Analytic Geometry and Calculus 1
Analytic Geometry and Calculus 1 MATH 1300
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This 11 page Class Notes was uploaded by Cydney Conroy on Thursday October 29, 2015. The Class Notes belongs to MATH 1300 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/231835/math-1300-university-of-colorado-at-boulder in Mathematics (M) at University of Colorado at Boulder.
Reviews for Analytic Geometry and Calculus 1
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Date Created: 10/29/15
Math 1300 Fall 2005 Review Sheet for Midterm Exam 3 1 Use linear approximation to nd sin47 2 Use an appropriate local linear approximation to estimate 48 3 Assume the earth is a perfect sphere and that the radius of the earth at the equator is 3960 miles or 20908800 feet Imagine that a string is wrapped tightly around the earth at the equator Then imagine that the string is lengthened by an amount that allows it to be strung all the way around the earth at the equator on short poles that are feet above the ground feet is a little less than of a foot or about 4 inches Use differentials to approximate the increase in the length of the string 4 Find for the following functions some of which may be de ned implicitly a y lnsinz 13 b y 5sinrwg c y tan 1ln I WM 5 Evaluate the following limits 39 Te 14 T a 11m 4 716273 574g1101 b limggnlr00 11 E C limx7oltgt 7 e214 d limgcnlr00 11 e limgcnir00 1 ln a mom i am g lim nmz sin I1 mmwm 6 Given 13 1212 3 t 7 213712 352 WWW ww 1W f I nd a z and y intercepts b vertical and horizontal asymptotoes c critical points classify each as a relative maximum relative minimum or neither d intervals where f is increasing and decreasing e in ection points f intervals where f is concave up and concave down g Sketch the curve 7 Same as 6 given I4 7 4x2 71 413 7 81 f z 1212 7 8 8 What is the length of the shortest line segment lying wholly in the rst quadrant tangent to the graph of y i and with its endpoints on the coordinate axes 9 A closed cylindrical can is made using 2 sq ft of material for the sides top and bottom What height and radius would maximize the volume of the can 10 Let z521 1 1 Use the Intermediate Value Theorem to show that the equation 0 has at least one solution on the interval 170 2 Use the Mean Value Theorem to show that there is exactly one solution to the equation 0 on the interval 170 11 Let 12 on the interval a7b for a lt 12 Which of the following values of c satisfy the conclusion of the Mean Value Theorem for f 7a2b2 2 251227a2 ab 3i 7 C 2 4lcbia 12 Use the Mean Value Theorem to prove Rolle7s Theoreml 13 Suppose Harold and Kumar race each other to White Castle and arrive at the same time Use the MVT to show that there was at least one moment when they had the same velocity Math 1300 Calculus I Fall 2006 Answer Sheet for Review 3 1 Your graph should look something like this 01 0 No limgt0 does not exist 3 3 115 and 115 4 Answer 6 01 A 33 M3 pilH V 00 H M3 02H k0 70 8 8n 1 8 b 1 3 6 Solution 1 Increasing oo 2 U 0 oo Decreasing 20 2 Relative Maximum 24c2 Relative Minimum 00 3 Concave Up 00 2 U 2 00 Concave Down 2 2 2 ln ection Points 1 2 l 2 1 2 7 f attains its minimum of 1 at z 0 By the de nition of an absolute minimum f 2 1 gt 0 for all x The desired inequality follows directly from this 8 To prove the rst statement in the Hint suppose that the graph of f has a point P b fb with bin I such that P lies below or on the tangent line I and a lt I Then w 3 the slope of Z fal Since f is differentiable on the open interval I and a b are in I we get that f is continuous on ab and differentiable on ab Hence by the Mean Value Theorem 1 7 w fc for some 0 in ab 7 a Thus f a 2 f c holds for this 0 in ab The proof of the second statement in the