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**Unformatted text preview: **Math 21B
Kouba
Discussion Sheet 3 1.) Use any method to determine the following indeﬁnite integrals (antiderivatives). gig/ﬁlm b.)/%dm c.)/1::xd:r (1.)/Jam e.) /¢%ﬁdm f.) /cos2a:dac g.) /c0t$d$ h.) /cot2$ csczmda: . 2 5 6 2
1.) fﬁii—m j.) /5’3—“:—5§+—6d:c k.) /($2+1)($3+3m)10d$ 1:2 a: + 1
x + 6 (ln :1:)4 2 t
l. — d . . “(338)
) (a: + 5)2 x m) f a: (1110 n) /sec (3:17) 2 dm 2.) Use any method to compute the area of the region bounded by the graphs of
a.)y=$,y:0,andy:4—g;. b.)y=\/1ﬁ$2andy:0. 3.) Assume that snow is falling at the rate of t + \/t in./hr. at time 25 hours. SET UP a
deﬁnite integral and compute the total amount of snowfall between t : 0 and t : 4 hours. 5
1 .
4.) Use the limit deﬁnition of the deﬁnite integral to evaluate / 7 d3: ; use an arbitrary
1 :c partition 1 : 3:0 < 101 < :02 < < :17an < mn : 5 0f the interval [1, 5] and use sampling
points Ci : ,/.17i_111i for i : 1,2,3,...,n. 5.) Determine an equation of the line tangent to the graph of IE
F(a:) 23+2m+$/ arctantdt atm: 1.
1 6.) Each of the following limits is equal to a deﬁnite integral. Determine a deﬁnite integral
for each. Do not evaluate the deﬁnite integral. n a.) ”13130 23(Hiil4él b-) .11.“; :1n(3+%l(§l , i , " (n+2i—1)2
6-) .132. Zin+n2 d) #32. ET 1:]. i=1 1
' 7.) Assume that f is an odd function and / f(:I;)dw : 3 . What is the value of
-2 1 —2 f(ac) da: ?
—1 8.) A thin rod lies along the x-axis between a: 2 1 and :c : ln 5. Its density at :1: cm. is given by e”w(1 — e‘I)5 gm./cm. SET UP a deﬁnite integral and compute the exact mass
of the rod. 9.) Find all values of c guaranteed by the Mean Value Theorem for Integrals for f(ac)— rc+1, if—1£w<0
_ 1—m2, ingmgl 10.) The total distance 5 (in miles) traveled by a hiker at time t (in hours) is
3(t) = t+ ln(1 + (%)t). Find the hiker’s average hiking speed between t : 0 hrs. and t : 4
hrs. 11.) Determine a function having the following properties : f"(:c) =1+ 6%,f’(0) = —1, and f(0) 2 3
THE FOLLOWING PROBLEM IS FOR RECREATIONAL PURPOSES ONLY. 12.) Two bicyclists are twelve miles apart. They begin riding toward each other, one
pedaling at 4 mph and the other at 2 mph. At the same time a bumblebee begins flying
back and forth between the riders at a constant speed of 10 mph. How far does the
bumblebee travel by the time the riders meet ? ...

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- Winter '09
- Kouba
- Calculus