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# Calculus with Applications MATH 11

UCSC

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This 11 page Class Notes was uploaded by Sienna Bradtke III on Monday September 7, 2015. The Class Notes belongs to MATH 11 at University of California - Santa Cruz taught by Staff in Fall. Since its upload, it has received 37 views. For similar materials see /class/182213/math-11-university-of-california-santa-cruz in Mathematics (M) at University of California - Santa Cruz.

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Date Created: 09/07/15

Math 11b Review Glenn Gray ggray ucscedu Fall 2007 1 Review This compilation of de nitions and theorems is meant to be a useful reference for Math llbi It should also be useful for students who will eventually take Math 22 This is by no means a complete compilation but should give some useful things to think about All of this material was taken from Calculus for Biology and Medicine by Claudia Neuhauser and should be read over in the book1 look carefully over examples Also these de nitions and theorems can often be visualized with graphs and pictures seeing these pictures is a great way to understand the de nitions so look in the book at the examp esl 2 Review of Derivates Here are some basic de nitions and properties of derivatives these are necces sary when checking to see if you have the correct antiderivatieve and some of thier properties are the same as integralsi 21 Properties of Derivatives Property 1 Scalar Multiplication d d C39f1 c Ewan c e 9 lt1 Property 2 Derivative of a sum is the sum of the derivatives lm 91 fil z lm lt2 Property 3 Product Rule lm gltzgti imam gltzgt fltzgt lm lt3 1There are many books in the library that cover the same material sometimes it is good to look over another book to get someone elsels take on the same materi V V V Property 4 Chain Rule gmi f 9rl g ltzgt lt4 3 Review of Integrals 31 De nite Integrals De nition 1 De nite Integral Let P zohhrgpiqzn be a partition of ab and set Ark rk 7 1161 and ck 6 164116 The de nite integral of f frama to b is n b dz HIlDiHmO Z fckAzk 5 a P k1 if the limit exists in which case f is said to be Riemann integrable 0n the interval a7 b 32 Properties of Integrals Assume that f and g are integrable on a7 by Property 1 161 dz or 6 Property 2 b a fltzgtdz 7 fltzgt dz lt7 a 12 Property 3 Scalar Multiplication b b hfzdzh fzdz7 hegti 8 Property 4 Integral of the sum is the sum of the integrals b b b W gzl dz fltzgt dz gltzgtdzl lt9 Property 5 If f is integrable over an interval containing the three numbers a b and c then Abfzdz 161 dzbez dzl 10 Property 6 Iffz 2 0 an ab then bfzd120i ll Property 7 Iff1 g 91 on ab then Abf1 d1 S 11191 d1l 12 Property 8 fin S f1 S M on an then b mb7a f1d1 Mb7al 13 33 The Fundamental Theorem of Calculus Fundamental Theorem of Calculus 1 Part I ff is continuous on an then the function F de ned by F1fudu7 a 1 b 14 is continuous on abwith d EFW fltzgt 15gt Theorem 1 Leibniz s Rule If 91 and h1 are di erentiable functions and is continuous for u between 91 and h1 then h1 fudu mowz e momz 16gt Fundamental Theorem of Calculus 2 Part II Assume that f is contin uous on a7b then b M dz M e Fltagt 17gt where F1 is an antiderivaZive of that is F1 Theorem 2 Area If f and 9 are continuous on a z with 2 91 for all 1 E a z then the area of the region between the curves 9 and y 91 from a to b is equal to 12 Area 7 d1l 18 34 Applications of Integration Theorem 3 Average Value Assume that f1 is a continuous function on a7b The average value of f on the interval a7b is 1 1 fav Ea d1l 19 4 Integration Techniques Here are some of the most Widely used integration techniques 41 Methods of Integration Rule 1 Substitution Rule for Inde nite Integrals Ifu 91 then fmmwwzfwww om Rule 2 Substitution Rule for De nite Integrals Ifu 91 then 5 95 fwmwmm fowl on a 9a Rule 3 Integration by Parts Rule Ifur and 91 are di erentiable func tions then uzvzdz 7 u zvzdz 22 or in short form udv uv 7 vdui 23 Method 1 Partial Fraction Decomposition For integrands of the form 131 QI 7 where 131 and are polynomials of degree n and m respectivly one might use PFD Ifn 2 m then use long division Ifn S m then use partial fraction decomposition There are four possibilities for the denominator 24 1 Linear Factors m i i i 25 azbczd7azb czd 2 Repeated Linear Factors Pr 7 A B C D az bcz d3 7 am b cz d cz d2 cz d3 26 5 Irreducible Quadratic Factors 131 7 AIB CID 27 az2 bc12 d 7 a12 12 c12 d 2Remember that one should always try to simplify the integrand algebraicly rst then use the following methods ie try something easy and then something har er 4 Repeated Irreducible Quadratic Factors 131 7AIB CID EerF G az2 b2c12 d er 7 az2 12 az2 b2 c12 d ez f 28 Next nd the coe cients ie A B C by the method of comparing coef cients 42 Improper Integrals 421 Type 1 Unbounded Intervals Theorem 4 Unbounded Intervals o fz dz z113010 2 fz dz 29 a dzlimzgt700afz dz 30 Theorem 5 Let be continuous on the interval aoo If z131010 2 fz dz 31 exists and has a nite value we say that the improper integral fltzgt dz 32gt converges and de ne dz 33 otherwise we say that the improper integral diverges Theorem 6 fzdzofzdz1mfzdzg 34 3This can also be understood fromthe