Calculus Placement Review. x x. =. Find each of the following. 9 = 4 ( )

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1 Calculus Placement Review I. Finding dmain, intercepts, and asympttes f ratinal functins 9 Eample Cnsider the functin f ( ). Find each f the fllwing. (a) What is the dmain f f ( )? Write yur answer in interval ntatin. (b) What are the intercepts f f ( )? Label apprpriately. (c) What are the asympttes f f ( )? Label apprpriately. Slutin (a) The denminatr cannt be zer, s we need t slve 0. This gives ±,,,.. Hence, the dmain is ( ) ( ) ( ) (b) The -intercepts are fund by slving ( ) 0 are in the dmain f f()) ( + )( ) 0 ± S, the -intercepts are (,0) and (,0). The y-intercept is fund by evaluating f ( 0). f ( ) f (and checking the slutins S, the y-intercept is 9 0,. (c) Since this ratinal functin is in lwest terms (the numeratr and denminatr d nt share a cmmn factr), the vertical asympttes ccur when the denminatr is equal t zer. Therefre, the vertical asympttes f y f ( ) are and. The degree f the numeratr and the degree f the denminatr are equal (they are bth in this case), which means this ratinal functin has a hrizntal asymptte. When this is the case, the hrizntal asymptte can be fund

2 evaluating the rati f the leading cefficients. In this case, this rati is, s y f is y. the hrizntal asymptte f ( ) In rder fr a ratinal functin t have an blique (r slant) asymptte, the degree f the numeratr must be eactly ne greater than the degree f the denminatr. This is nt the case fr y f ( ), s this functin des nt have an blique asymptte. II. Slving epnential equatins Eample Slve the fllwing equatin: 8 + Slutin ( ) ( ) III. Using prperties f lgarithms Eample Rewrite lg 6lg y + lg z as a single lgarithm with a cefficient f ne. Slutin lg 6lg y + lg z lg lg lg y 6 z 6 y lg + lg y z 6 + lg z

3 IV. Finding eact values f trignmetric functins Eample Withut using yur calculatr, find the eact values f the si trignmetric functins f π. Slutin The reference angle fr in the third quadrant. Therefre π π is (which is π sin π cs π tan π ct π sec π csc 60 ), and the terminal side f π is V. Applying sum, difference and duble angle identities. Eample Find the eact value f each f the fllwing belw given that π π tan α, < α < π and cs β,0 < β <. (a) csc β (b) sin ( α + β ) (c) cs ( α )

4 Slutin Using reference triangles, we see Fr the triangle invlving α, the missing side is the hyptenuse, which we can find using the Pythagrean therem. + c ( ) c c c Fr the triangle invlving β, the missing side is the ppsite side, which we can again find using the Pythagrean therem. + b + b b b ± Since the triangle is in Quadrant I, we use the psitive slutin f b. (a) (b) sin ( α β ) csc β + sinα cs β + sin β csα + 0

5 (c) cs ( α ) cs α sin α 7 VI. Graphing trignmetric functins Eample Graph ne perid f scale accurately. Slutin y cs and ne perid f y tan. Be sure t label yur y cs

6 y tan VII. Prving trignmetric identities Eample Prve the fllwing identity: csc ( + tan sin tan ) csc cs Slutin csc ( + tan sin tan ) csc + csc tan sin ( cs ) sin csc + sin cs csc + ( cs ) cs csc + cs cs cs csc cs csc tan cs sin sin cs

7 VIII. Slving trignmetric equatins Eample Withut the use f a calculatr, slve the fllwing equatins. Slutins shuld be in radians. (a) sin sin 0 (c) csθ sin θ Slutin (a) (b) sin sin sin 0 ( sin ) 0 sin 0 OR sin sin 0 OR sin 0 sin 0 OR sin ± π π kπ OR + kπ OR + kπ OR π π + kπ OR + kπ csθ sin csθ ( csθ )( csθ + ) ( cs θ ) csθ cs cs θ + csθ 0 0 csθ 0 OR csθ + 0 csθ OR csθ π π θ + k π OR θ + kπ Nte that the equatin csθ has n slutin. θ θ IX. Slving triangles using Law f Sines and Law f Csines Eample The fllwing infrmatin refers t a triangle ABC. Find all the slutins, if there are any, t the fllwing triangle. Rund yur answers t fur decimal places. c.6, γ 8., b.6

8 Slutin Using the Law f Sines sinγ sin β c b sin 8. sin β.6.6.6sin 8. sin β.6 sin β 0.60 β sin 7. α ( 0.60) β γ sinγ sinα c a sin 8. sin.7.6 a.6sin.7 a sin Checking fr a secnd triangle, we see that β culd als be , and since β + γ < 80, there is a secnd triangle. α sinγ sinα c a sin 8. sin a.6sin 8.9 a sin β γ.9 8.