1 Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo with questions.
2 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse function for ll 6= 0. Find f lgericll nd show tht f is self-inverse function. 8 Show tht f : 7! 8, 6= is self-inverse function : reference to its grph using lger. 9 Consider the function f() =. Find f (). Find: i (f ± f )() ii (f ± f)(). 0 Consider the functions f : 7! + nd g : 7! 8. Find g ( ). Solve for if (f ± g )() =9. Consider the functions f : 7! nd g : 7! p. Find: i f() ii g (): Solve the eqution (g ± f)() =. Given f : 7! nd g : 7!, show tht (f ± g )() =(g ± f) (). Which of these functions is self-inverse function, so f () =f()? f() = f() = c f() = d f() = e f() = 6 The horizontl line test ss: For function to hve n inverse function, no horizontl line cn cut its grph more thn once. Eplin wh this is vlid test for the eistence of n inverse function. Which of the following functions hve n inverse function? i ii iii - REVIEW SET A NON-CALCULATOR If f() = find: f() f( ) c f( ) If f() = + where nd re constnts, find nd for f() = 7 nd f() =. If g() =, find in simplest form: g( + ) g( )
3 66 FUNCTIONS (Chpter ) For ech of the following grphs determine: i the rnge nd domin ii the nd -intercepts iii whether it is function. -\Wl_T_ (, ) Drw sign digrm for: ( + )( ) If f() = +, f() = nd f () =, find nd. 7 Cop the following grphs nd drw the inverse function on the sme set of es: 8 Find f () given tht f() is: + 9 Given f : 7! +6 nd h : 7!, show tht (f ± h )() =(h ± f) (). REVIEW SET B For ech of the following grphs, find the domin nd rnge: ( )( ) (, ) CALCULATOR If f() = nd g() = +, find: (f ± g)() (g ± f)() Drw sign digrm for:
4 FUNCTIONS (Chpter ) 67 Consider f() =. c For wht vlue of is f() meningless? Sketch the grph of this function using technolog. Stte the domin nd rnge of the function. + Consider the function f() =. Find nd given tht = f() hs smptotes with equtions = nd =. Write down the domin nd rnge of f (). + 6 Consider the function f : 7!. Determine the equtions of the smptotes. c d Discuss the ehviour of the function s it pproches its smptotes. Determine the es intercepts. Sketch the grph. 7 Consider the functions f() = + nd g() =. Find (g ± f)(). Given (g ± f)() =, solve for. c Let h() =(g ± f)(), 6=. i Write down the equtions of the smptotes for the grph of h(). ii Sketch the grph of h() for 6 6. iii Stte the domin nd rnge of h(). 8 Consider f : 7! 7. On the sme set of es grph =, f nd f. c Find f () using vrile interchnge. Show tht f ± f = f ± f =, the identit function. 9 The grph of the function f () =, is shown longside. Sketch the grph of = f (). Stte the rnge of f. c Solve: i f () = 0 ii f () = (, ) (, )
5 68 FUNCTIONS (Chpter ) REVIEW SET C For ech of the following grphs, find the domin nd rnge: ( ) ( ) ( ) If h() =7 : find in simplest form h( ) find if h( ) =. If f() = nd g() = p, find in simplest form: (f ± g)() (g ± f)() Find, nd c if f(0) =, f( ) = nd f() = nd f() = ++c. Cop the following grphs nd drw the grph of ech inverse function on the sme set of es: 6 For ech of the following functions f() find f () : f() =7 f() = + 7 Given f : 7! nd h : 7!, show tht (f ± h )() =(h ± f) (). 8 Given f() = + nd g() =, find (g ± f )().
