(1) (Pre-Calc Review Set Problems # 44 and # 59) Show your work. (a) Given sec =5,andsin <0, find the exact values of the following functions.
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1 () (Pre-Clc Review Set Problems # 44 n # 59) Show your work. () Given sec =5,nsin <0, fin the ect vlues of the following functions. sin( ) =, cos( ) =, tn( ) =, cot( ) =,ncsc( ) =. (b) Use the Lws of Logrithms to rewrite the following eression in terms of simle logrithm functions like ln(), not ln(y), ln( ), or ln( ). r q ln y w! z
2 () () (8 oints) Do the following sequences converge or iverge? Circle your choice. If the sequence converges, fin the ect vlue to which the sequence converges. You o not nee to elin. (i) n = 3n n +4 n Diverges, or, Converges to n + (ii) n =ln n Diverges, or, Converges to (iii) n =+e n Diverges, or, Converges to (iv) n = n/3 + n / Diverges, or, Converges to (b) (3 oints) If is constnt n <, then the infinite series Show work: (converges / iverges). If it converges, its sum is. (c) (4 oints) If the infinite geometric series = 4 5,thenwhtisthe sum ? Answer:. Show work:
3 () (5 oints) Fill in the blnks. You o not nee to elin. Consier three oints A =(0, 90, 50), B =( 00, 0, 0), n C =(50, 5, 00). The one tht is closest to the yz-lne is. The one tht lies on the z-lne is. The one tht is frthest from the origin is. (e) Fin n eqution for the liner function with the following contour igrm. Show your work. (f) Determine whether the following limit eists. If so, fin it. If not, elin why it oes not eist. Show work. lim (,y)!(0,0) 3 3 y y 6.
4 (3) Let f be twice i erentible function. Some of the vlues of f n its erivtives re given in the tble below. f() f 0 () f 00 () () Evlute the integrl (4 +3)f 00 (). (b) Evlute the integrl / 0 cos()f 0 sin().
5 (4) () Determine whether the imroer integrl e 8 e 3 ln converges or iverges. Evlute its vlue if it is convergent. Show your work. (b) Let R be the close region boune by the curve y = ntheliney =. Fin the ect volume of the soli obtine by rotting R bout the line y =3.
6 (5) () Fin the first 4 nonzero terms of Tylor series for y = your work. 4 bout =0. Show (b) Fin the first four nonzero terms in the Tylor series roun =0forthefollowing function. (You cn use the Tylor Series of the known functions.) y = z z
7 (6) Determine whether ech of the following series converges or iverges. Plese stte which test you use to mke your ecision. () X n= 5 cos(n) n (b) X ( ) n ln n n n=
8 Useful formuls Integrtion by Prts: uv = uv vu or uv 0 = uv vu 0 Useful Integrls for Comrison: 0 0 converges for >nivergesfor le. converges for <nivergesfor. e converges for >0. nth egree Tylor Polynomil of f() centeret = : f() =f()+f 0 ()( )+ f 00 ()! Tylor series of f() centeret = : f() =f()+f 0 ()( )+ f 00 ()! Tylor Series of imortnt functions: ( ) + f 000 () 3! ( ) + f 000 () ( ) 3 + 3! ( ) f (n) () ( ) n n! sin() = cos() = 3 3! + 5 5!! + 4 4! 7 7! + 6 6! + e =+ +! + 3 3! + 4 4! + = for << ln( + ) = for <le ( + ) =+ + ( ) +! ( )( ) 3 + for << 3!
9 Finite Geometric Series: n = ( n ) for 6=. Infinite Geometric Series: = for <. Rtio Test: For the series P n,suose, n+ lim n! n If L<, then the series converges. If L>, then the series iverges. If L =,thenthetestfils. Di erentition formuls = L. (n )=n n (e )=e (ln ) = (sin()) = cos (rcsin()) = (tn()) = sec (sec()) = sec tn (rccos()) = ( )=(ln) (cos()) = sin (cot()) = (csc()) = csc csc cot (rctn()) = + Aformulfortheistncebetweentheoints(, y, z) n(, b, c) in3-sceis = ( ) +(y b) +(z c). If lne hs sloe m in the irection, sloe n in the y irection, n sses through the oint ( 0,y 0,z 0 ), then its eqution is z = z 0 + m( 0 )+n(y y 0 ). The Lgrnge error boun for P n (): Suose f n ll its erivtives re continuous. If P n () isthen th Tylor olynomil for f() bout, then E n () = f() P n () le M (n +)! n+, where m f (n+) lem on the intervl between n.
10 Here, b, c, re constnts. A Short Tble of Inefinite Integrls I. Bsic Functions. n = n + n+ + C, (n 6= ) 5.. =ln + C = ln + C ln = ln + C sin = cos + C cos = sin + C tn = ln cos + C II. Proucts of e, cos, n sin 8. e sin(b) = + b e [ sin(b) b cos(b)] + C 9. e cos(b) = + b e [ cos(b)+bsin(b)] + C 0. sin()sin(b) = [ cos()sin(b) b sin()cos(b)] + C, 6= b b. cos()cos(b) = [b cos()sin(b) sin()cos(b)] + C, 6= b b. sin()cos(b) = [b sin()sin(b)+cos()cos(b)] + C, 6= b b III. Prouct of Polynomil () with ln,e, cos, n sin 3. n ln = n + n+ ln (n +) n+ + C, n 6=,>0 4. ()e = ()e 0 ()e ()e + C 5. ( )(signslternte) ()sin = ()cos()+ 0 ()sin() ()cos() + C 6. ( )(signslternteinirs) ()cos = ()sin()+ 0 ()cos() ( )(signslternteinirs) 3 00 ()sin() + C
11 IV. Integer Powers of sin n cos 7. sin n = n sinn cos + n sin n, nositive n 8. cos n = n cosn sin + n cos n, nositive n 9. sin m = cos m sin m + m m sin m, m 6=,m ositive 0. sin = cos ln cos + + C. cos m = sin m cos m + m, m 6=,m ositive m cos m. cos = sin + ln sin + C 3. sin m cos n : If n is o, let w =sin. If both m n n re even n non-negtive, convert ll to sin or ll to cos (using sin +cos = ), n use IV-7 or IV-8. If m n n re even n one of them is negtive, convert to whichever function is in the enomintor n use IV-9 or IV-. The cse in which both m n n re even n negtive is omitte. V. Qurtic in the Denomintor 4. + = rctn + C, 6= 0 b + c 5. + = b ln + + c rctn + C, 6= 0 6. ( )( b) = (ln ln b )+C, 6= b ( b) c + 7. ( )( b) = [(c + )ln (bc + )ln b ]+C, 6= b ( b) VI. Integrns involving +,,,>0 8. =rcsin + C 9. ± =ln + ± + C 30. ± = ± + ± + C 3. = + C
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