Supplementary Materials
Be able to work any part of any problem from the following Worksheets
possibly with different functions:
Approximation Problems
Finding over and underestimates and bounds for error for the slope
of a tangent line at a point on a given graph.
Finding over and underestimates and bounds for error for distance
traveled given velocity.
Finding over and underestimates and bounds for error for an infinite
sum with alternating terms.
Continuity Problems
Determining limits of continuous functions.
Interpretation of the Extreme Value Theorem.
Interpretation of the Intermediate Value Theorem.
Interpretation of the Squeeze Theorem.
proof that sin(x)/x --> 1 as x --> 0 using the squeeze theorem.
(Don't worry about problem #6)
Section 2.1
Definition of a tangent line as a limit of secant lines.
Be able to estimate the slope of a tangent by finding the slope
of a secant from
a graph
a table of data
a formula
Be able to say whether the estimate is an overestimate or underestimate.
Be able to find a bound for the error and say how to make the approximation
better.
Know the relationship between the slope of a tangent line and instantaneous
velocity. Be able to relate both directly to their limit definitions to show
that they are the same. Know what each part of the limit definition refers
to in each context.
the height of the graph at two different x-values (the position
at two different times)
change in y (change in distance)
change in x (change in time)
slope of a secant (average velocity)
slope of the tangent (instantaneous velocity)
Section 2.2
Be able to guess a limit given
a table of values
a graph
an equation from which you create either a table or graph
limit notation
one-sided and two-sided limits
vertical asymptotes
definition
be able to analyze an equation to determine whether values are
approaching positive or negative infinity for each side of an asymptote (remember,
the two sides may be different).
Section 2.3
Know, be able to use, and be able to identify the use of all of
the limit laws (11 of them).
Direct substitution property (also know what polynomials and rational
functions are).
Be able to find limits of piecewise defined functions.
Be able to compute limits given an equation (especially for rational
functions and functions with radicals).
Section 2.5
Know the three-part definition of continuity at a point.
Know the definition of continuity on an interval.
Be able to determine if a function is continuous at a point given
a graph or an equation.
Be able to say which parts of the three-part definition is satisfied
or not satisfied in any given example.
Understand and be able to apply the Intermediate Value Theorem.
Section 2.6
Know the meaning of limits at infinity / horizontal asymptotes.
Be able to guess a limit at infinity from a graph or table.
Be able to compute limits at infinity similar to homework problems.
Section 2.7
See Section 2.1
Section 2.8
Know the limit definition of the derivative of a function at a
point.
Be able to compute derivatives at a given point using the definition.
Be able to find the equation of a tangent line to the graph of a
function at a point.
Know the relationship between the derivative and slope of a tangent
and instantaneous velocity (see Section 2,1)
Section 2.9
Be able to compute a derivative using x as an arbitrary point
to get the derivative function.
Be able to prove Theorem 4: if a function is differentiable at a
point, then it is continuous at that point.
Be able to give a counterexample to the converse of Theorem 4.
Be able to recognize from a graph or give examples of points where
a function is not differentiable
corner/cusp
discontinuity
vertical tangent
Proof that if f(x)=sinx, then f '(x)=cosx.
There will be six questions as follows:
One question from the Approximations Problems Worksheet or the Continuity
Problems Worksheet
One question from the Chapter 2 homework (possibly with different numbers)
One question from Chapter 2 WeBWork (with different numbers)
One question from the Preliminaries Quiz
One proof or definition
One question applying this information in a new way