The thin lens equation in physics is

$$\frac { 1 } { s } + \frac { 1 } { S } = \frac { 1 } { f }$$

where s is the object distance from the lens, S is the image distance from the lens, and f is the focal length of the lens. Suppose that a certain lens has a focal length of 6 cm and that an object is moving toward the lens at the rate of 2 cm/s. How fast is the image distance changing at the instant when the object is 10 cm from the lens? Is the image moving away from the lens or toward the lens?

Fill in the blanks. If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, then the _______ of f(x) as x approaches c is L.

Compute the derivative of the following functions.

$$f ( x ) = 15 e ^ { 3 x }$$

Simplify. Express your answer in lowest terms. $8 \frac { 1 } { 2 } \div 2 \frac { 1 } { 8 }$

Use an identity to solve each equation on the interval

$$[ 0,2 \pi )$$

$$\cos 2 x + \cos x + 1 = 0$$

Graph each equation.

Solve each problem. Round to the nearest tenth, if necessary. Use 3.14 for

$$\pi.$$

What is the length of a rectangle with width 10 in. and area

$$45\ in.^2?$$

Given that ΔABC ≅ ΔDEF, m∠ A = 70°, m∠ B = 60°, m∠ C = 50°, m∠ D = (3x + 10)°, m∠ E = (y/3 + 20)°, and m∠ F = (z² + 14)°, find the values of x, y, and z.

In Gulliver's Travels, by Jonathan Swift, Gulliver first traveled to Lilliput. The Lilliputian average height was one twelfth of Gulliver's height. a. How many Lilliputian coats could be made from the material in Gulliver's coat? b. How many Lilliputian meals would be needed to make a meal for Gulliver?

What is an even function? An odd function? What symmetry properties do the graphs of such functions have? What advantage can we take of this? Give an example of a function that is neither even nor odd.

Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Which statements are true? For a given confidence level, halving the margin of error requires a sample twice as large.

State the amplitude, period, and phase shift for each function. Then graph the function.

$$y = \tan \frac { 1 } { 2 } \left( \theta + 30 ^ { \circ } \right)$$

Find the mechanical energy of a block-spring system with a spring constant of 1.8 N/cm and an amplitude of 2.4 cm.

Determine whether the given binomial is a factor of the polynomial P(x).

$$( x - 1 ) ; P ( x ) = 4 x ^ { 4 } - 5 x ^ { 2 } + 3 x – 2$$