Hint is similar These two statements show that if some point of the graph of f on the interval I is below or on the line Z then f is not increasing on I that is f is not concave up on I Hence if f is concave up on I then every point of the graph of f on I is above the line I 9 Note that f is not continuous on 712 Now f 1 7 4 and W g 2 in the real numbers but there is no solution to 1 7 z 2 10 False as f cm is a counterexample 11 11 1 11 f1 fzd 0 fzd 7 f0 fzd 29 7 77 36 12 fog el dx 13 Critical points at z 0 and z 1 absolute maximum at z 0 absolute minimum at z 71 14 Answer 1 15 f is increasing on 7oooo since f is always positive 16 f has an in ection point at z a if n is odd and 2 3 17 SYMMETRIES none INTERCEPTZ 0 y INTERCEPT y 0 VERTICAL ASYMPTOTE The line 95 1 3 Homomut ASYMPTOTE Since hnn M 0 and hm m on the hne n 0 1s a honzontal asymptote 12e 2525wr Dmvmvas f m and f m came poms e 712 and e 0 mohmsmsDmmusms summon at 1s decreasing on the 1ntem1 700 7121 and Increasing on the mtexwls 712 1 and 1 00 RELATIVE EXTREMA me has a nelatwe mlmmum at e 712 coucuvm The numbers for whxch f 0 are 7 an s f s unde ned for e 0 an 1scont1nuous at e 1 Checking the sly o1 f on the mtexwls 1 a 5 determined by these numbers we get that ab 1s concave down on 700 7 a T A 1 a 5 1 a 5 1 concave up on 77 F41 concave down on 0 i le concave up on 7 3 5 and concave down on 1oo WE W 1 a 5 Imaonou poms 7 e w on and w Your gaph should look hke ths ANSWER TO 18 IN SECTION 53 2 a 2 L rHUasw ioo e e The graph should look hke where the dotted green curve ls the curvrlrnear asymptote 18 30 km lrorn B 19 Here are the answers 1 The rlntexcep39s are u and 7amp0 There are no yrlntelc pts 2 There are no crrtrcal pornts srnce f w rs never zero and e 0 where there s unde ned s not ln the dornarn ol 3 f rs lncreasmg on 7000 and 0oo f s concave up on 7000 and concave down on o oo 4 There s a vertrcal asymptote e 0 and an obllque asymptote y e s i E The graph should look lrke Answers for Tyne or False a False esrn For example llm 7 rs an rndeterrnrnate lorrn ol type oooo such that M c 1 um e 1 but um e um dos not me am at am Ew tam 1 b Fals e For example my a0 rs lncreasmg and dl elentlable on em 00 but f 0 0 C d e f True lff is not increasing on I7 then there exist a lt bin I such that fa 2 fb Since f is differentiable on I7 it follows that f is continuous on 1 b and differentiable on 17 b As f is not increasing on 1 b we get from Theorem 512 that f 4 0 for some z in I False The function f x3 does not have a relative extremum7 but f 0 0 False The function f 5 has an in ection point at z 07 but f 0 is not de ned True lf f x is continuous on I and f x 31 0 for all x in I7 then by the Intermediate Value Theorem f x does not change sign on I Hence either f z gt 0 for all z in I7 so f is concave up on I7 or f z lt 0 for all z in I7 so f is concave down on I True If we and Mm such that Pz qzQ Mm and the degree of Mm is less than the degree of Hence QWQW Wt PW is a rational function7 then long division yields polynomials q where i r s 31330 m Thus y qx is an asymptote of f False f 5 has a vertical tangent line at z 07 but it has no vertical asymptote since it is continuous everywhere r 0 lim 7 maioo True De ne a function h by h g 72fz for all z in foo7 00 The assumptions imply that h0 0 and h z g z 7 2f x 0 for all z in 70000 Therefore by the Constant Difference Theorem h 0 for all z in 70000 Hence g 2fx for all z in 70000 False Let 1z if z 31 07 f0 ifx0 Then f is differentiable on 017 f lt 0 for all z in 017 and w 1 7 0 1 gt 0 Therefore there is no 0 in 01 such that f 0 Math 1300 Fall 2005 Review Sheet for Final Exam 1 Evaluate the following de nite and inde nite integrals 2 3z5 7z2 z a dz 1 z b tan2 z dz og 4a 2 g 4 d 1 J dz e z4z5 7 232 dz M4 f