Pre Cal

Short Answer: Type your answer below each question. Show your work.

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. 

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) =  {33-(10.45+10sqrtv-v)(33-t)/33-1.5958(33-t)

if 0 <=v<1.79

if 1.79<=v<20

if v>=20

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.) Show your work.

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. Show your work.

(c)           Find the y-intercept. Show your work.

f(x) = x^2(x + 2)

(a).

(b).

(c).

6.  Verify the identity. Show your work.

(1 + tan^2u)(1 - sin^2u) = 1 

7. Verify the identity. Show your work.

cot^2x + csc^2x = 2csc^2x - 1 

8.  Verify the identity. Show your work.

1 + sec^2xsin^2x = sec^2x 

9.  Verify the identity. Show your work.

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

12.  Find the vertical asymptotes, if any, of the graph of the rational function. Show your work.

f(x) = x-4/x(x-4)

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? Show your work.

14.  Find the specified vector or scalar. Show your work.

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

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

cot(-5pi/4)

16.

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

6x^2 + 12x + 5 = 0

Give your answers in exact form. Show your work.

17.  Use the compound interest formulas A = Pe^pi and A = P(1+r/n)^nt 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? Show your work. 

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 describe the graph the given function.

h(x) = 2 | x | + 2

20.  Find the reference angle for the given angle. Show your work.

-404°