Calculus

1

Verify the identity.

cot θ ∙ sec θ = csc θ

2

A gas company has the following rate schedule for natural gas usage in single-family residences:

                Monthly service charge                  $8.80

                Per therm service charge

                                1st 25 therms                      $0.6686/therm

                                Over 25 therms                  $0.85870/therm

What is the charge for using 25 therms in one month?

What is the charge for using 45 therms in one month?

Construct a function that gives the monthly charge C for x therms of gas.

3

The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is

W(t) =  

where v represents the wind speed (in meters per second) and t represents the air temperature . Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second. (Round the answer to one decimal place.)

4

Complete the following:

(a)           Use the Leading Coefficient Test to determine the graph’s end behavior.

(b)           Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept.

(c)           Find the y-intercept.

f(x) = x²(x + 2)

(a).

(b).

(c).

5

For the data set shown by the table,

 a. Create a scatter plot for the data. (You do not need to submit the scatter plot)

 b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best choice for modeling the data.

Number of Homes Built in a Town by Year

6

Verify the identity.

(1 + tan²u)(1 – sin²u) = 1

7

Verify the identity.

cot²x + csc²x = 2csc²x – 1

8

Verify the identity.

1 + sec²xsin²x = sec²x

9

Verify the identity.

cos(α – β) – cos(α + β) = 2 sin α sin β

10

The following data represents the normal monthly precipitation for a certain city.

Draw a scatter diagram of the data for one period. (You do not need to submit the scatter diagram). Find the sinusoidal function of the form  that fits the data.

_____11.

The graph below shows the percentage of students enrolled in the College of Engineering at State University. Use the graph to answer the question.

Does the graph represent a function?

_____12.

Find the vertical asymptotes, if any, of the graph of the rational function.

f(x) =

_____13.

The formula A = 118e^0.024t models the population of a particular city, in thousands, t years after 1998. When will the population of the city reach 140 thousand?

_____14.

Find the specified vector or scalar.

u = -4i + 1j and v = 4i + 1j; Find .

_____15.

Find the exact value of the trigonometric function. Do not use a calculator.

_____16.

Find the x-intercepts (if any) for the graph of the quadratic function.

6x² + 12x + 5 = 0

Give your answers in exact form.

_____17.

Use the compound interest formulas A = Pe^rt and A = P to solve.

Suppose that you have $11,000 to invest. Which investment yields the greater return over 10 years: 6.25% compounded continuously or 6.3% compounded semiannually?

_____18.

Find functions f and g so that h(x) = (f ∘ g)(x).

h(x) = (6x – 14)⁸

_____19.

Begin by graphing the standard absolute value function f(x) = | x |. Then use transformations of this graph to graph the given function.

h(x) = 2 | x | + 2

_____20.

Find the reference angle for the given angle.

-404°