Calculus
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m166calculusprojectfall2015.docxM166
“Calculus” Project
Due: Wednesday, December 9, 2015
PROJECT WORTH 50 POINTS –
1) NO LATE SUBMISSIONS WILL BE ACCEPTED
2) COMPLETED PROJECTS NEED TO BE LEGIBLE
I. Computing Derivatives (slope of curve at a point) of polynomial functions.
For each of the following functions in a.-e. below perform the following three steps:
1. compute the difference quotient
2. simplify expression from part 1. such that h has been canceled from the denominator
3. substitute and simplify
a.
b.
c.
d.
e. consider , using the results from parts a. through d.,
f. find a general formula for (steps 1 through 3 performed).
II. Show that
Consider the unit circle with in standard position in QI.
a. show that the area of the right triangle (see diagram) is
b. show that the area of the sector (see diagram) is
c. show that the area of the acute triangle (see diagram)
d. set up the inequality
e. multiply the inequality in part d. by . (direction of inequalities is unchanged)
f. take the reciprocal of each term from part e. The direction of the inequality must be reversed because .
g. plug in 0 for for only. The result should be
III. Show that
a. multiply by
b. use trigonometric identity to rewrite the numerator of the expression in part a. in terms of
c. factor the expression in part b. with one factor equal to . (find remaining factor).
d. use the fact that and substitute in the second factor (result is 0)
IV. Show that derivative of
a. find the difference quotient for
(use sum angle formula )
b. factor out of the two terms in the numerator with in part a
c. split up the expression in part b with each term over the denominator h
