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# Review Sheet for MATH 250A at UA

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Date Created: 02/06/15

MATH 250a Fall Semester 2007 Section 2 J M Cushing Thursday September 13 httprnathariz0naeducushing250ahtrnl Some Basic Facts About Sine and Cosine sin2 x cos2 x 1 for all x 61 sin aux l c2 coswx Asinltwx gt Some Basic Facts About Sine and Cosine sin2 x cos2 x 1 for all x 61 sin aux l c2 coswx Asinltwx gt Some Basic Facts About Sine and Cosine sin2 x cos2 x 1 for all x 61 sin aux l c2 coswx Asinltwx gt 2 2 Cl 62 Some Basic Facts About Sine and Cosine sin2 x cos2 x 1 for all x 61 sin aux l c2 coswx Asinltwx gt c A 612622 surl z 2 2 2 cl 62 2 2 61 62 Cl 62 Some Basic Facts About Sine and Cosine sin2 x cos2 x 1 for all x 61 sin wx 02 cos wx Asinltwx 6 2 c A2461 622 alrl z 2 or Cl 3612 6 6 Cl 62 Example sinxc05xAsinltwx6gt Example sinx I cosszsin wx I Example sinx l cosx Asinltx6gt Example sinx l cosx Asinltx6gt Example sinx I cosx sinltx gt Example sinxcosx xEsin 7T x 4 J Example sinx I cosx xEsinx period 27 amplitude 2 J5 phase period 27139 amplitude J5 phase J 1 4 sinx l cosx Exam le xEsinx 7T 4 Example sinx I cosx xEsinx period 27 amplitude 2 J5 phase y sinxcosx Other Trig Functions Slnx COSX tanx j COSX Slnx 1 1 secx cscx COSX Slnx have singularities at roots of denominators Other Trig Functions s1nx cosx tanx cotx cosx s1nx 1 1 secx cscx cosx s1nx have singularities at roots of denominators tan x cot x are periodic with period 7r sec x CSC x are periodic with period 27r Other Trig Functions ytanx 722 T Basic Facts About Trig Functions dsinx dx Basic Facts About Trig Functions dsinx cos x Basic Facts About Trig Functions d sin x d cos x cos x dx Basic Facts About Trig Functions dsinx dcosx cosx sinx Basic Facts About Trig Functions dsinx dcosx cosx sinx Differentiate other trig functions using differentiation rules Basic Facts About Trig Functions dsinx dcosx cosx sinx Differentiate other trig functions using differentiation rules dtanx dx Basic Facts About Trig Functions dsinx dcosx cosx sinx Differentiate other trig functions using differentiation rules dsinx dcosx s1nx cosx dtanx dx cos2 x Basic Facts About Trig Functions dsinx dcosx cosx sinx Differentiate other trig functions using differentiation rules dsinx dcosx s1nx cosx dtanx dx cos2 x 2 39 2 cos xs1n x cos2 x Basic Facts About Trig Functions dsinx dcosx cosx sinx Differentiate other trig functions using differentiation rules dsinx dcosx cosx s1nx dtanx dx cos2 x 2 39 2 cos xs1n x cos2 x 1 COS2 x Basic Facts About Trig Functions dsinx dcosx cosx sinx Differentiate other trig functions using differentiation rules d sin x d cos x cosx s1nx dtanx dx cos2 x 2 39 2 cos x l srn x cos2 x 1 2 sec2 x COS2 x Inverse Trig Functions ysinx is not monotone for all x Inverse Trig Functions y sin x is not monotone for all x but is monotone on appropriate intervals of x Inverse Trig Functions y sin x is not monotone for all x but is monotone on appropriate intervals of x for example it is increasing on 7r 2 g x 3 7r 2 withrange 1 y 1 Inverse Trig Functions y sin x is not monotone for all x but is monotone on appropriate intervals of x for example it is increasing on 7r 2 g x 3 7r 2 with range 1 g y 31 and therefore has an inverse on this interval Inverse Trig Functions y sin x is not monotone for all x but is monotone on appropriate intervals of x for example it is increasing on 7r 2 g x 3 7r 2 with range 1 g y 31 and therefore has an inverse on this interval yarcsinx on 1 x 1 withrange 7r2 x 7r2 Inverse Trig