Midterm 1 information/Learning goals
From UBCMATH WIKI
- 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.
- Approximate a polynomial or rational function (especially Hill functions) with simpler functions for sufficiently large/small values of the independent variable.
- Explain the connection between rate of change of a function and the slope of a secant line.
- Calculate the slope of a tangent line (when one exists) as a limit of the slopes of secant lines approaching the tangent line.
- Define "limit of a function" unambiguously.
- 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.
- Calculate finite and infinite limits of any power function and of functions comprised of the summation, difference, product, quotient, and composition of power functions.
- 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).
- Define "derivative of a function" in terms of secant and tangent lines.
- State the limit definition of the derivative.
- Compute the derivative of a power function, polynomial or a function involving a square root using the definition of the derivative.
- Compute the derivative of any polynomial using the power and sum rules.
- 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.
- Use given information about the tangent lines of a parametrized function (a function defined with unspecified constants) to identify the unknown constants.
- Find the velocity and acceleration given the displacement function (graph or formula), and vice versa.
- Calculate the derivative of products and quotients of power and rational functions.
- Given the graph of a function, sketch the graph of the derivative function.
- 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.