The period of a pendulum is given by $T = 4 \sqrt { \frac { \ell } { g } } \int _ { 0 } ^ { \pi / 2 } \frac { d \theta } { \sqrt { 1 - k ^ { 2 } \sin ^ { 2 } \theta } } = 4 \sqrt { \frac { \ell } { g } } F ( k )$, where $\ell$ is the length of the pendulum, $g \approx 9.8 \mathrm { m } / \mathrm { s } ^ { 2 }$ is the acceleration due to gravity, k=sin $\left( \theta _ { 0 } / 2 \right)$, and $\theta _ { 0 }$ is the initial angular displacement of the pendulum (in radians). The integral in this formula F(k) is called an elliptic integral, and it cannot be evaluated analytically.

Approximate F(0.1) by expanding the integrand in a Taylor (binomial) series and integrating term by term.

Draw a valid Lewis structure for each species:

$$\mathrm { a. } \mathrm { CH } _ { 3 } \mathrm { CH } _ { 3 }\quad \mathrm { b } . \mathrm { CH } _ { 5 } \mathrm { N } \quad \mathrm { c } . \mathrm { CH } _ { 3 } ^ { - }\quad \mathrm { d } . \mathrm { CH } _ { 3 } \mathrm { Cl }$$

In an auto race, a car with an average speed of 200 mi/h takes an average of 31.5 s to complete one lap of the track. How many seconds does it take the car to complete one lap at an average speed of 210 mi/h?

Polar coordinates of a point are given. Use a graphing utility to find the rectangular coordinates of each point to three decimal places.

$$\left( 4 , \frac { 2 \pi } { 3 } \right)$$

Write an equation of the line that passes through the given point and has the given slope. (3, 1), slope 2

Write the decimal number one hundred twenty-three thousandths.

The percent contribution currently of natural gas to electricity generation in the United States is ______.

Given

$$\overline { A B }$$

with endpoints A(0, 5) and B(5, 5), let

$$\overline { A ^ { \prime } B ^ { \prime } }$$

with endpoints A'(0, 3) and B'(3, 3) be the image of

$$\overline { A B }$$

after a dilation. What is the scale factor of the dilation?

(a) Is there a choice of axes of rotation for a rotating bowling ball and a rotating baseball such that the baseball has a greater rotational inertia? If so, describe the axes. If not, explain why not. (b) Repeat for a revolving bowling ball and a revolving baseball.

Simplify each expression.

$$\frac { \sqrt { 13 } + \sqrt { 10 } } { \sqrt { 13 } - \sqrt { 5 } }$$

The work required to launch an object from the surface of Earth to outer space is given by $W=\int _ { R } ^ { \infty } F ( x ) d x$, where R=6370 km is the approximate radius of Earth, $F(x)=\mathrm{G M m} / x ^ { 2 }$ is the gravitational force between Earth and the object, G is the gravitational constant, M is the mass of Earth, $m$ is the mass of the object, and $\mathrm{GM}=4 \times 10 ^ { 14 } \mathrm { m } ^ { 3 } / \mathrm { s } ^ { 2 }$.

Find the work required to launch an object in terms of $m$.

Name the property illustrated by each equation.

$$( 12 + 5 ) 6 = 12 \cdot 6 + 5 \cdot 6$$

Use symmetry to evaluate the following integrals.

$$\int _ { - 10 } ^ { 10 } \frac { x } { \sqrt { 200 - x ^ { 2 } } } d x$$

State the contrapositive and converse. If today is Tuesday, then we are not in Belgium.