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1. Verify the identity. Show your work.
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? Show your work.
What is the charge for using 45 therms in one month? Show your work.
Construct a function that gives the monthly charge C for x therms of gas.
- 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:
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 ºC and a wind speed of 12 meters per second. (Round the answer to one decimal place.) Show your work.
- 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. Show your work.
(c) Find the y-intercept. Show your work.
- For the data set shown by the table,
- Create a scatter plot for the data. (You do not need to submit the scatter plot)
- Use the scatter plot to determine whether an exponential or a logarithmic function is the best choice for modeling the data.
Number of Homes Built in a Town by Year
Year Number of Homes
1985 12
1991 91
1994 145
1997 192
2002 224
(a) Scatter plot:
(b) The graph appears to be leveling out as the year increases beyond the year 2000. This is characteristic of a logarithmic function, where the difference from year to year is diminishing. An exponential curve would have an increasing difference from year to year.
- Verify the identity. Show your work.
(1 + tan2 u)(1 – sin2 u) = 1
- Verify the identity. Show your work.
cot2 x + csc2 x = 2csc2 x – 1
- Verify the identity. Show your work.
1 + sec2 x sin2 x = sec2 x
- Verify the identity. Show your work.
cos(α – β) – cos(α + β) = 2sin α sin β
- The following data represent the normal monthly precipitation for a certain city.
Month, x Normal Monthly Precipitation, inches
January, 1 3.91
February, 2 4.36
March, 3 5.31
April, 4 6.21
May, 5 7.02
June, 6 7.84
July, 7 8.19
August, 8 8.06
September, 9 7.41
October, 10 6.30
November, 11 5.21
December, 12 4.28
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 y = A sin(ω x – ϕ) + B that fits the data. Show your work.
- 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? Explain.
- Find the vertical asymptotes, if any, of the graph of the rational function. Show your work.
- The formula A = 118e0.024t models the population of a particular city, in thousands, t years after 1998. When will the population of the city reach 140 thousand? Show your work.
- Find the specified vector or scalar. Show your work.
- Find the exact value of the trigonometric function. Do not use a calculator.
- Find the x-intercepts (if any) for the graph of the quadratic function:
6x2 + 12x + 5 = 0
- Use the compound interest formulas A = Pert and A = P(1 + r/n)nt to solve.
- Find the function f and g so that h(x) = (f ° g)(x)
- Begin by graphing the standard absolute value function f(x) = |x|. Then use transformations of this graph to describe the graph of the given function.
- Find the reference angle for the given angle. Show your work.
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