Pre calculus mid term help

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Pre-Calculus Midterm Exam Score: ______ / ______ Name: ____________________________ Student Number: ___________________ Short Answer: Type your answer below each question. Show your work. 1 Verify the identity. Show your work. cot θ ∙ sec θ = csc θ cotΘ=1/tanΘ=cosΘ/senΘ secΘ=1/cosΘ cscΘ=1/senΘ (cosΘ/senΘ)∗ (1/cosΘ)=1/senΘ=cscΘ 2 A gas company has the following rate schedule for natural gas usage in single-family residences: Monthly service charge Per therm service charge 1st 25 therms Over 25 therms $8.80 $0.6686/therm $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. 8.8 + 25*0.6686 16.715+8.8 = 25.52 8.8 + 25*.6686 + 20*.85870 25.52+17.174 = 42.694 C(x) = { 8.8 + 0.6686x if 0 ≤ x ≤ 25 { 25.515 + 0.85870(x - 25) if x > 2tex 1 Pre-Calculus Midterm Exam 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 𝑡 W(t) = {33 − (10.45+10√𝑣−𝑣)(33−𝑡) 2204 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. v=12 t=15 33-((10.45+10*sqrt(12)-12)(33-15))/2204 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) = x2(x + 2) (a). End behavior is decreasing to the left, and increasing to the right. (b). ZEROS: at 0 and -2. Those are the x-intercepts. This by using your function in its factored form for f(x)=0. (c). y-intercept: x=0, what is f(0)? 2 Pre-Calculus Midterm Exam 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 LOGARITHMIC FUNCTION 6 Verify the identity. Show your work. (1 + tan2u)(1 - sin2u) = 1 (1 + tan^2u)(1 - sin^2u) = 1 (1 + sin^2u/cos^2u)(1 - sin^2u) = ((cos^2u + sin^2u)/(cos^2u)) (1 - sin^2u) = (1/cos^2u)(1 - sin^2u) = Notice that 1 - sin^2u is the same as cos^2u (1/cos^2u)(cos^2u) = cos^2u / cos^2u = 1 7 Verify the identity. Show your work. cot2x + csc2x = 2csc2x - 1 3 Pre-Calculus Midterm Exam cos²x 1   2 = ▬▬▬ + ▬▬▬ = ▬▬▬ − 1  sin²x     sin²x    sin²x multiply both sides by sin²x cos²x + 1 = 2 − sin²x  cos²x + sin²x = 1 8 Verify the identity. Show your work. 1 + sec2xsin2x = sec2x 1+ Sec^2 X. Sin^2 X = 1 + 1/(cos^2 X). Sin^2 X = 1 + Sin^2 X/cos^2 X = 1+ tan^2 X =Sec^2 X 4 Pre-Calculus Midterm Exam 9 Verify the identity. Show your work. cos(α - β) - cos(α + β) = 2 sin α sin β cos(α-β) = cos(α)∙cos(β) + sin(α)∙sin(β) cos(α+β) = cos(α)∙cos(β) - sin(α)∙sin(β) cos(α-β) - cos(α+β) = (cos(α)∙cos(β) + sin(α)∙sin(β)) - (cos(α)∙cos(β) - sin(α)∙sin(β)) cos(α-β) - cos(α+β) = 2∙sin(α)∙sin(β) cot(θ) ∙ sec(θ) = cos(θ) 1 ---------- ∙ ---------- = sin(θ) cos(θ) 1 -------- = sin(θ) 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. Show your work. 5 Pre-Calculus Midterm Exam 10 Answer The plot appears to maximize at around 8.25 in, and might minimize around 3.75 in, meaning the amplitude A = ½(8.25 - 3.75)in = 2.25 in with an average around ½(8.25 + 3.75)in = 6 in It looks like we need a phase shift as well: the maximum occurs at t = 7, and the minimum at t = 1. (If it wasn't phase-shifted, the minimum would be at t = 0 and t = 12.) So, f(t) = 6in - 2.25in*cos(π(t-1)/6) . 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. 6 Pre-Calculus Midterm Exam Does the graph represent a function? Explain 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 = 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. 7 Pre-Calculus Midterm Exam 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 (− 5𝜋 ) 4 16. Find the x-intercepts (if any) for the graph of the quadratic function. 8 Pre-Calculus Midterm Exam 6x2 + 12x + 5 = 0 Give your answers in exact form. Show your work. 17. 𝑟 𝑛𝑡 Use the compound interest formulas A = Pert and A = P(1 + 𝑛) 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 9 Pre-Calculus Midterm Exam 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° Score: ______ / ______ 10

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