Download Math 005A Prerequisite Material Answer Key

Math005APrerequisiteMaterialAnswerKey
1. a) P = 4s (definitionofperimeterandsquare)
b) P = 2l + 2w (definitionofperimeterandrectangle)
c) P = a + b + c (definitionofperimeterandtriangle)
d) C = 2π r (definitionofcircumference)
e) s = rθ (definitionofarclength,radius,andcentralangle)
2. a) A = s 2 (definitionofareaofasquare)
b) A = lw (definitionofareaofarectangle)
c) A = π r 2 (definitionofareaofacircle)
1
d) A = r 2θ
2
(definitionofareaofasector,radius,andcentralangle)
1
1
a+b+c
e) A = bh , A = absinC , A = s ( s − a ) ( s − b) ( s − c ) ,where s =
2
2
2
(standardareaofatriangleformula,SASareaformula,Heron’sFormula)
B+b
f) A = h
2 (Areaofatrapezoidformula)
g) S = 4π r 2 (Surfaceareaofsphereformula)
3. a) V = lwh (definitionofvolumeofrectangularsolid)
1
b) V = bhl
2
(Volumeofrightprismwithtriangularbaseformula)
B+b
c) V = lh
2 (Volumeofrightprismwithtrapezoidalbaseformula)
d) V = π r 2 h (Volumeofacylinderformula)
1
e) V = π r 2 h
3
(Volumeofaconeformula)
1
f) V = lwh
3
(Volumeofapyramidformula)
4
g) V = π r 3
3
(Volumeofasphereformula)
(note:3a,b,c,dareallexamplesofrightprisms,whichhavethegeneral
formulaVolume=AreaofBase×Heightoftheprism.In3bandctheheightof
theprismisdenotedwith𝑙andtheℎdenotestheheightofthebase)
4. a) −x 3 + 4x 2 − 4x − 9 (distributiveproperty,combiningliketerms)
b) 8x 3 − 24x 2 − 48x + 4 (distributiveproperty,combiningliketerms)
c)
4x − 5
(distributiveproperty,combiningliketerms)
3x + 4 d)
4x 2 +10x − 4
( x + 2) ( x − 2) ( x + 3)
(factoring,findingacommondenominator,equivalent
fractions,distributiveproperty,combiningliketerms)
e) 4x 2 −12x + 9 (FOILordistributiveproperty,combiningliketerms)
f)tan 𝑥 sec 𝑥 identity)
(Pythagoreantrigidentity,quotienttrigidentity,reciprocaltrig
g) 2sin x (doubleangleformulaforsine)
h) x − 2 (factoring)
1
23
5. a) y = x +
5
5 (slopeofaline,pointslopeformulafortheequationofaline)
1
3
b) y = − x −
2
2 (slopesofperpendicularlines,pointslopeformulaforthe
equationofaline,isolatingavariable)
c) y = 2x + 4 (slopesofparallellines,pointslopeformulafortheequationof
aline)
6. a) 29 (distanceformula)
b) d(x) = x 2 − 5x −10 x + 34 (distanceformula,substitution–youare
substitutingsquarerootxforyinthedistanceformula,FOIL/distributive
property,combiningliketerms)
