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Worksheet5
1. Differentiate each of the following. X
a. F(x)=arcsin(ex2) b. F(x)=ln{1+_[\}t3+5dt}
o
3
c. Fix) = m d. F(x) =l cosx2. 63in? x2 dx
ln(tanx) 0
Sinx ‘lxl
e. F(x)=le1tdt r. F(x)=lcos(t2+1)dt
1 3
3 x3
g. F(x)=l cos(t2+1)dt h. F(x)=l(3t5)100dt
Xl i
x
x
i. F(x) =x5l t2 dt
0 t2+1 2. Find the xvalue(s) for which each of the following functions has a
global minimum value. a. F(x) = ex2'7
X b. F(x) = 9 +1 (t1)(2—t)6 dt for x_>_0
o 3. Evaluate the following definite integrals. Think carefully. Nothing
sophisticated is needed to solve these problems. .9. o
a. l(2/x2 + x2/2)dx b. l(1+x)2 dx
l ~l
1r
° I
C I (1+X)200 dx Cl. I smx dX
‘l 0 coszx 1r
3 i
e. I (sinx+cosx)2dx f. l x2(1+x3)1/1° dx
1 0
4
w 5%,
g. I coszx dx h. I 5e025xdx
o o
1 if:
i. I2xsec2(x2)dx j. I (xcosx+sinx)dx
o o 4. a. Make a sketch of the region bounded by the graphs of y = 1/2 x ,
y = 2 , and x = 0 . b. Set up definite integrals which represent the volumes of the
solids formed by revolving the region in part a. around i. the xaxis ii. the y—axis 5. A large bucket full of water weighing 100 lbs. is slowly lowered 25
feet by a rope and pulley. a. How much work is done in lowering the bucket ?
b. How much work is done in lowering the bucket if the bucket is leaking , losing 4 in.3 of water per foot lowered. (Assume that one cubic
foot of water weighs 62.4 lbs.) 6. Find the average value of each function over the indicated interval.
a. f(x) = x 3 on [0, 1] b. f(x)=sinx on [0,1t/2] c. f(x)=2xex2 on [xlln2', \lins'] d. f(x) = x sec 2 x + tan x on [0, 1/2] ...
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 Winter '07
 MAT21B

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