Section 103 WebAssign Web Student Assignment

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Analyze the provided mathematical problems involving parametric equations, tangent lines, derivatives, power series, and convergence intervals. The tasks include finding slopes, concavity, tangent line equations, approximating integrals using power series, deriving Taylor series, and determining convergence intervals with endpoint checks.

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The set of problems presented centers around advanced calculus concepts, especially those related to parametric equations, derivatives, Taylor and power series, and the analysis of convergence intervals. These topics are fundamental in understanding the behavior of functions and their approximations, particularly within the context of calculus and mathematical analysis.

Parametric Equations: Slope and Concavity at a Specific Parameter Value

The initial problem involves analyzing a parametric curve defined by certain functions at a specific parameter value, t=2. The goal is to determine the slope of the tangent line at this point and then assess whether the curve exhibits concavity or convexity at this point.

For parametric equations, the slope at t=2 is given by dy/dx, which can be calculated using the chain rule: dy/dx = (dy/dt) / (dx/dt). Computing dy/dt and dx/dt at t=2 involves differentiating the parametric functions with respect to t and then substituting t=2. The resulting derivative ratio provides the slope.

Concavity involves finding the second derivative d²y/dx², which can be expressed in terms of derivatives with respect to t. Specifically, d²y/dx² = (d/dt)(dy/dx) / (dx/dt). Calculating this at t=2 allows us to determine whether the curve is concave up, concave down, or has DNE (does not exist) based on the second derivative's sign and existence.

Finding Equation of the Tangent Line at a Given Point

The next set of problems pertains to deriving the tangent line equations at specified points for different functions, including parametric curves and explicit functions like y = x² or y = 2x+9. For parametric curves, the process involves calculating dy/dx at the point of interest, then applying the point-slope form of a line, y - y₁ = m(x - x₁), where m = dy/dx at t corresponding to that point.

For explicit functions, differentiation is straightforward, and substituting the point's coordinates yields the slope. The tangent line equation can then be written accordingly.

Additionally, the second derivative d²y/dx² is sometimes requested to understand the curvature at that point, which involves differentiating dy/dx with respect to t and then converting that into terms of x to assess concavity further.

Power Series Approximation of an Integral with Error Bound

The problem asks for approximating a definite integral using a power series expansion, ensuring the approximation error remains below 0.0001. This involves expanding the integrand, possibly as a Taylor series or other series, and summing terms up to the point where the remainder term or error estimate falls within the specified tolerance.

Specifically, techniques such as the Taylor series remainder theorem may be employed to bound the error, guiding how many terms must be calculated. Approximations require carefully calculating each term and summing them to achieve the desired accuracy.

Deriving Taylor Series Using Definition

Another task involves deriving the Taylor series for a given function centered at a specific point c, using the definition of Taylor series. This process involves computing derivatives of the function at c and constructing the series as a sum of terms (f(c) + f'(c)(x - c) + f''(c)(x - c)²/2! + ...).

In the provided example, the integral of e^(-x⁸) from 0 to 1 is approximated using Taylor series expansion, perhaps by expressing the exponential as a power series and integrating term-by-term.

Finding Interval of Convergence of Power Series

The final problem seeks the interval over which a power series converges. This includes determining the radius of convergence using techniques such as the ratio or root test, and verifying convergence at boundary points (endpoints). For the provided example, the convergence interval is analyzed considering the power series' behavior at the boundary points, ensuring convergence criteria are satisfied via tests like the alternating series test or direct evaluation.

Understanding the domain of convergence is essential for applying power series successfully to approximate functions and evaluate integrals or derivatives within this interval.

Conclusion

These problems collectively reinforce core concepts in calculus related to parametric derivatives, tangent lines, series expansions, and convergence analysis. Mastering these techniques enables a deeper comprehension of the behavior of functions, their graphical representations, and their approximations, which are foundational for advanced mathematical analysis and applications.

References

  • Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendentals (11th ed.). Wiley.
  • Thomas, G. B., & Finney, R. L. (2002). Calculus and Analytic Geometry (9th ed.). Pearson.
  • Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning.
  • Larson, R., Edwards, B. H., & Hostetler, R. P. (2017). calculus (11th ed.). Cengage Learning.
  • Spivak, M. (1994). Calculus. Publish or Perish Press.
  • Moore, R. L. (2011). Power Series and Their Applications. Academic Press.
  • Royden, H. L., & Fitzpatrick, P. M. (2010). Real Analysis (4th ed.). Pearson.
  • Rudin, W. (1987). Real and Complex Analysis (3rd ed.). McGraw-Hill.
  • Isaacs, L. R. (2014). Advanced Calculus. Pearson.
  • Ayoub, R., & McShane, E. J. (1995). Introduction to Power Series. Springer.

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