7. Forarighttrianglewithlegsoflengthaandbandhypotenusec, a 2 + b 2 = c 2 .
(Pythagoreantheorem,definitionofrighttriangle,definitionofhypotenuse)
8. a) d(t) = 60t (distance=ratextime)
b) d(t) = 20t 13 (modelingwithfunctionsbyplacingtheinitialstartingpointat
theoriginandexpressingthepositionofeachcarasapointwithtwo
coordinateswithtasavariable,distanceformula,simplifyingpolynomials,
commonfactor,simplifyingexpressionsinvolvingsquareroot)
P2
9. a) A(x) =
(isolatingforavariable,areaofasquare,substitution,
16
simplifying)
8
b) h = r
5 (similartriangles,proportions,solvingforavariable)
6
c) h = b
5 (similarity,proportions,solvingforavariable)
10. a) x 5 (Multiplicationlawofexponents)
b) x 6 (Powerofapowerlawofexponents)
c) x (Divisionlawofexponents)
d)isalreadysimplified
2
e) 2
x (negativeexponents,multiplyingafractionbyaconstant)
2
f)
x2
2 (Divisionlawofexponents)
11. a) x (Fractionalexponents)
b) 3 x 2 1
c)
x
(Fractionalexponents,indexofaroot)
(Fractionalexponents,negativeexponents)
2
x (Fractionalexponents,negativeexponents,multiplyingafractionbya
constant)
12. a) x1/2 (Fractionalexponents)
d)
b) x −1/4 (Fractionalexponents,negativeexponents)
c) x 3/2 (Fractionalexponents,divisionlawofexponents,fractionsubtraction)
x +1
d) 1/2
x (Fractionalexponents,findingacommondenominator,Multiplication
lawofexponents)
13. a) 3 (1+ 3x ) (1− 3x )
(GCF,differenceoftwosquares)
b) x ( x +10 ) ( x − 2 )
(GCF,factoringquadratics)
c) ( 2x + 5) 4x 2 −10x + 25
(grouping,sumoftwocubes)
d) x 3 +1 ( x +1)
(grouping,GCF)
e) − ( x + 2 ) ( 5x + 9 )
(GCF,distributivelaw,combiningliketerms)
(
)
(
)
f) (3x − 5) ( 5x + 9 )
(Differenceoftwocubesusingsubstitution,
FOIL/distributivelaw,combiningliketerms)
3
g) 6 ( x + 5) ( x −1) ( x +1)
(GCF,distributivelaw,combiningliketerms,GCF)
−3x 2 − 8x + 3
(3x −1) ( x + 3)
14. a)
or −
2
2
2
x 2 +1
x
+1
( ) (distributivelaw,combiningliketerms)
(
b) −
c) −
d)
)
2 ( x 2 − 5x − 9 )
(x
2
+ 9)
2
(distributivelaw,combiningliketerms,factoringGCF)
4 (3x 2 −1)
( x 2 +1)
3
(factoringGCF,distributivelaw,combiningliketerms)
x+2
2 (1+ x )
3/2
(LCD,multiplyingrootswiththesamebase,distributivelaw,
combiningliketerms,convertingradicalstoexponentialform,multiplicationlaw
ofexponents)
e) −
x−4
( x + 4)
1
a 2 + a
3/2
− 12
=a
− 12
[a + 1] ,combining
(eitherLCDorfactoringtrick
liketerms,lawsofexponents)
f)
1
1/2
( x +1) (5x + 2)
2
g) 4x −2/3 ( 2x −1)
1
1
h) x 2/3 − x 5/3
2
2
(factoringGCF,combiningliketerms)
(factoringGCF)
(distributivelaw,lawsofexponents)
15. a)Nosolution
(rationalexpression=0whennumeratoris0,common
denominator,multiplicationlawofexponents,distributivelaw,combininglike
terms,solvingalinearequation,checkanswers,cannothavenegativeunder
squareroot)
b) x =
1
3 (rationalexpression=0whennumeratoris0,common
denominator,multiplicationlawofexponents,combiningliketerms,solvinga
quadratic,checkanswers,cannothavenegativeundersquareroot)
c) x = −5, −1, 1 (factoroutGCF,distributivelaw,combiningliketerms,solving
linearequations)
5
d) x = − , 1
2 (commondenominatorforRHS,multiplybothsidesbyLCMof
denominators,distributivelaw,combiningliketerms,factoring,solvinglinear
equations)
1
2 (moveeverythingtooneside,factorbygrouping,solvinglinear
equation)
e) x =
π
π
5π
+ 2kπ , + kπ ,
