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Disc2 - Math 21B Kouba Discussion Sheet 2 1 Determine the...

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Unformatted text preview: Math 21B Kouba Discussion Sheet 2 1.) Determine the following indeﬁnite integrals(a (antiderivatives). x2d+1 x2+1 aW/ xb.)/x+3dx c.)/m3mx:1d d.)/\/—1_1—:_—_;\$d\$ e.)/—1—j—w(;;)—2dm f.)/1::mdm g.)/1:emdx dm i.) /c0s2xdx j.) [cotmdx k.) /c0t2m csczwdx a: V1 —:v2 2 l.) fwdm m.) /L——2 +——5—-—-\$+6da: n.) /(:c2+1)(:1:3+3\$)10dm 11:2 :I:+1 as n 4 o.) /(\$:56)2 dx p.) [(1:3) d3: q.) /sec2(3w) 2tan<3\$)dm 2.) Evaluate the following deﬁnite integrals. 7T .. 1 7r a.) /2 cosy: 65in“ d2: b.) / 5”: (1:12 C.) / 5 sec2 3m dx 0 —1 0 sin 31; arctana: 2 3.) Differentiate each : a.) F(:r) = 1 + t2 dt b.) F(x) = / 5t dt —100 tanz 4.) Use the limit deﬁnition of the deﬁnite integral to evaluate /5\$12 — dz: , use an arbitrary partition 1— — 3:0 < 1:1 < .732 <. .< xn_1 < \$1,: 5 of the interval \$[21, 5] and use sampling points c, = Jam--1512, for i = 1,2,3, ...,n E 5.) Determine an equation of the line tangent to the graph of F (:12) = 3 + 2x + a: / arctan t dt 1 at\$=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. 211.13%. ismm me. 2M 3356) “ 2‘ (n+2i—1)2 “.11.“; Zin+n2 (”.13er 1:1 7.) Find the average value of each of the following functions over the given interval. Draw a sketch showing the connection between your answer and the deﬁnite integral. a.) ﬂat) 21:3 +1 on [—1,1] b.) f(:I:) = 5+\/E on [0,4] 1 1 3 1 8.) If / f(:c) dx = 3 and / f(a:) d3: = —2 . What is the value of / f(:c) dx ? ._2 —-2 3 9.) A thin rod is 16 cm. long. Its density 1: cm. from its left end is given by 2 + ﬂ gm./cm. a. Estimate the rod’s total mass by using four equal subdivisions and mid oints to p estimate densities. b.) Set up a deﬁnite integral and compute the exact mass of the rod. 10.) Consider the following sequence of numbers 2 1, 5, 9, 13, 17, 21, 25, Find the sum of the 615t through the 127th numbers in this sequence. 1 11.) Use the limit deﬁnition of a deﬁnite integral to evaluate / (m2 -— 2x + 3) dz): 1 12.) Find the area of the region bounded by the graphs of the given equations. a.)y=m,y=2x,andx=2 b.)y=e”,:z:=0,andy=2 c.)m=y2andm=9 1 13.) Assume that f is an odd function and / f(\$) d3: = 3 . What is the value of —2 —2 f(:c) dz ? —1 THE FOLLOWING PROBLEM IS FOR RECREATIONAL PURPOSES ONLY. 14.) Connect 6 toothpicks end-to—end to form 4 triangles. ...
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Disc2 - Math 21B Kouba Discussion Sheet 2 1 Determine the...

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