Functions y sin x is not monotone for all x but is monotone on appropriate intervals of x for example it is increasing on with range 1 g y 31 and therefore has an inverse on this interval y on 1 g x g l withrange 7r2 x 7r2 the principle branch of the inverse Inverse Trig Functions 7I2 I Inverse Trig Functions Inverse Trig Functions y y arcsin x Inverse Trig Functions y y arcsin x 72 1t2 Inverse Trig Functions Review inverses of other trig functions on page 3334 Inverse Trig Functions Differentiation of inverse functions Apply the chain rule to page 140 ff1xx Inverse Trig Functions Differentiation of inverse functions Apply the chain rule to page 140 ff1xx f 39f1x Inverse Trig Functions Differentiation of inverse functions Apply the chain rule to page 140 ff1xx f39f1xf1x Inverse Trig Functions Differentiation of inverse functions Apply the chain rule to page 140 ff1xx f39f1xf1x 1 Inverse Trig Functions Differentiation of inverse functions Apply the chain rule to page 140 ff1xx f39f1xf1x 1 1 d 1 3f xf39f 1x Inverse Trig Functions Example fx sin x f1x arcsinx Inverse Trig Functions Example fx sin x f1x arcsinx d 1 1 3f xf39f1x Inverse Trig Functions Example fx sin x f1x arcsinx d 1 1 3f xf39f1x d arcsin x dx Inverse Trig Functions Example fx sin x f1x arcsinx d 1 1 3f xf39f1x d 1 arcs1n x dx cos Inverse Trig Functions Example fx sin x f1x arcsinx d 1 1 3f xf39f1x d 1 arcs1n x dx cos arcsin x Inverse Trig Functions Example fx sin x f1x arcsinx d 1 1 3f xf39f1x d 1 arcs1n x dx cos arcsin x Inverse Trig Functions Example fx sin x f1x arcsinx d 1 1 if x f39f 1x d 1 arcs1nx dx cosltarcsinxgt Inverse Trig Functions Example fx sin x f1x arcsinx d 1 1 if x f39f 1x d 1 arcs1nx dx cosltarcsinxgt Inverse Trig Functions Example fx sin x f1x arcsinx d 1 1 3f xf39f1x d 1 arcs1n x dx cos arcsin x Inverse Trig Functions Example fx sin x f1x arcsinx d 1 1 3f xf39f1x d 1 arcsin x cos arcsin x dx 1 d 1 arcs1n x x dx 1 x2 1 x2 Chapter 6 Integration Techniques rx 50 A x u 5 9 k 2 390 g 40 quot w Western hequck m A a 7 WhiLE spruce 20 v 4 Western larch whim and Shasta red fir Mountain hwn ock p Engelmann spruce C Ponderma pinr HIacK spute 0 I I I I I I I LmIgepnppino 0 20 40 60 80 100 1 20 DIsranm from med sourm m FIGURE 42 Seed dispersal curves for nine Conifers of the Inland MourrraI39rI West MrCaughey el 1 1986 Dr39spersal charm terisfics of differentspecies can be compared by showing the relative number percentage of seeds found at different disrances from me seed source C L Leadem et al FieldStudiex ofSeedBiology BC Ministry ofForests Crown Publications Inc Victon39a BC 1997 Seed Dispersal Eriogonumfasciculatum 500 i A N S 400 51 O c V g 300 Q3 Q m 5 h 200 Q a S 100 Z 0 n 0 I I c I o 1 2 3 4 Distance m Distance Number of Seeds 111 per 111 0 467 1 74 2 7 3 1 J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 500 NA 1 Some Typical Questions E u u E 400 What 1s the proportion of seeds E 300 between a and 9 meters away E 200 What s the probability a seed will E fall between a and 9 meters away a 5 Z 10 0 What 1s the mean dispersal distance 0 I c I O 1 2 3 4 Distance m Distance Number of Seeds 111 per 111 0 467 l 74 2 7 3 l J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 500 A 1 Some Typical Questions N E 400 3 What 1s the proportion of seeds gm between 0 and 2 meters away 0 E 3 2 39 Total number of seeds 549 E z 100 o 0 I c I O 1 2 3 4 Distance m Distance Number of Seeds 111 per 111 0 467 l 74 2 7 3 l J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 100 80 a o I 0o seeds per m2 20 I 8506 0 399 C 0 1 2 3 Distance m Distance