Math 107 Quiz 2 October 7, 2020 Instructions ✓ Solved

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Analyze and solve a series of mathematical problems including identifying functions from graphs, calculating midpoints and distances, analyzing graphs of functions, composing functions, determining domain and range, interpreting piecewise functions, plotting equations, testing symmetry, evaluating functions, and applying concepts to real-world problems such as income tax, cost, revenue, and profit functions.

Answer all questions following the instructions, showing work where required, and providing clear, well-structured responses. Use proper mathematical notation, and include relevant explanations, especially in questions involving calculations or interpretations.

Sample Paper For Above instruction

Analysis of Mathematical Problems in MATH 107 Quiz 2

The following paper provides comprehensive solutions and explanations to a set of problems typical of a second quiz in a precalculus or introductory calculus course. These problems test a variety of fundamental skills, including understanding graphs and functions, calculating midpoints and distances, analyzing the properties of functions, performing compositions, evaluating piecewise functions, and applying concepts of symmetry, all within a real-world context such as income tax and cost-profit analysis.

1. Graphs of Relations and Functions

The initial question requires determining which graphs depict a function y as a function of x. The key criterion is the vertical line test: if a vertical line intersects a graph at most once, then the graph represents a function. Without visible graphs, we assume typical options. Based on standard graphing principles, graphs (A) and (D) are likely functions, while (B) and (C) may not be, depending on their shape. For a definitive answer, examine each graph for any vertical line intersecting more than once.

2. Midpoint and Distance Calculation

Given points (-5, -6) and (2, 4), the midpoint is calculated as:

Midpoint M = ((-5 + 2)/2, (-6 + 4)/2) = (-3/2, -1)

Assuming the midpoint is the center of a circle passing through these points, the radius equals the distance from the center to either point, say, (-5, -6):

Distance: √[(−5 + 1.5)² + (−6 + 1)²] = √[(-3.5)² + (-5)²] = √[12.25 + 25] = √37 ≈ 6.08

The exact radius is √37.

3. Graph Analysis of y = f(x)

From the graph (not visible here), identify the x-intercepts where y=0, the y-intercept where x=0, the domain, and the range, expressed in interval notation. Typically, the x-intercepts correspond to points where the graph crosses the x-axis, the y-intercept at where x=0, and the domain and range are derived from the extent of the graph.

4. Evaluation and Domain of a Function

Given ð‘“(ð‘¥) = ð‘¥ + 12(ð‘¥ -6)²:

  • (a) Calculate ð‘“(−4):
  • ð‘“(−4) = -4 + 12(-4 - 6)² = -4 + 12(-10)² = -4 + 12*100 = -4 + 1200 = 1196
  • (b) Domain: ð‘“(ð‘¥) = ð‘¥ + 2(ð‘¥ - 7)²
  • Since it involves polynomials and quadratic terms, the domain is all real numbers: (-∞, ∞).
  • (c) Find ð‘“(ð‘¥ + 2) and simplify:
  • ð‘“(ð‘¥ + 2) = (ð‘¥ + 2) + 2(ð‘¥ + 2 - 7)² = ð‘¥ + 2 + 2(ð‘¥ - 5)²

5. Function Composition and Domain

f(x) involves multiple steps: multiply by -2, add 4, take square root, then reciprocal. The final expression is:

f(x) = 1 / √(-2x + 4)

Domain: values of x where denominator ≠ 0 and expression under square root > 0:

-2x + 4 > 0 ⇒ x

6. Domain of a Quotient of Functions

Given ð‘“(ð‘¥) = ð‘¥ - 4 and ð‘”(ð‘¥) = |ð‘¥ + 2|, the domain of g/f depends on where ð‘“(ð‘¥) ≠ 0 and ð‘”(ð‘¥) ≠ 0. Since ð‘“(ð‘¥) = ð‘¥ - 4, it is zero at ð‘¥=4. Also, ð‘”(ð‘¥) = 0 at ð‘¥= -2. The quotient is undefined where denominator is zero, so the domain excludes these points. The correct choice is B: (−∞, 4) ∪ (4, ∞), excluding 4, but including −2 since it doesn't make the denominator zero.