6 GRAPHING AND TRANSFORMING FUNCTIONS (Chpter ) For the grph of = f() given, sketch the grph of: =f() = f() c = f( + ) d = f() e = f( ) ( ) g() For the grph of = g() given, sketch the grph of: = g()+ = g() c = g( ) d = g( + ) 6 For the grph of = h() given, sketch the grph of: = h()+ = h() c = h( ) d = h h() ( ) REVIEW SET A NON-CALCULATOR If f() =, find in simplest form: f() f( ) c f() d f( ) e f() If f() =, find in simplest form: ³ f() f( ) c f( ) d f e f() f( ) Consider f : 7!. Sketch the function f. Find lgericll the i -intercept ii -intercept iii grdient of the line. c i Find when =0:: ii Find when =0:7: The grph of f() = + + is trnslted to its imge g() the vector. Write the eqution of g() in the form g() = + +c+d. ³
7 GRAPHING AND TRANSFORMING FUNCTIONS (Chpter ) The grph of = f() is shown longside. The -is is tngent to f() t = nd f() cuts the -is t =. On the sme digrm sketch the grph of = f( c) where 0 <c<. Indicte the coordintes of the points of intersection of with the -is. z ( ) 6 For the grph of = f(), sketch grphs of: = f( ) = f() c = f( + ) d = f()+ ( ) ( ) ( ) ( ) 7 The grph of = f() is shown longside. Sketch the grph of = g() where g() =f( + ). Stte the eqution of the verticl smptote of = g(). c Identif the point A 0 on the grph of = g() which corresponds to point A. ( ) A (, ) 8 Consider the function f : 7!. On the sme set of es grph: = f() = f( ) c =f( ) d =f( ) + REVIEW SET B If f() =, find in simplest form: f( ) f( + ) c f()+ Consider f() =( + ). Use our clcultor to help grph the function. Find: i the -intercepts ii the -intercept. c Wht re the coordintes of the verte of the function? CALCULATOR Consider the function f : 7!. On the sme set of es grph: = f() = f( + ) c =f( + ) d =f( + )
8 GRAPHING AND TRANSFORMING FUNCTIONS (Chpter ) Consider f : 7!. Use our clcultor to help grph the function. True or flse? i As!,! 0: ii As!,! 0: iii The -intercept is : iv > 0 for ll. The grph of the function f() =( + ) + is trnslted units to the right nd units up. c Find the function g() corresponding to the trnslted grph. Stte the rnge of f(). Stte the rnge of g(). 6 For ech of the following functions: i ii iii iv Find = f(), the result of trnsltion ³. Sketch the originl function nd its trnslted function on the sme set of es. Clerl stte n smptotes of ech function. Stte the domin nd rnge of ech function. = = c = log 7 Consider the function g() =( + ). Use technolog to help sketch grph of the function. Find the es intercepts. c Find the coordintes of the verte of g(). 8 Sketch the grph of f() = +, nd on the sme set of es sketch the grphs of: f() f() c f()+ REVIEW SET C If f() =, find in simplest form: ³ f( ) f() c f d f( + ) Sketch the grph of f() =, nd on the sme set of es sketch the grph of: = f( ) = f() c = f() d = f( )
9 GRAPHING AND TRANSFORMING FUNCTIONS (Chpter ) The grph of cuic function = f() is shown longside. ( ) Sketch the grph of g() = f( ). Stte the coordintes of the turning points of = g(). (, ) (, ) The grph of f() = is trnsformed to the grph of g() reflection nd trnsltion s illustrted in the digrm longside. Find the formul for g() in the form g() = + + c. V (, ) ( ) g() Given the grph of = f(), sketch grphs of: f( ) f( + ) c f(). ( ) (' Ow_\) 6 The grph of f() = + + is trnslted to its imge, = g(), the vector. Write the eqution of g() in the form g() = + + c + d. 7 Find the eqution of the line tht results when the line f() = + is trnslted: i units to the left ii 6 units upwrds. Show tht when the liner function f() =+ is trnslted k units to the left, the resulting line is the sme s when f() is trnslted k units upwrds.
10 Grphing Rtionl Functions Worksheet - All grphs must e done on grph pper. No Clcultor: For ech of the following rtionl functions ou should ) find n horizontl smptotes, verticl smptotes or olique smptotes; ) find n -intercept(s) nd the -intercept; c) find the coordintes of n hole(s) in the grph; d) write down the end ehvior of the function; e) grph the functions without clcultor.. f ( ) = 9. g( ) =. ( + ) h:. + i: + Clcultor Allowed: For ech of the following rtionl functions ou should ) find n horizontl smptotes, verticl smptotes or olique smptotes; ) find n -intercept(s) nd the -intercept; c) find the coordintes of n hole(s) in the grph; d) write down the end ehvior of the function; e) grph the functions with the id of our clcultor.. f ( ) = 6. g : + 7. h( ) + + = + 8. i( ) = + 8 8
11 QUADRATIC EQUATIONS AND FUNCTIONS (Chpter 6) 8 0 The totl cost of producing tosters per d is given C = euros, nd the selling price of ech toster is ( ) euros. How mn tosters should e produced ech d in order to mimise the totl profit? A mnufcturer of reques knows tht if of them re mde ech week then ech one will cost ( ) pounds nd the totl receipts per week will e (000 ) pounds. How mn reques should e mde per week to mimise profits? INVESTIGATION Answer the following questions: + + c =0 hs roots p nd q. Prove tht p + q = nd pq = c. SUM AND PRODUCT OF ROOTS +=0 hs roots p nd q. Without finding the vlues of p nd q, find: p + q pq c p + q d p + q Find ll qudrtic equtions with roots which re: one more thn the roots of +=0 the squres of the roots of +=0 c the reciprocls of the roots of +=0. REVIEW SET 6A NON-CALCULATOR Consider the qudrtic function = ( + )( ). Stte the -intercepts. Stte the eqution of the is of smmetr. c Find the -intercept. d Find the coordintes of the verte. e Sketch the grph of the function. Solve the following equtions, giving ect nswers: =0 0 = 0 c = 60 Solve using the qudrtic formul: + +=0 + =0 Solve the following eqution completing the squre : +7 =0 Use the verte, is of smmetr nd -intercept to grph: =( ) = ( + ) +6 6 Find, in the form = + + c, the eqution of the qudrtic whose grph: touches the -is t nd psses through (, ). hs verte (, ) nd psses through (, ).
