution: V= the number of millionvolts.

(a)V =100 million volts

(b)V =500 million volts

(c)V =25,000 million volts

(d)V = 100 gigavolts ($\mathbf{100}\mathbf{脳}\mathbf{109}\mathbf{}\mathit{v}\mathit{o}\mathit{l}\mathit{t}\mathit{s}\mathbf{=}\mathbf{105}\mathbf{}\mathit{m}\mathit{i}\mathit{l}\mathit{l}\mathit{i}\mathit{o}\mathit{n}\mathbf{}\mathit{v}\mathit{o}\mathit{l}\mathit{t}\mathit{s}$)

Show that if $\mathit{p}$is a positive integer, then $\left(\frac{p}{n}\right)\mathbf{=}\mathbf{0}$ when $\mathit{n}\mathbf{>}\mathit{p}$,so $\mathbf{(}\mathbf{1}\mathbf{+}\mathit{x}{\mathbf{)}}^{\mathbf{p}}\mathbf{=}\mathbf{鈭�}\left(\begin{array}{l}p\\ n\end{array}\right){\mathit{x}}^{\mathbf{n}}$is just a sum of $\mathit{p}\mathbf{+}\mathbf{1}$terms, from $\mathit{n}\mathbf{=}\mathbf{0}$to $\mathit{n}\mathbf{=}\mathit{p}$. For example, ${\mathbf{(}\mathbf{1}\mathbf{+}\mathit{x}\mathbf{)}}^{\mathbf{2}}$has $\mathbf{3}$terms, ${\mathbf{(}\mathbf{1}\mathbf{+}\mathit{x}\mathbf{)}}^{\mathbf{3}}$has $\mathbf{4}$terms, etc. This is just the familiar binomial theorem.

Find the work done by the force is $\mathit{F}\mathbf{=}\left(2xy-3\right)\mathit{i}\mathbf{+}{\mathit{x}}^{\mathbf{2}}\mathit{j}$ in moving an object from (1,0) to (0,1) long each of the three paths shown:

(a) straight line,

(b) circular arc,

(c) along lines parallel to the axes.