tanzdz 0 g z5l 7 z25 dz 2 e in z2 h T dz 2 2 i 2zex dz 0 j4z3 l cosz4 z dz 4 z 1 1W 4 2 Sglve the following initial value problems w 7 iww b 3z 172 yl2 3 Find the exact area under the curve z 7 1 over the interval 13 using Riemann sums With righthand endpoints ie7 z zk a kAz You may nd the follWing formulae useful n n nnl ln7 ZkT k1 k1 2 4 Evaluate the integral dz7 given that 72 fzz2 zgt0 z zSO 5 De ne Fz by Fz 6 dt 0 a Use Part 2 of the Fundamental Theorem of Calculus to nd Fz b Check the result in part a by rst integrating and then differentiating 6 Find the area of the region enclosed by the curves y z2 and y 7 Find the area of the region enclosed by the curves y z7 y 4z7 and y 2 7 z 8 Find the volume of the solids that result When the region enclosed by the curves is revolved about a the zaxis b the y axis 9 Set up but DO NOT EVALUATE integrals that express the volume of the solids that result When the region enclosed by the curves is revolved about a the zaxis b the y axis Multiple Choice and TrueFalse questions d 10 Find i if 13y4 I7 dz A B y 2y4 c 715 31 D 715 7 312w E 0 ll Than 6173 Agto Bw 07w Dgt1 E71 12 The function 714 7 612 is concave up on mcmaw mewaw Pw4 ammmm mew 13 The function 3sin12 has an absolute minimum of A75 373 00 D7 E7 14 Express the number 60 as the sum of two nonnegative numbers Whose product is as large as possible A 555 B 1050 C 30 30 D 159 None of the above 15 True or false Given 12 7 9 on 73 3 the value c that satis es the conclusion of Rollels Theorem is c TRUE FALSE 16 True or false Given 13 on 02 the value c that satis es the conclusion of the Mean Value Theorem is c l TRUE FALSE Math 1300 Spring 2006 Review Sheet Answer Key for Midterm Exam 2 The solutions were typed up fast and very well may contain errors Hope this helps 1 32 Use the de nition of the derivative to nd 37 if y Tilt dy 1 h1 7 11 1 ans 7 hm 1 h7gt0 h 7 39W 2 32 Let y Find the derivative of y using i the limit de nition of the derivative 1 W 1 h ansi 37y lim 7 h7gt0 ii the power rule 1 ans e ewe77 d 3 Find d7 d 7 1 33 y 62 i 7 2 35 y 2Sig2sx 2sec221 3 36 y sin2sin2z 27 2 sinsin2 z cossin2 I2 sinz cos z 4 4 1 siny lnz y 2 if cosy7 zy 5 42 y 111COSI jig isinx cos as 7 1233wsinx72 dy Be 051 6 42 y 7 In W a E 123 sinx72 2e7 7393 W 7 43 y W7 320 2 Wee 2e 8 43 y 5inm 7 5inltgt cosz 4 35gt Find 1 and f E if fltzgt sine 0031 fI sinQI cos2 I f 7 5 41 Find all values of I at which the curve y3 yIQ 12 7 3y2 0 has a horizontal tangent linei ansi 07 07 07 3 6 33 Find an equation for the tangent line to the graph of y 512 7 3713 I at z 4 ans y 7 34804 44029z 7 4 l 7 4 4 Calculate lim 1 7f acan 1 ans 6 1 3 7 Erika is cleaning the gutters on her house using a 10ft ladder propped up against the wall Emily pulls the base of the ladder away from the wall at a rate of 3 ftseci how fast is the top of the ladder falling down the wall when it is 6 ft from the ground possible hint your answer should be negative 74 ftseci 2 37 The public health spending in dollars of a certain town is given by the equation 5 7500 ln2p 50007 where p is the population of the town If the population of the town is growing at a steady rate of 100 people per year7 at what rate is the public health spending of the town increasing when the population is 5000 people 100 dollars year 3 41 The minute hand of a watch begins melting at a certian rate when it reaches the top of the hour As the hand begins rnoving7 what is the formula for the rate at which the area swept out by the hand is changing hint The area for the whole circle is 7T7 2 nd a relation between any generic angle 9 and the rotation around a whole circle to nd the area of a portion of a circle in terms of 7 and t9 2 9 9r2 dAird A T7SOT 33
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