of Seeds 111 per 111 0 8506 1 1348 2 1275 3 01821 Some Typical Questions What is the proportion of seeds between 0 and 2 meters away J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 100 8506 80 a o I 0o seeds per m2 20 1348 1 1 A V 3 2 Distance m Distance of Seeds 111 per 111 0 8506 1 1348 2 1275 3 01821 Some Typical Questions What is the proportion of seeds between 0 and 2 meters away J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 100 80 a o I 0o seeds per m2 20 8506 1348 0 1 2 3 Distance m Distance of Seeds 111 per 111 0 8506 1 1348 2 1275 3 01821 Some Typical Questions What is the proportion of seeds between 0 and 2 meters away 8506 l 1348 9854 J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 100 I 8506 Some Typical Questions 80 N What is the proportion of seeds E 60 between 0 and 2 meters away 850613489854 20 13 48 Most 11kely an over est1mate o 1 c 0 1 2 3 4 Distance m Distance of Seeds 111 per 111 0 8506 1 1348 2 1275 3 01821 J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 100 I 8506 Some Typical Questions 80 N What is the proportion of seeds E 60 between 0 and 2 meters away 850613489854 20 13 48 Most 11kely an over est1mate O I A It is unlikely that the density 0 1 2 I 2 remains constant from 0m to 1m Distance 0quot and instantaneously drops at 2m Distance of Seeds 111 per 111 0 8506 1 1348 2 1275 3 01821 J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 100 I 8506 Some Typical Questions 80 39 N What is the proportion of seeds E 60 between 0 and 2 meters away 850613489854 20 13 48 Most hkely an over est1mate O I A It is unlikely that the density 0 1 2 I 2 remains constant from 0m to 1m Distance 0quot and instantaneously drops at 2m Distance 0f Seeds More likely the density 1 per 111 continuously decreases 0 8506 1 1348 2 1275 3 01821 J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 100 0 Some Typical Questions 80 What is the proportion of seeds between 0 and 2 meters away 8506 l 1348 9854 a o I 0o seeds per m2 20 Most likely an over estimate O 0 n A It is unlikely that the density 0 1 2 4 remains constant from 0m to 1m Distance 0quot and instantaneously drops at 2m More likely the density suppose we had more data continuously decreases J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 100 80 a o I 0o seeds per m2 20 l l A I V I o 1 2 3 Distance In Some Typical Questions What is the proportion of seeds between 0 and 2 meters away 8506 l 1348 9854 Most likely an over estimate It is unlikely that the density remains constant om Om to 1m and instantaneously drops at 2m More likely the density continuously decreases J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 100 I Some Typical Questions 80 N What is the proportion of seeds E 60 between 0 and 2 meters away 8 8 4o 20 0 39 I C l O 1 2 3 4 Distance m 12 X 8506 etc J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 100 I Some Typical Questions 80 N What is the proportion of seeds E 60 between 0 and 2 meters away 8 8 4o 20 0 I c I O 1 2 3 4 Distance m Answer sum of four rectangular areas J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 100 Some Typical Questions 80 What is the proportion of seeds between 0 and 2 meters away a o I 0o seeds per m2 20 001 A I v I O 1 2 3 Distance m This procedure suggests that we are approximating the area under a continuous density function that the data is a sampling J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Review Of Definite Integrals fbftdt1imftjAt gtOO Review Of Definite Integrals fbftdt1imftjAt y ft HO T base ti At a t ti At b Review Of Definite Integrals