Quiz 2 Professor Ms. Chowdhury Math 107

Quiz 2 Professor Ms Chowdhury Math 107

The quiz is worth 35 points. There are 10 multiple choice questions (2 points each) and 5 short answer problems (3 points each). The quiz is open book and open notes. You may refer to your textbook, notes, and online classroom materials, but you may not consult anyone else. Show all your work to receive full credit. Answers should be typed or scanned handwritten and include your name. Submit your completed quiz by the due date posted in the course schedule. Contact the instructor via email if you have questions.

Paper For Above instruction

This paper addresses various mathematical problems from a quiz designed for a math course, specifically focusing on algebraic functions, graphing, and applications. The first part covers multiple-choice questions involving evaluating functions, composing functions, and calculating distances, centers, and radii of circles. The second part involves short-answer exercises such as graphing equations, solving practical problems related to cost and revenue, and modeling investments using linear equations.

Introduction

The quiz in question assesses students' understanding of fundamental algebraic concepts, including function evaluation, composition, graphing, and application in real-world contexts. These concepts are essential in developing a thorough understanding of algebra and its applications in fields such as economics, engineering, and social sciences. Correctly solving these problems requires knowledge of function operations, graphing techniques, and interpreting algebraic models.

Function Evaluation and Composition

The first set of questions focuses on evaluating functions at specific points and understanding composite functions. For example, evaluating f(x) = x + 13 at x = -4 yields a straightforward calculation: f(-4) = -4 + 13 = 9. This underscores the importance of substituting values accurately. Additionally, understanding how to determine functions f and g such that their composition h(x) = (f g)(x) is essential. For instance, if h(x) = 3/8x + 6, then f and g must be defined in such a way that their combination produces h(x).

Graphing and Geometric Problems

Graphing equations such as circles involves identifying centers and radii from their standard forms. For example, the circle (x + 7)^2 + (y + 8)^2 = 49 has center at (-7, -8) and radius 7, since the radius squared is 49. Calculating distances between points employs the distance formula: d = √[(x₂ - x₁)^2 + (y₂ - y₁)^2], which yields the exact length between two points, such as (6, 4) and (-3, -3), resulting in a distance of 7√2 or approximately 9.899.

Function Domains and Compositions

Understanding the domain of functions like f(x) = x/(x^2 + 18) involves identifying values where the denominator is not zero. In this case, the domain excludes x = 0 because it makes the denominator zero. For compositions, such as (g f)(x) with given functions, substitution and simplification lead to new functions that combine their properties, demonstrating operations like composition and algebraic manipulation.

Applications in Cost, Revenue, and Investment Modeling

Addressing real-world problems, the quiz includes calculating costs of producing a specific number of jars through functions like C(x) = 0.7x + 38,000, which requires substitution to find the total cost at a given production volume. Similarly, analyzing profit by subtracting cost from revenue, such as (R - C)(948), provides insight into profitability. The weekly car rental cost as a function of miles driven exemplifies translating problem statements into algebraic models: f(x) = 200 + 0.45x.

Finally, exponential growth modeling of investments using linear equations captures the trend over years. For example, a linear model y = mx + b relates the investment values at different years, translating real-world financial growth into mathematical expressions.

Conclusion

The comprehensive analysis of the quiz problems illustrates the application of algebraic principles to evaluate functions, graph geometric figures, determine distances, and model economic scenarios. Mastery of these topics enables students to approach diverse problems analytically, applying mathematical techniques to interpret real-world data and phenomena effectively. Developing proficiency in these areas is fundamental for advancing in mathematics and related disciplines.

References

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  • Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
  • Swokowski, E. W., & Cole, J. A. (2011). Algebra and Trigonometry with Analytic Geometry. Cengage Learning.
  • Blitzer, R. (2018). Algebra and Trigonometry. Pearson.
  • Princeton Review. (2020). Algebra Workbook for Dummies. Wiley.
  • Knuth, D. E. (1997). Mathematics and Computer Science. American Mathematical Society.
  • Bateman, D. R. (2014). Applied Mathematics: Methods, Models, and Computing. McGraw-Hill.
  • Agarwal, R. P. (2012). Differential Equations & Boundary Value Problems. Springer.
  • Harrison, M. (2015). Mathematical Modeling: A Chemical Engineering Perspective. CRC Press.
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