Unformatted text preview: 04/10/2001 TUE 16:53 FAX 6434330 MOFFITT LIBRARY .001 Department of Mathematics, University of California, Berkeley Math 1A
Alan Weinstein, Fall 2000 First Midterm Exam, Thursday, September 28, 2000 Instructions. Be sure to write on the front cover of your blue book: (1) your name, (2)
your Student ID Number, (3) your TA's name (Eric Antokoletz, Victor Deletang, Matthieu
Hamel, Andre Henriques, Di—An Jan, Chu—Wee Lim, Russell O’Connor, Alf Onshuus, Em-
manuel Py, Shahed Sharif, Dan Stevens, Karla Westphal, or Alexander Woo). Read the problems very carefully to be sure that you understand the statements. Show
all your work as clearly as possible, and circle each ﬁnal answer to each problem. Remember:
if we can’t read it, we can’t grade it. 1. [12 points] Let ﬁx) 2 m/(w+ l) and g(:r) = —1/(:c + 1). (A) Find f'(:17). (B) Find g'($).
(C) Find the equation of the tangent line to the graph y = :1: / (a: + 1) which goes through
the point (1,1/2).
(D) Find the horizontal and vertical asymptotes of the graph of f.
(E) Find four different functions such that the derivative of each function is equal to 32:2.
(F) Look at your answers in parts (A), (B), and (E); what do they suggest about the relation
between the functions f and g? 2 . [6 points] Find the following limit. (Hint: write %em as a. limit]
em — e 3 . [8 points] If f(2) = 5 and f’(2) = 3, ﬁnd each of the following, or say if there is
insuﬁicient information to ﬁnd it. (A) (l/f)’(2)- (B) %(%f($)2+17) at w = 2- (C) f(3)- (D) f(2-001)- 4 . [6 points] A round solar panel is to be constructed with radius 10 meters, but errors
in construction are inevitable. Using the properties of inequalities to justify your
answer, ﬁnd a number 5 with the property that, if the radius of the collector is within
6 meters of the speciﬁed size, the area of the collector will be within 2 square meters of
the desired area of 1001f square meters (Assume that, even with errors, the collector is still
perfectly round.) 5 . [13 points] Let Q be the function deﬁned by the equations Q(t) = 0 for t S 0 and
Q(t) = 16t2 for t 2 0.
(A) Sketch the graph of Q.
(B) Find Q’(t) for all those 15 where Q is diﬁerentiable.
(C) Sketch a graph of Q'.
(D) Where is Q continuous? Why?
(E) Where is Q’ differentiable?
(F) Find a function R such that R'(t) = Q(t) for all t, and sketch its graph. ...
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