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Worksheet 3 1. Assume thatf is a function satisfying f '(x) > 0 for all values of x in
the interval [a, b]. Let [a, b] be partitioned by a =x 0 < x1 < x2 < < x n = b . Determine which of the following quantities is largest, or
state that it is not possible to determine. Vl Vt ‘ Xi‘1)and .Z " Xi_1) (it (:1 2. Use equal subdivisions and the limit definition of a definite integral to
evaluate each of the following. lo 10
a. [1:3 dx b. Jx3 dx
I 1 3. Assume that a thin rod lies along the interval [a, b] and has density f(x) grams per centimeter at pointx . Let x 0, x1 , x 2 , . . ., x n be a
partiton of [a, b] and c1 , c 2, c 3 , . . . , c n corresponding sampling
numberswith ci in [xi,1, xi] for i=1,2,3,...,n. Whatisthe physical interpretation of each of the following quatities ? a. Xi'Xi_1 b‘ n
C. f(Ci)(Xi-Xi_1) I f(Ci)(Xi'Xi_1)
n h b
e. lim .2f(ci)(xi-xi_1) f. Jf(x)dx
mesh-*0 5' a
b 4. Each of the following limits is equal to a definite integral J f(x) dx .
Determine the definite integral for each. a in
3. lim ’Esin(i/n)-1/n “+00 133 V\
b. Iim '2 sin(3 + i/n)-1/n “+00 L=|
I. El” W) 3
c. rm . ’ —~
haw all W
d. lrm 2, +14.
“900 C! 3n L 5. Let f be a function which is differentiable for all values of x. In
addition, the function satisfies f'(x) = 5 f(x) for all values of x. Show
that there exists at least one number 0 satisfying 10 f(c) = f(3) - f(1) ...
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