fbftdt1imftjAt gtOO y t 39 height ti At a t ti At b Review Of Definite Integrals fbftdt1imftjAt y ft HO T area of approximating rectangle ti At a t ti At b Review Of Definite Integrals fbftdt1imftjAt gtOO y ft sum of areas of all approximating rectangles ti At a t ti At b Review Of Definite Integrals fbftdt1imftjAt gtOO y ft 39 limit as number of approximating rectangles increases Without bound ti At a t ti At b Seed Dispersal Eriogonumfasciculatum 10 1 Some Typical Questions 08 N What is the proportion of seeds Eoe between 0 and 2 meters away 8 8 04 3 02 O 00 I I O 1 2 3 4 Distance m For example fx 2 2086 202 J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 101 08 A quots 5 u o 06 Q V m 3 o 04 m C 02 O 00 I I 0 1 2 3 4 Distance m For example fx 2 2086 202 J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Some Typical Questions What is the proportion of seeds between 0 and 2 meters away 2 2 f fxdx f 2086 208de 0 0 Seed Dispersal Eriogonumfasciculatum 101 08 A quots 5 u o 06 Q V m 3 o 04 m C 02 O 00 I I 0 1 2 3 4 Distance m For example fx 2 2086 202 J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Some Typical Questions What is the proportion of seeds between 0 and 2 meters away 2 2 f fxdx f 208e 208xdx 0 0 Seed Dispersal Eriogonumfasciculatum o 08 A quots 5 u o 06 Q V m 3 o 04 m C 02 O 00 I I 0 1 2 3 4 Distance m For example fx 208e2 08x Some Typical Questions What is the proportion of seeds between 0 and 2 meters away 2 2 f fxa x f 208e23908xa x 0 0 Can evaluate using the Fundamental Theorem of Calculus J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Fundamental Theorem 0f Calculus mew F02 Fm where dFm dt f0 Seed Dispersal Eriogonumfasciculatum 10 1 Some Typical Questions 08 on What is the proportion of seeds Eoe between 0 and 2 meters away 04 2 2 2 08x f fxdx f 2086 dx c 0 0 02 fx 2082ng Fx o 3 2 Distance m For example fx 2 2086 202 J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 10 1 Some Typical Questions 08 N What is the proportion of seeds Eoe between 0 and 2 meters away g 04 2 2 2 08x f fxdxf 2086 dx c 0 0 02 O 2086 208x gt e 208x o 3 2 Distance m For example fx 2 2086 202 J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum o 08 A quots 5 u o 06 Q V m 3 o 04 m C 02 O 00 I I 0 1 2 3 4 Distance m For example fx 2 2086 202 Some Typical Questions What is the proportion of seeds between 0 and 2 meters away 2 2 f fxdx f 2086 208de 0 0 fX 2 2082ng gt FOO e 208x 2 f 2085208 F2 FO 0 7416 x 09844 J M Mahaffy and A ChavezRoss Calculus A Modeling Approach for the Life Sciences Volume I Pearson Custom Publishing 2004 Seed Dispersal Eriogonumfasciculatum 1 Some Typical Questions 08 N What is the proportion of seeds Eoe between 0 and 2 meters away 8 8 04 3 02 O 00 I I Y I O 1 2 3 4 Distance In Another example 2567 gtlt104612395xl10 Seed Dispersal Eriogonumfasciculatum 1 Some Typical Questions 08 N What is the proportion of seeds Eoe between 0 and 2 meters away 0 4 2 f fx dx e O 02 O 00 I I Y I O 1 2 3 4 Distance In Another example 2567 gtlt104612395xl10 Seed Dispersal Eriogonumfasciculatum 1 Some Typical Questions 08 N What is the proportion of seeds Eoe between 0 and 2 meters away 0 4 2 r39 ffmw e O 02 F x 00 I I Y I O 1 2 3 4 Distance In Another example 2567 gtlt104612395xl10 Seed Dispersal Eriogonumfasciculatum 1 Some Typical Questions N I What is the proportion of seeds EOG between 0 and 2 meters away 2 f fx dx a 0 F x 00 I Y Distance m Turns out it s not 4 125x110 possible to find a formula Another example 2567 X 10 e for FCC What to do

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