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Unformatted text preview: Workshop 3
1) Suppose f is deﬁned by f (x) = 3ecos x .
Maple produced graphs of f and its ﬁrst
four derivatives on the interval [2, 7] (be
careful when examining the derivative graphs
look carefully at the vertical !). The graph
of f is to the right, and the graphs of the
ﬁrst four derivatives of f are on the back
of this page. You should assume that the
graphs are correct for this problem.
Suppose I is the value of 7 2 f (x) dx. a) Use the graph of f alone to estimate I .
b) Use the information in the graphs to tell how many subdivisions N are
needed so that the Trapezoid Rule approximation TN will approximate I
with error < 10−5 .
c) Use the information in the graphs to tell how many subdivisions N are
needed so that the Simpson’s Rule approximation SN will approximate I
with error < 10−5 . Graph of f ′ Graph of f (3) Graph of f ′′ Graph of f (4) 2) The only information known about a function T and its derivatives is
contained in this table:
x T (x) T ′ (x) T ′′ (x)
a) Compute 23 T ′ (x) dx.
b) Compute 23 T ′′ (x) dx.
c) Compute 2 xdx.
d) Compute 3
2 xT ′′ (x) dx. Don’t look at b) and c)! Integrate by parts. e) Compute 3
2 x2 T ′′′ (x) dx. And again and again. 3) Calculate four of the following integrals:
x cos x2 dx ; x cos2 x2 dx ; x2 cos x dx ; x2 cos2 x dx ; x cos2 x dx . Comment Most people use many parentheses and rewrite the integrands
to decrease possible confusion. So x2 cos2 x becomes x2 (cos x)2 and x cos2 x2
becomes x cos2 (x2 ). ...
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