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Unformatted text preview: Midterm Exam 1, Wednesday, Feb 14, 2007
Lev Borisov, Math 221, Lecture Section 1‘ DO NOT OPEN THE EXAM BEFORE THE START
ANNOUNCEMENT 2 Please write your name and your TA’s name below.
Name: TA: Each problem is worth 20 points, for a total of 100 points. Cal
culators are not allowed on this test. Please read each
question carefully, it also helps to check afterwards that you have
answered each part of each question. You must show all your
work to receive credit. When you turn in your paper after
the test, make sure the TA checks your name in their list or writes
your name doWn. Good luck! [1] Sketch the graph of the function f = 2sin(7ra:) + 1. What is the period of this
function? [2] Calculate the following limits. (a)[10PtS] .
hm x3 + sin 30
cc—>+oo (1:2 +1)(2:c +1)
(b)[10ptsl
, 1 —— cos a:
11m —————————
z—>O $2 Hint: Multiply numerator and denominator by 1 + 00815. You are not allowed to use
L’Hospz'tal’s Rule for this exam. ~[2] (20 pts) Please leave blank! [3] For which values of c is the function 5132, 3:20
fﬁx)~{ 2+:v, a:<c continuous for all x? You must justify your answer. [3] (20 pts) Please leave blank! [4] Use the deﬁnition of derivative as a limit to. ﬁnd the equation of the tangent
line to the curve y = x2 + :1: at the point (—3;6). ‘ Please leave  ank! [5] Let f be a function, which is diﬁerentiable at $0. Consider the function
go) = we)? I
Use the deﬁnition of derivative as a limit to ShOW that
91m) = 2f(wo)f'(xo)~ Remarki This is a particular case of the Chain Rule. However, you are not allowed to
use the Chain Rule to solve this problem. Please leave blan ! ...
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 Summer '07
 Denissou
 Calculus, Geometry, Chain Rule, Derivative, Continuous function

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