Final Exam Notes
The final exam will be on Thursday June 15 1:00-4:00
in Bldg 360 Rm 323 (our usual classroom). As with the
tests, you may use a non-programmable, non-graphing
scientific calculator. You should also bring a ruler,
some reliable pencils and an eraser. Scrap paper will be
supplied if needed.
No notes or formula sheets will be allowed; however,
less familiar formulas (the surface area of a sphere for
example) will be supplied if needed. You should know
simple, standard formulas such as those for the area and
perimeter of rectangles, circles and triangles, and also
the volume of a box. You are expected to know the
quadratic formula and the basic information on trig
functions contained in useful
facts & formulas.
The exam will consist of ten questions of a style
similar to those of the four term tests. The topics to
focus on are:
- Limits and Continuity
- Be able to calculate limits (including limits from
left, right) and use limits to determine continuity of
a function at a point.
- Definition of the Derivative
- Know how to use the limit definition of the
derivative to calculate the derivative of a
function.
- Derivative Rules
- There will be one or two questions which test your
ability to apply the differentiation rules
(+/-/×/÷/power/trig/chain). These may include
some questions on computing antiderivatives and solving
simple differential equations as well.
- Interpretations of the Derivative
- Be familiar with the two interpretations of the
derivative we have seen: f'(x) represents the slope of
the tangent line to the graph of f(x), and also the
rate of change of f(x) with respect to x. For the
second of these, recall that if s(t)=displacement of an
object at time t, then s'(t)=velocity=v(t), and
s''(t)=acceleration=v'(t). You should also be able to
determine the units for the derivative given the units
for the original function.
- Tangent Lines & Approximation
- You should know how to find the equation of the
tangent line to a curve at a point, and how to then use
this tangent line to approximate values the function
near the point. These problems often require you to
decide what the function being approximated is, and the
point near which the approximation is being made. For
example, if you are to approximate (9.1)1/2
using a tangent line approximation,
f(x)=x1/2 while c=9.
- Implicit Differentiation
- Be able to determine the first and second
derivative of functions defined implicitly, and use the
derivative to construct tangent lines to implicitly
defined curves. In connection with tangent line
approximation, these tangent lines may then be used to
approximate the implicitly defined function near the
point of tangency.
- Related Rates
- You can expect a word problem on related
rates.
- Optimization
-
You can expect one or two word problems on
optimization. Recall the two classes of problems we
saw:
- Find the absolute min/max of a continuous
function on a closed interval. These involved
testing the function at the critical numbers as
well as the end points to determine the maximum and
minimum values of the function.
- Find the absolute min/max of a continuous
function subject to a constraint. These generally
involved first using the constraint to reduce the
objective function to a single variable, and then
testing the critical number(s) using either the
first or second derivative test to determine the
absolute min or max.
- Curve Sketching
-
For a given function f(x), be able to
- determine intervals of increase/decrease and
determine relative max/min using first
derivatives;
- determine intervals where function is concave
up/down and determine inflection points using
second derivatives;
- determine horizontal, vertical and slant
asymptotes;
- determine x and y intercepts;
- put all of these items together to sketch the
graph of f(x)
- Newton's Method
-
You should know how to implement Newton's Method to
solve an equation of the form f(x)=0. In particular,
be able to
- State the equation to be solved in the form
f(x)=0. (This gives you f(x)).
- Determine the starting value x1
using either a graph or table.
- Implement the method. For this you must know
the formula
xn+1=xn-f(xn)/f'(xn)
(the formula will not be given).
You should also know how Newton's Method is
derived from the graph of the function the way I
showed in class.
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