No calculators permitted on this test.
Show all work to receive full credit.
(7 questions, 100 points total)
Some helpful formulas:
sin 2x = 2 sin x cos x
' 7! x/E
sin:cosz
4 4 2
1. (16 pts.) 21. Find the centroid of the region below y : e'" for 0 s x s 1
October 25, 2007
Three Applications of the Derivative Math 315
These notes describe the three applications of the derivative that were discussed in
class: Taylor polynomial approximation; convergence of sequences dened itera-
tively; Newtons method. The r
SHOW ALL WORK. No credit will be given for answers only. Simplify your answers as much
as possible. Complete all work on separate sheets of paper and make sure your name is on
each sheet.
1. Suppose that a savings account originally contains $100 and earn
SHOW ALL WORK!
1. (Spts) Below are for directions fields. Circle the direction eld that goes with the
differential equation y' = 1+l .
Try sketching the direction eld.
2. Consider the initial value problem ty' 3y = 2t4 sec2(2r) withy{%] = O .
a 15 ts Find
. Find the derivative of the following functions.
(a) f(w)1n(m2_1>
(b) 9(m) : sin2 a: l sin($2)
(0) FM : was? m)
(d) (Km) : a7:
. Find the equation of both tangent lines to the ellipse 1:2 | 43;? : 36 that pass through
the point (12,3)
1 2
. Use the den
Here is a list of what weve done this semester along with some exercises that you should be able to do.
Of course, you are responsible for any homework problems weve done, so know how to do them, too.
(Exam 1 stu)
Section 5.5 & 5.6: The Fundamental Theore
The arc length function 5 of the curve
C={(x,y):a:c£b, y=f(:c)}
is the distance along the curve C from the ini
tial point P0 = (a,f(a) to Q : (x,f(ac), i.e.
son) = x «1 + [ow]? cit.
By the FTC
Thus
(1) (ds)2 = (dz/22)2 + (d202-
Symbolically
L=/ds. If the
Math 1205 Trigonometry Review
We begin with the unit circle. in s
The definition of a unit circle is: x2 + y2 = 1
Where the center is (0, 0) and the radius is 1.
x
u)
(U. 1)
An angle of 1 radian is an angle at the center of a circle measured in the counte
PROBLEM 1
Note: There are a number of ways of solving this problem, so your method might
be valid even if it isnt the same as the one presented here. But the solution should
be the same.
This is a 2-step problem. We first nd a point where tie two planes