Midterm 1 information/Learning goals

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Week 1

  1. Explain that a power function with a greater power is greater than one with a lower power for sufficiently large values of the variable and vice versa for sufficiently small values of the variable.
    • Equivalently: Explain how the plot of the power function $y=x^n$ changes as $n$ changes.
  2. Approximate a polynomial or rational function (especially Hill functions) with simpler functions for sufficiently large/small values of the independent variable.

Week 2

  1. Explain the connection between rate of change of a function and the slope of a secant line.
  2. Calculate the slope of a tangent line (when one exists) as a limit of the slopes of secant lines approaching the tangent line.
  3. Define "limit of a function" unambiguously.
  4. Define "continuous function" in terms of left and right-hand limits and use this definition to determine if a given function is or is not continuous.
  5. Calculate finite and infinite limits of any power function and of functions comprised of the summation, difference, product, quotient, and composition of power functions.
  6. Define "asymptote" as a line to which a function gets arbitrarily close in either a finite limit (for a vertical asymptote) or an infinite limit (for a horizontal asymptote).
  7. Define "derivative of a function" in terms of secant and tangent lines.

Week 3

  1. State the limit definition of the derivative.
  2. Compute the derivative of a power function, polynomial or a function involving a square root using the definition of the derivative.
  3. Compute the derivative of any polynomial using the power and sum rules.
  4. Find the equation of a tangent line to any type of function discussed so far that passes through a point, either on or off the graph of the function.
  5. Use given information about the tangent lines of a parametrized function (a function defined with unspecified constants) to identify the unknown constants.
  6. Find the velocity and acceleration given the displacement function (graph or formula), and vice versa.

Week 4

  1. Calculate the derivative of products and quotients of power and rational functions.
  2. Given the graph of a function, sketch the graph of the derivative function.
  3. Given a function, calculate its intervals of increase/decrease, concavity, inflection points, and local and global maximum/minimum. Use this information to plot the function.