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Worksheet 7 1.) Initially, a flock of blackbirds numbers 750 in December 1988. Two
years later there are 1000 blackbirds. How many blackbirds will there be
in December 1995 if the rate at which the number of birds changes is
proportional to the number of birds present ? 2.) What definite integral is closely approximated by each of the
following sums ? a) i Ll+ gag . O % £3 (AJAXL  7mm) bl w '0 c) g .5. Q d) E00 L:3oo (‘00) I ‘00 . 'LZISOI 500 3.) a.) Let f(x)=3. i. Sketch the graph of f. i
ii. Evaluate J0f(x)dx. 3 for 0 5 x <1
6 for x=1
i. Sketch the graph off . 1
ii. Evaluate Jo f(x)dx. b.) Let f(x): i 0.) Let f(x) = " the first digit in the decimal expansion of x " for x in
[0, 1] . For example, f(0.713) = 7 and f(1/3) = 3 . \J i. Sketch the graph of f. i ii. Evaluate I f(x)dx.
o 4.) Asume that y is a differentiable function of x, and compute
y ' = dy/dx for each of the following. 7  ex
a.) y = IO ex9 dx b.) y =10 arctanx/Tdt
CD‘QX
c.) y=,[ (t2+5)10dt d.) y2x+cos(3y)=tan2x
MX
85'
e.) I [7+cos(t2)] dt=lnyx5
3
5.) Integrate.
\ l
a) 3 33 b) 3—,“er M
c) X M d) 3 xi M
l—er‘l “l' [+X1
9) X3 f ‘
Sl+xl M ) SkiPX;
\ X
g) g (“01 M h.) S (“01 M
x1 x1
I) SCXHV M J) Smﬁwl M
k) M I) S I”)? M m.) S UMX)1M n_) SX‘MX M X
0') 8x lynx M p') S x('7\+ﬂmx)'
9.
q') 8 4? ghtmmpw q I x'“ dx
5.) JR.In(x3) dx t.) Jlmf? dx
u.) I (lnx)2 dx . ,v.) I xex dx
w.) I arctanx dx x.) I sin I)? dx
y.) IeW‘dx z.) I Xx}??? dx 6.) Let S(x) be the square of the distance from (O, O) to the point (x, y)
on the graph of y = e x . Compute the average value of S for x in [0, 2] . ...
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 Spring '07
 MAT21B
 Derivative, 1m, 01 m, 7mm

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