Math 151

  • Solve first-order separable differential equations and initial value problems.
  • Solve application problems involving first-order separable differential equations, such as exponential growth and decay.
  • Solve integral problems by first examining the integral, then selecting and applying the appropriate technique of integration.
  • Apply integration to physics problems relating to mass, centers of mass, work, and fluid force.
  • Identify, analyze and evaluate improper integrals.
  • Evaluate the limits of functions which have the indeterminate forms "zero/zero" and "infinity/infinity" using L'Hopital's Rule.
  • Transform the other indeterminate forms into those which L'Hopital's Rule can be implemented.
  • Define an infinite sequence; analyze and assess the monotonicity and convergence of a given sequence.
  • Identify some basic series, including the geometric series, harmonic series and a telescoping sum.
  • Compare the different convergence tests, including the Integral Test, the Ratio Test, the Root Test, the Comparison Test, the Limit Comparison Test, the Alternating Series Test, and the Divergence Test.
  • Assess the convergence of a series by formulating the comparison of the given series to a known series.
  • Assess if an alternating series converges absolutely, converges conditionally or diverges.
  • Analyze a series, assess which convergence tests can be applied in determining its behavior, and apply this test to show the series convergence or divergence.
  • Derive the Taylor series of a given function using a variety of techniques.
  • Calculate the radius of convergence of a given power series.
  • Apply Taylor's Theorem and Taylor polynomials to approximate to a certain degree of accuracy, the values of functions at non-trivial points.
  • Apply the known power series expansions of important functions to generate the series expansion of other functions.
  • Express a given second degree equation in the form of its standard conic equation and sketch the standard conic sections.
  • Analyze a conic section by rotating it to a standard position.
  • Sketch the graphs of functions in polar coordinates, including cardiods, lemniscates, and limacons.
  • Calculate the areas of polar regions.
  • Calculate the arclength of polar curves and the surface area bounded by polar curves.
  • Calculate the equation of tangent lines to polar curves.
  • Express a curve with parametric equations.
  • Calculate the tangent lines and arclength of parametrized curves.