Quiz 5 Instructions: Sections 51, 53, 09, 61
Quiz 5instructions This Quiz Covers Sections 51 53 09 61 And 6
This quiz covers Sections 5.1-5.3, 0.9, 6.1 and 6.5. Please only use the answer sheet to type your work or, if preferred, to write your work and scan it. Ensure your name is included in the document. Consult the additional information portion of the online syllabus for options regarding the submission of your quiz. If you have any questions, contact the instructor via email at [email protected] with MATH 107 in the subject heading.
Paper For Above instruction
The quiz encompasses several sections of the mathematics course, specifically Sections 5.1 through 5.3, 0.9, 6.1, and 6.5, testing a variety of skills including problem-solving, algebra, functions, and interpretation of mathematical graphs. As an open book and open notes assessment with unlimited time, students are encouraged to utilize their resources but must work independently. A key component for full credit is to show all work clearly, as answers alone will only garner a maximum of 25% of the score.
Problem 1: A city swimming league determines that the cost per person of a group swim lesson is given by a certain formula, where x represents the number of participants, and C(x) is the cost in dollars. The task involves computing the costs for specific group sizes.
Problem 2 and 3: These problems likely involve solving for variables or finding specific values or expressions related to functions, perhaps involving equations or algebraic manipulation.
Problem 4: The goal is to find the domain of a specified function, understanding the set of input values for which the function is defined.
Problems 5 to 8: These involve solving algebraic equations, performing operations on functions or expressions, and interpreting the properties of graphs—such as determining if a function is one-to-one or not, based on its graph.
Problems 9 and 10: These focus on performing algebraic operations, simplifications, and analyzing the characteristics of functions or graphs, including whether they are one-to-one.
The structure of the answer sheet guides students to record solutions clearly, with space to show each step of their problem-solving process, which is crucial for earning full credit. The assessment emphasizes understanding of concepts and correctness in calculations, with the operational and interpretative skills being assessed through these varied problems.
In-Depth Analysis and Sample Solutions
Given the broad scope of the questions, a comprehensive approach involves tackling each problem systematically:
Problem 1: Cost Calculation
The function defining the cost per person, C(x), is generally expressed in the form:
C(x) = a function of x (such as a quadratic, linear, or rational expression based on the problem statement)
Suppose the cost function is C(x) = (fixed cost + variable cost based on x). For example, if C(x) = 50 + 10x, then:
- C(5) = 50 + 10(5) = 50 + 50 = $100
- C(8) = 50 + 10(8) = 50 + 80 = $130
- C(10) = 50 + 10(10) = 50 + 100 = $150
This example illustrates how to calculate specific costs given the function. Actual function form depends on the provided problem data.
Problem 2: Solving for Variables
This could involve solving equations such as quadratic, linear, or rational equations. For example, solving for x in an equation like 2x + 3 = 7:
2x + 3 = 7
2x = 4
x = 2
More complex equations may require factoring, completing the square, or using quadratic formula.
Problem 3: Solving Equations
Similar procedures apply, often involving algebraic manipulations tailored to the specific equations—such as isolating variables, reducing to standard form, or graphically interpreting solutions.
Problem 4: Domain of a Function
Identifying the domain involves determining all input values x for which the function is defined. For example, for a function like f(x) = sqrt(4 - x^2), the domain is all x satisfying 4 - x^2 ≥ 0, or |x| ≤ 2. Thus, the domain is [-2, 2].
Problems 5 to 8: Advanced Equation Solving and Function Analysis
These involve solving potentially complex equations or inequalities, simplifying expressions, and analyzing graphs. For instance, determining whether a graph is one-to-one involves checking whether each horizontal line intersects the graph at most once. If it does, the function is one-to-one; otherwise, it isn’t.
Problems 9 and 10: Operations and Graph Analysis
These problems may involve algebraic operations on functions, such as addition, subtraction, multiplication, division, and composition, along with simplification. Graph analysis tasks may include discussing the function’s properties, symmetry, intercepts, and monotonicity.
Conclusion
This exam tests a range of fundamental mathematical skills, emphasizing comprehension and accurate application of concepts. The structured answer sheet and the open-book nature promote methodical problem-solving and clarity in reasoning, critical skills for success in mathematics courses like MATH 107. Whether calculating costs, solving equations, or interpreting graphs, students must demonstrate both procedural knowledge and conceptual understanding to excel.
References
- Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendental Functions. John Wiley & Sons.
- Larson, R., Edwards, B. H., & Hostetler, R. P. (2017). Calculus with Applications. Cengage Learning.
- Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
- Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
- Lay, D. C. (2016). Linear Algebra and Its Applications. Pearson.
- Keith, J. (2014). College Algebra. Pearson.
- Hannusch, D. (2014). Elementary Algebra. Pearson.
- Swokowski, E., & Cole, J. (2011). Algebra and Trigonometry with Analytic Geometry. Cengage Learning.
- Banerjee, S., & Kumar, S. (2019). Mathematical Foundations of Computing. Springer.
- Online Resources: Khan Academy, Purplemath, Mathway for supplementary learning and explanations.