Pre-Calculus Midterm Exam Score: ______ / ______ Name 273179

Pre- Calculus Midterm Exam Score: ______ / ______ Name: ____________________________

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 cot= cos/ sin sec= 1/cos cos/sin1/cos 1/sin x 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. 8.8+250..715+8.8 25.52 What is the charge for using 45 therms in one month? Show your work. 8.8+250.6686+200..52+17..694 Construct a function that gives the monthly charge C for x therms of gas. C(x)=8.8+0.6686x If 0.515= 8.8 + 250. 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.) Show your work. 6.0c 4 Complete the following: (a) Use the Leading Coefficient Test to determine the graph's end behavior. The leading coefficient is positive x->infinity (b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept. 0 with multiplicity 2 and -2 with multiplicity 1 this it touches at 0 and goes through at 2 (c) Find the y-intercept. Y intercept= the values you get when x=0, therefore the y-intercept is 0 f(x) = x2(x + 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. Logarithmic function Number of Homes Built in a Town by Year 6 Verify the identity. Show your work. (1 + tan2u)(1 - sin2u) = 1 Sec^2ucos^2u=1/cos^2ucos^2u 7 Verify the identity . Show your work. cot2x + csc2x = 2csc2x – 1 Csc x is cosecx Cot^2x+1=cosec^2x Lhs=cot^2x+cosec^2x+cosec^2x Cot^2x+cosec^x=1+2cosec^2x 8 Verify the identity. Show your work. 1 + sec2xsin2x = sec2x Sec^2x=1/cos^2x Sec^2xsin^x=sin^2x/cos^2x=tan^2x 1+tan^2x=sex^2x 9 Verify the identity. Show your work. cos(α - β) - cos(α + β) = 2 sin α sin β cos(a-b)=cos(a)cos(b)+sin(a)sin(b) cos(a+b)=cos(a)cos(b)-sin(a)sin(b) cos(a-b)-cos(a+b)=(cos(a)cos(b)+sin(a)sin(b))-(cos(a)cos(b)-sin(a)sin(b)) cos(a-b)-cos(a+b)=2sin(a)sin(b) 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. A=(max-min)/2 A=2.14 B=(max+min)/2 B=6.05 W=2pi/Twhere t is the period W=2pi/t W=2pi/12 W=pi/6 Y=2.14sin((pi/6)x-phi)+6.05 6.21=2.14sin((pi/6)(4)-phi)+6.05 6.21=2.14sin(2pi/3-phi)+6.05 6.21-6.05=2.14sin(2pi/3-phi) 0.16=2.14sin(2pi/3-phi) 0.16/2.14=sin(2pi/3-phi) Sin(2pi/3-phi)=0.pi/3-phi=arcsin(0.pi/3-phi=4. -phi=4.pi/3 Phi=-2. . 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? Explain No, A vertical line intersects the relation more than once 12. Find the vertical asymptotes, if any, of the graph of the rational function. Show your work. f(x) = vertical asymptotes: f(x)=(x-4)/(x(x-4)) f(x)=x x=0 or x=. 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. A=140=118e^0.024t 0.024t=ln(1.185441) T=ln(1.185441)/0.024 T=0./0.024=7.12 years The populations would reach 140,000 sometime in . Find the specified vector or scalar. Show your work. u = -4i + 1j and v = 4i + 1j; Find . U+v where u=-4i+j,v=4i+j 2j 15. Find the exact value of the trigonometric function. Do not use a calculator. 180=pl radians 90=pl/2 radians 45=pl/4 radians (-5*=. Find the x-intercepts (if any) for the graph of the quadratic function. 6x2 + 12x + 5 = 0 Give your answers in exact form. Show your work. -12-squareroot24/12=-1.+squareroot24/12= -0.. Use the compound interest formulas A = Pert 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? Show your work. Investment= 11000 R=6.25=0.0625 T=10 For the interest Investment= 11000 R=6.3=o.o63 T=10 N= periods, semiannual means twice a year so periods only invested at 6.3%compinded semiannually over 10 years yields the greater return 18. Find functions f and g so that h(x) = (f ∘ g)(x). h(x) = (6x - 14)8 H(x)=(6x-14)^8 F(x)=x^8,g(x)=6x-.