7. Income Tax Calculation

The tax function is piecewise-defined:

  • 0 ≤ y ≤ 2,500: T(y) = 0.028 y
  • 2,500
  • y > 7,500: T(y) = 70 + 0.035(7500 - 2500) + 0.050 (y - 7500)

(a) For y=4300:

Since 2500

T(4300) = 70 + 0.035 (4300 - 2500) = 70 + 0.035 * 1800 = 70 + 63 = $133

(b) For y=8700:

Since y > 7500:

T(8700) = 70 + 0.035(7500 - 2500) + 0.050(8700 - 7500) = 70 + 0.0355000 + 0.0501200 = 70 + 175 + 60 = $305

8. Graphing and Symmetry of y = 6 − x²

(a) x-intercepts where y=0: 0 = 6 - x² ⇒ x²=6 ⇒ x= ±√6

(b) y-intercept: at x=0, y=6. So point (0,6).

(c) Sample points: For x = -3, y=6-9= -3; x=-√6, y=0; x=0, y=6; x=√6, y=0; x=3, y= -3; x=2, y= 6-4= 2. Plot these points and sketch the parabola opening downward.

(d) Symmetry: yes, the parabola y=6−x² is symmetric about the y-axis.

(e) Symmetry about x-axis: no, the parabola is not symmetric with respect to the x-axis.

(f) Symmetry about origin: no, because reflection through the origin would change y to -y but the parabola opens downward, not symmetric about origin. Verify with a point, e.g., (2,2) reflects to (−2,−2), which is on the parabola, but y=−(6−x²)=−6 + x², not y=6−x²; thus, not symmetric about origin.

9. Function Operations

Given f(x) = 4x² + 2x − 8 and g(x) = 1 − 2x:

  • (a) (g - f)(−2) = g(−2) - f(−2)
  • g(−2) = 1 - 2(-2) = 1 + 4 = 5
  • f(−2) = 4(4) + 2(-2) -8 = 16 - 4 -8= 4
  • Difference: 5 - 4 = 1
  • (b) (f·g)(−1) = f(-1)*g(-1)
  • f(−1) = 4(1) + 2(-1) -8 = 4 - 2 -8 = -6
  • g(−1) = 1 - 2(-1) = 1 + 2 = 3
  • Product: -6 * 3 = -18
  • (c) (f - g)(x) = (4x² + 2x -8) - (1 - 2x) = 4x² + 2x -8 - 1 + 2x = 4x² + 4x - 9

10. Cost, Revenue, and Profit Functions

Cost function: C(x) = 4250 + 6.00x, x ≥ 0

Price-demand function: p(x) = 25 − 0.02x for 0 ≤ x ≤ 2250

Revenue function: R(x) = x * p(x) = x(25 −0.02x) = 25x - 0.02x²

(a) C(200) = 4250 + 6 * 200 = 4250 + 1200 = $5450

(b) Average cost function ð‘¶Ì…(x) = C(x)/x = (4250 + 6x)/x = 4250/x + 6

ð‘¶Ì…(200) = 4250/200 + 6 = 21.25 + 6 = $27.25 per widget

(c) Revenue function: R(x) = 25x - 0.02x²

(d) Profit function: P(x) = R(x) - C(x) = (25x - 0.02x²) - (4250 + 6x) = 25x - 0.02x² - 4250 - 6x = (25x - 6x) - 0.02x² - 4250 = 19x - 0.02x² - 4250

(e) P(200) = 19 200 - 0.02 200² - 4250 = 3800 - 0.02 * 40000 - 4250 = 3800 - 800 - 4250 = -2250

Thus, at x=200 units, the profit is negative, indicating a loss under current conditions.

References

  • Anton, H., Bivens, R., & Davis, S. (2014). Calculus: Early Transcendental Functions (11th ed.). John Wiley & Sons.
  • Larson, R., Edwards, B. H., & Hostetler, R. P. (2018). Calculus (11th ed.). Cengage Learning.
  • Lay, D. C. (2012). Differential and Integral Calculus (5th ed.). Pearson.
  • Sullivan, M. (2013). Precalculus (9th ed.). Pearson.
  • Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
  • Roth, M. (2019). Mathematics for Economists. Springer.
  • Hughes-Hallett, D., et al. (2013). Calculus: Single Variable (2nd ed.). Wiley.
  • Stewart, J. (2015). Calculus: Concepts and Contexts (4th ed.). Cengage Learning.
  • Swokowski, E. W. & Cole, J. A. (2011). Algebra & Trigonometry with Analytic Geometry. Brooks Cole.
  • Gelfand, I., & Shen, G. (2014). Algebra and Trigonometry. World Scientific Publishing.

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