1.1-1.2 (considered review) Know the definition of
a function. Be able to
interpret the function if it is given algebraically, numerically or graphically .
Review piecewise given functions and basic function properties such as
symmetries, even and odd properties, and increasing and decreasing
properties. Review function transformations and know their effect on the
graph of a function . Understand the composition of functions. Know the
properties of the following families of functions: linear functions , power
functions, polynomial functions , rational functions and trigonometric
functions.
1.3 Know the intuitive definition of the limit of a function at a point
a. Be
able to find limits graphically and numerically. Be able to find right
and
left limits of a function at a given point.
1.4 Be familiar with the various limit laws. Be able to apply them to a
specific limit calculation . Know the algebraic techniques that help to
find
limits. Know and be able apply the Squeeze theorem.
1.5 Know and understand the definition of continuity of a function
for
both a point a and for and interval [a,b] (remember the 3 conditions). Be
able to decide if a function is continuous if the graph or the formula of the
function is given. Be familiar with the Theorems that follow from the limit
laws . Know how to use continuity to evaluate limits (we called it the
substitution property ). Be able to determine the points where a function has
discontinuities and if the discontinuity is removable, be able to remove
it.
Know and be able apply the Intermediate Value Theorem to determine if
a function has roots in a given interval.
1.6 Be able to find limits involving infinity. Know the intuitive
definition
of the limit of a function at infinity. Be able to find such limits
algebraically, graphically and numerically. Know what a vertical and a
horizontal asymptote of a function are and how to find them using limits.
Know how to graph polynomial and rational functions using these limits and
asymptotes.
2.1 Know the definition of the derivative and be able to find
derivatives
using the definition. Be able to compute average and instantaneous rates of
change and know the difference between the two. Be able to find the
equation of the tangent line at a given point. Understand the meaning of the
derivative at a point and as a function and be able to interpret derivatives in
the context of real world applications.
2.2 Know what it means for a function to be differentiable at a point
and over an interval. Be able to determine points or intervals where a
function is not differentiable and be able to give justify why the function is
not differentiable. Be able to draw graphs of derivative functions from
the graph of a function. Understand the definition for higher order
derivatives.
2.3 Know the basic rules for derivatives and be able to apply them to
find derivatives of given functions.