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Kyle Taitt CSU Webwork MATH 160 WeBWorK assignment M-FA-1.4 due 09/14/2016 at 11:59pm MDT

Paper For Above instruction

The provided set of questions and data involve various mathematical principles primarily centered on the concepts of continuity, limits, and asymptotic behavior of functions. These problems are designed to assess understanding of the ε-δ definition of continuity, the nature of removable discontinuities, applications of the Intermediate Value Theorem (IVT), limits at specific points, and the analysis of function behavior via graphing and algebraic techniques. This comprehensive review emphasizes the importance of foundational calculus concepts essential for analyzing and understanding the behavior of functions and their properties.

Firstly, the problems related to continuity focus on the precise ε-δ definition. The task is to match functions with appropriate δ choices to demonstrate continuity at x=0. For example, for the function f(x) = x, choosing δ = e suffices because |f(x) - f(0)| = |x|, which can be made less than e by ensuring |x|

Next, the demonstration of removable discontinuities involves establishing points where the function is not initially continuous and then redefining the function at those points to restore continuity. The function involving piecewise definitions with a specific value at x=3 illustrates this, where the limits from both sides at x=3 are equal, indicating no jump or asymptotic behavior. Redefining f(3) at the limit value results in a continuous function, which is fundamental in understanding how functions can be modified for continuity.

The properties of limits and the IVT are central to several questions. For instance, questions ask whether, for the given function f(x) = 3x^3 + 3x^2 + 2, the IVT guarantees a value c in [0,1] such that f(c) = k for certain k values. The IVT states that if a function is continuous on a closed interval and a value k lies between the function’s values at the endpoints, then there exists at least one c within that interval such that f(c) = k. This is extended to questions about the interval where a root must exist based on the values given in the table, requiring identification of the smallest interval where the function transitions from positive to negative or vice versa, indicating the presence of a root.

Various questions analyze the behavior of functions as variables approach infinity or specific points, emphasizing the significance of limits. For example, the limit of the Lorentz contraction L = L₀ √(1 - v^2/c^2) as v approaches c requires understanding that as v approaches c from below, L tends to zero, indicating the object appears contracted to a point. Similarly, the behavior of relativistic mass m = m₀ / √(1 - v^2/c^2) becomes unbounded as v approaches c because the denominator approaches zero, leading to an infinite limit, reflecting relativistic effects.

Graphical analysis problems involve interpreting limit behavior from graphs, such as the limits of functions f and g at various points, understanding the impact of discontinuities, and asymptotic behaviors. Calculations of asymptotes via algebraic division, as in polynomial and rational functions, reinforce the connection between algebraic manipulation and geometric interpretation. For example, dividing a cubic polynomial by a quadratic reveals the slant asymptote and behaviors at infinity, fundamental in the analysis of rational functions.

Finally, conceptual questions about limits and function behavior include understanding what it signifies when limits approach infinity, which in the context of relativity, mirrors the phenomenon of objects approaching the speed of light, or when limits at specific points do not exist, indicating discontinuities or unbounded behavior. The importance of these concepts lies in their application to real-world physics, engineering, and advanced mathematics, illustrating how theoretical principles translate to observable phenomena.

References

  • Anton, H., Bivens, I., & Davis, S. (2013). Calculus: Early Transcendentals. John Wiley & Sons.
  • Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
  • Thomas, G.B., & Finney, R.L. (2002). Calculus and Analytic Geometry. Pearson.
  • Lay, D. C. (2012). Analysis with an Introduction to Proof. Pearson.
  • Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
  • Abbott, S. (2010). Understanding Analysis. Springer.
  • Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
  • Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications. Wiley.
  • Jerison, D. (2013). The Geometry of Function Spaces. American Mathematical Society.
  • Oppenheim, A.V., & Willsky, A.S. (1997). Signals and Systems. Prentice Hall.

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