+ 2kπ , wherekisaninteger
(doubleangle
6
2
6
formulaforsine,factor,solvingatrigequation,periodicityofsineandcosine)
f) x =
π
(1+ 2k ) π , 5π + 2kπ , wherekisaninteger (doubleangle
+ 2kπ ,
6
2
6
formulaforcosine(theonethathassine),distributivelaw,factoringwith
substitution,solvingtrigequations,periodicityofsineandcosine)
g) x =
1
x + h + x (multiplyingby1(i.e.topandbottombytheconjugate),FOIL
ordifferenceoftwosquaresformula,combiningliketerms)
1
b) −
x x+h x + x+h
(findingacommondenominator,multiplyingtop
16. a)
(
)
andbottombytheconjugate,differenceoftwosquaresformula,distributivelaw,
combiningliketerms)
2
17. a)
xy
(negativeexponents,findingacommondenominator,factoring)
x
b)
2
(LCM,multiplyingby1,distributivelaw,combiningliketerms,
( x +1) factoring)
5x − 4
c) −
( x + 3) ( x +1) ( x − 2) (LCM,distributivelaw,dividingx+3isthesameas
multiplyingbythereciprocal1/(x+3))
18. a) −27, (3, − 27)
(substitution,usingafunctiontofindtheycoordinateof
apointonthegraphofafunction)
b) 2 5/3 or 4 3 2 , 4, 2 5/3
(substitution,usingafunctiontofindtheycoordinateof
apointonthegraphofafunction)
19. a) −4x − 2h + 3 (substitution,distributivelaw,combiningliketerms,factoring)
2x + h + 4
b) −
(substitution,distributivelaw,combiningliketerms,
2
2
( x + 2) ( x + h + 2) factoring)
1
c)
(substitution,multiplyingtopandbottombyconjugate,
x+h−3+ x −3 distributivelaw,combiningliketerms)
20. Usedesmos.comorwolframalpha.com
#−x if x < 0
21. a) f (x) = $
% x if x ≥ 0 (definitionofabsolutevalue)
#
5
%%−2x − 5 if x < −
2
b) f (x) = $
%2x + 5 if x ≥ − 5
%&
2
(solvinglinearinequalitiesinvolvingabsolute
(
)
value)
#
π
%%cos2x − sin x if 0 ≤ x < 6
c)(see#15g) h(x) = $
%sin x − cos2x if π ≤ x ≤ π
%&
6
2
(solvingatrigonometric
equation,doubleangleformulaforcosine,factoringwithsubstitution)
&$ x 2 − 4 if x ≤ −2 or x ≥ 2
d) y = %
&' 4 − x 2 if − 2 < x < 2
(factoring,solvingquadraticinequalitiesinvolving
absolutevalue)
22. a) { x | x ≠ ±2}
(factoring,can’thave0indenominator)
b)allrealnumbers
(understandinganirreduciblepolynomialhasnoreal
roots,can’thave0indenominator)
c) { x | x ≥ 3}
(solvingalinearinequality,can’thavenegativenumbersunder
thesquareroot)
d) { x | x > 3}
(solvingalinearinequality,can’thavenegativenumbersunder
thesquareroot,can’thave0indenominator)
e) { x | x > 4}
(domainofln,solvingalinearinequality)
f) { x | x < −1 or x > 6}
(factoring,solvingaquadraticinequality,can’thave
negativenumbersunderthesquareroot)
⎧
( 2k +1) pi, where k is an integer ⎫
g) ⎨ x | x ≠
⎬
2
⎩
⎭ (domainoftangentfunction,
periodicityoftangent,beingabletocomeupwithanalgebraicdescriptiongiven
apattern)
h) { x | x ≠ 3}
(denominatorcan’tbe0)
i)allrealnumbers
(domainofalinearfunction)
23. a) y = 25 − x 2 (understandingthedifferencebetweenthepositiveand
negativesquareroot)
b) y = − x − 2 + 3 (understandingthedifferencebetweenthe
positiveandnegativesquareroot)
24. a) sin 2 θ + cos2 θ = 1, tan 2 θ +1 = sec 2 θ , 1+ cot 2 θ = csc 2 θ b) sin 2x = 2sin x cos x (doubleangleformulaforsine)
cos2x = cos2 x − sin 2 x
c)
2 cos2 x −1
2
1− 2sin x
(doubleangleformulaforcosine)
d) c = a + b − 2ab cosC e) sin 0 = 0 f) sin (α + β ) = sin α cos β + cos α sin β
(sumformulaforsine)
g) cos (α + β ) = cos α cos β − sin α sin β
(sumformulaforcosine)
h)seenextpage
2
2
2