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 F(2)=l2l=2 2lxl. Find the reference angle for the given angle. Show your work. -404° Reference angle is 44* and In the 4th quadrant Score: ______ / ______ 1 A change to society due to change in values, behaviors, etc. In order for social change to occur, there must be people such as Myles Horton whom act as this driving force of change. A behavior that leads towards organized action and change. These interactions involves others in society (large groups or other individuals). In Chapters 1-4 of The Long Haul, you will see that Myles Horton participated in several social actions in which he promoted social change. You will focus upon one of these social actions this week in order to write a short summary of how he used the four key components of: Story, Shared Story, Risk-taking, and collective action to promote social change. An account of Myles Horton’s past experiences and events that evolve into something more. How has Myles Horton’s story transcended to others? What does his story teach others? An act of doing something which involves risk in order to achieve a goal or objective. Action that is taken by a group of people whom have a common goal and objective. Through the combining of each individuals strengths and knowledge, this group of people work to achieve a common goal that is shared by all.

Paper For Above instruction

The provided material encompasses a wide range of mathematical topics, including algebraic identities, functions, transformations, logarithmic and exponential modeling, and real-world application problems. These problems require verification of identities, construction and analysis of functions, and interpretation of data within mathematical contexts. Additionally, the content ventures into social change, using Myles Horton’s activism as a case study to illustrate the power of stories, shared experiences, risk-taking, and collective action to foster societal transformation.

Mathematical Concepts and their Applications

Many of the problems involve verifying identities using fundamental trigonometric and algebraic properties. For example, the identity (1 + tan^2 u)(1 - sin^2 u) = 1 demonstrates the deep interconnectedness between tangent, sine, and cosine functions, rooted in Pythagorean identities. Such verification requires careful algebraic manipulation and understanding of basic identities, which are foundational in higher-level mathematics and essential for simplifying complex expressions.

Constructing functions based on real-world data exemplifies the practical application of mathematics. For example, the gas company’s billing model combines fixed and variable components, which aligns with linear and piecewise functions. Similarly, modeling wind chill effects involves the use of specific formulas linking wind speed and temperature to perceived coldness—highlighting the importance of understanding physical phenomena through mathematical expressions. Accurately calculating wind chill, population growth, or even financial investments relies on applying formulas like exponential growth models and compound interest principles, which are vital tools in economic and environmental sciences.

Function Behavior and Graphing

Analyzing a polynomial’s end behavior and roots provides insight into the graphing process. The Leading Coefficient Test helps determine the end behaviors of the polynomial function, which is positive as x approaches infinity, indicating the function rises indefinitely. Finding x-intercepts and y-intercepts involves solving equations and understanding multiplicity—such as a root with multiplicity 2 touching the x-axis and a simple root crossing through it. These skills are fundamental in graph interpretation, which is essential for scientific modeling and data analysis.

Trigonometry and Data Modeling

Applying trigonometric identities to simplify expressions enhances understanding of wave phenomena and oscillations, which are ubiquitous in physics and engineering. For example, the identity cos(α - β) - cos(α + β) = 2 sin α sin β plays a role in wave interference and signal processing. Moreover, modeling natural phenomena like monthly precipitation with sinusoidal functions demonstrates the usefulness of periodic functions in representing cyclical data—such as seasonal weather patterns—highlighting the importance of understanding amplitude, period, phase shift, and vertical shift.

Function Composition and Transformations

Combining functions through composition, as seen with h(x) = (f ◦ g)(x), enables complex transformations and modeling scenarios. For instance, defining f and g explicitly allows for the manipulation of data or functions to fit specific patterns, essential in advanced mathematics, computer science, and engineering applications. Understanding transformations of the absolute value function further exemplifies how basic functions can be shifted, stretched, or compressed to model different scenarios accurately.

Analyzing Society and Social Movements

Beyond mathematics, the discussion extends into the realm of social change, exemplified by Myles Horton's activism. His use of storytelling, shared experiences, willingness to take risks, and collective action underscores the profound impact that individual and group efforts can have on societal transformation. Horton’s example illustrates how personal narratives can inspire others, foster shared purpose, and mobilize collective efforts toward social justice. His story transcended personal history, becoming a catalyst for broader societal awareness and movement, emphasizing the importance of leadership, community engagement, and resilience in driving change.

Conclusion

The integration of mathematical principles with real-world problem-solving and societal understanding highlights the interconnectedness of these domains. Mastery of algebra, trigonometry, and modeling techniques is essential for interpreting data, designing solutions, and understanding natural phenomena. Simultaneously, recognizing the power of stories, risk-taking, and collective action in social movements demonstrates how individual efforts can inspire societal change. Both aspects rely on understanding, interpretation, and the courage to act—core elements in advancing knowledge and fostering a better society.

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