properties of Integrals 422 Type 2 Discontinuous Integrand Integrals With an integrand that has a discontinuity in the interval of integration have discontinuous integrands Theorem 7 b dz lim dz 35 Theorem 8 Ab 161 dz 013 0 161 dz 36 423 Comparison Rule for Improper Integrals Sometimes the convergence or divergence of an integral cannot be found simply by taking the limit because the integrand is complicated One method to deal With this is to compare the integrand and hence the integral to an integrand that is known to converge or diverge This is known as the comparison rule Theorem 9 Comparison Rule for Convergence We assume that fz 2 0 for z 2 a Suppose we wish to show that dz is convergent It is enough to nd a function gz such that gz 2 for all z 2 a and famgz dz is convergent 0 3 dz 3 gz dz 37 If gz dz lt 00 it follows that dz is convergent Theorem 10 Comparison Rule for Divergence We again assume that 2 0 for all z 2 a Suppose we now wish to show that dz is di vergent It is then enough to nd afunction gz such that 0 S gz S for all z 2 a and famgz dz is divergent 161 412 91 dzg 0 38 If gz dz is divergent it follows that face dz is divergent 43 Tables of Important Integrals l 1 n 7 z dzin1 1 dz lnlzl c 40 z zn1cn7 il 39 ex dxexc 4l sinx dx icosx c 42 l 17 dx arctanx c 43 44 Taylor and MacLaurin Expansions Taylor and MacLaurin series expansions are used to describe the behavior of functions locally around a certain point The error in the approximation grows as you move farther from the point of expansion7 adding more terms gives a more precise approximation for values far7 from the point of expansion De nition 2 MaCLaurin Series The Maclaurin Series Expansion of fx about x 0 is de ned as E Z fnTWx i 44 n0 The rst few terms look like this fltzgt M LSD LEW 45gt De nition 3 The Taylor Series Expansion of about x a is de ned as E anTW 7ayli 46 n0 The rst few terms look like this 439 fltzgt W 7 agt1 7 a 7 agt3 47gt Theorem 11 An expansion of a function can be written in terms of its n h degree Taylor Polynomial plus a remainder term fr Tnr Rn1r 48 Theorem 12 Remainder Term There exists a 5 between a andx such that the error term in Taylor s Formula from above is of the form Rn1x I an1 49 4By setting a 0 in the Taylor Series de nition you recover the MacLaurin Series de nition Math 11B Midterm 2 Review Problems Note these problems are not a representation of all topics that may appear on midterm 2 1 Evaluate the following inde nite integrals a N a 2 make 2x J3x 4 dx Ixe zdx Ix3e 2dx 1 Jxz 9dx I21 dx x 9 J10x6 x12 Jx43x3 2x2xl x22x1 Ie sinxdx dx Evaluate the following de nite integrals 2 1x2 lnx2dx 1 2 N dx xln x2 LAN szof 100 dx 0 7r Ie sinx dx 0 7H4 Itanx sec2 x e392m dx 0 dx x 9 I lldx i 23 x x2723 dx 1 0 gt1 9 50 Determine the partial fraction decomposition of the following rational functions Do not integrate x2 5 x lx 22 x2 x l x lx2 2 Determine the form of a partial fraction decomposition of the following rational functions determine the constants x5 x4 x2 xl x l2x523x 73 x10 x7 x3 x3 x2 xl2x2 52 x2 13 Do not 1 For which p does I ip dx converge 0 x For whichp does I LP dx converge 1 x Determine the convergence or divergence of the following improper integrals integrals Do not evaluate the 1lt1 5 1 lt7 1 xx J al x l x dx Hint Show that for all x Z l b 7 dx Hint Show that for all 0 lt x S1 V J dN 7 re t gt 0 Solve the following initial value problem dt 5 N 0 5 Determine the average value of the following functions on the indicated intervals a lnx on 1 e 1 x2 b on 0 l 10 Determine the area of the following plane regions 713 a The region in the 1st quadrant bounded by y x y 0 x 0 and x l b The region in the 2quotd quadrant bounded by y e y 0 and x 0 c The region in the 4th quadrant bounded by y lnx y 0 and x 0 Math 11B Midterm 2 Review Problems Note these problems are not a representation of all topics that may appear on midterm 2 1 Evaluate the following inde nite integrals a N a make 2x J3x 4 dx Ixe zdx Ix3e 2dx 1 Jxz 9dx I21 dx x 9 J10x6 C12 Jx43x3 2x2xl x22x1 Ie sinxdx Evaluate the following de nite integrals 2 1x2 lnx2dx 1 310 xln x2 szof 100 dx 0 I e sin x dx 0 7r Itanx sec2 x 6392 dx 0 dx 23 I W i x 2723 dx 1 0 gt1 9 50 Determine the partial fraction decomposition of the following rational functions Do not integrate x2 5 x lx 22 x2 x l x lx2 2 Determine the form of a partial fraction decomposition of the following rational functions determine the constants x5 x4 x2 xl x l2x 52 3x 73 x10 x7 x3 x3 x2 x l2x2 52 x2 13 Do not 1 For which p does I LP dx converge 0 x 1 For wh1chp does I 7 dx converge 1 x Determine the convergence or divergence of the following improper integrals integrals w l l l 1 l 0 x Do not evaluate the dx Hint Show that for all x Z l b 7 dx Hint Show that lt i W W4 forall 0ltxSl dN 7 te t gt 0 Solve the following initial value problem dt 5 N 0 5 Determine the average value of the following functions on the indicated intervals a lnx on 1 e l 2 x b on 0 l 10 Determine the area of the following plane regions 713 a The region in the 1st quadrant bounded by y x y 0 x 0 and x l b The region in the 2quotd quadrant bounded by y e y 0 and x 0 c The region in the 4th quadrant bounded by y lnx y 0 and x 0

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