12 86 QUADRATIC EQUATIONS AND FUNCTIONS (Chpter 6) 7 Find the mimum or minimum vlue of the reltion = + + nd the vlue of for which the mimum or minimum occurs. 8 Find the points of intersection of = nd =. 9 For wht vlues of k does the grph of = + + k not cut the -is? 0 Find the vlues of m for which + m =0 hs: repeted root two distinct rel roots c no rel roots. The sum of numer nd its reciprocl is 0. Find the numer. Show tht no line with -intercept of (0, 0) will ever e tngentil to the curve with eqution = +7. The digrm shows qudrtic f() = + m + n. (, ) Determine the vlues of m nd n. Find k given tht the grph psses through the point (, k). c Stte the verte of = g() given g() =f( ) +. d Find the domin nd rnge of f() nd g(). REVIEW SET 6B CALCULATOR Consider the qudrtic function = +6. Convert it into the form = ( h) + k completing the squre. Stte the coordintes of the verte. c Find the -intercept. d Sketch the grph of the function. e Use technolog to check our nswers. Use technolog to solve: ( )( + ) = = Drw the grph of = +. Find the eqution of the is of smmetr nd the verte of = Using the discriminnt onl, determine the nture of the solutions of: 7=0 + 8 = 0 6 If [AB] hs the sme length s [CD], [BC] is cm shorter thn [AB], nd [BE] is 7 cm in length, find the length of [AB]. A E B D C
13 QUADRATIC EQUATIONS AND FUNCTIONS (Chpter 6) 87 7 For wht vlues of c do the lines with equtions = + c intersect the prol = + in two distinct points? Choose one such vlue of c from prt nd find the points of intersection. 8 For the qudrtic = +, find: the eqution of the is of smmetr the coordintes of the verte c the es intercepts. d Hence sketch the grph. 9 An open squre continer is mde cutting cm squre pieces out of piece of tinplte. If the cpcit is 0 cm, find the size of the originl piece of tinplte. 0 Find the points where = + nd = + + meet. Find the mimum or minimum vlue of the following qudrtics, nd the corresponding vlue of : = + +7 = m of fencing is used to construct m 6 rectngulr niml pens s shown. m Show tht =. 9 Find the re A of ech pen in terms of. c Find the dimensions of ech pen if ech pen is to hve mimum re. d Wht is the mimum re of ech pen? Two different qudrtic functions of the form f() =9 k + ech touch the -is. Find the two vlues of k. Find the point of intersection of the two qudrtic functions. c Descrie the trnsformtion which mps one function onto the other. REVIEW SET 6C Consider the qudrtic function = ( ). Stte the eqution of the is of smmetr. Find the coordintes of the verte. c Find the -intercept. d Sketch the grph of the function. e Use technolog to check our nswers. Solve the following equtions: =0 7 =0 Solve the following using the qudrtic formul: 7 +=0 +=0
14 88 QUADRATIC EQUATIONS AND FUNCTIONS (Chpter 6) Find the eqution of the qudrtic reltion with grph: c Use the discriminnt onl to find the reltionship etween the grph nd the -is for: = + 7 = Determine if the qudrtic functions re positive definite, negtive definite or neither: = + + = Find the eqution of the qudrtic reltion with grph: (, ) 8 In right ngled tringle, one leg is 7 cm longer thn the other, nd the hpotenuse is cm longer thn the longer leg. Find the length of the hpotenuse. 9 Find the -intercept of the line with grdient tht is tngentil to the prol = +. 0 For wht vlues of k would the grph of = + k cut the -is twice? Find n epression for the qudrtic which cuts the -is t nd nd hs -intercept. Give our nswer in the form = + + c. For wht vlues of m re the lines = m 0 tngents to the prol = +7 +? The digrm shows prol = ( + m)( + n) where m>n. A B Find, in terms of m nd n, the: i coordintes of the -intercepts A nd B ii eqution of the is of smmetr. Stte the sign of: i the discriminnt ii.
15 ANSWERS 697 EXERCISE H i ii, iii f () = i f () = ii c i f () = c e -\Tw_ -\Qe_ f j 6 6 0g f j g c f j g d f j 6 6 0g f j 6 <g -\Tw_ -\Qe_ ii i ii, iii f () = i f () = + ii d f 6 Er_ Ew_ Er_ Qw_ Ew_ (, ) Qw_ 7 f : 7!, 6= 0 stisfies oth the verticl nd horizontl line tests nd so hs n inverse function. f () = nd f() = ) f = f ) f is self-inverse function 8 = 8 is smmetricl out =, ) f is self-inverse function. f () = 8 REVIEW SET A 0 c = 6, = i Rnge = f j > g, Domin = f j R g ii -int,, -int iii is function 9 i Rnge = f j =or g, Domin = f j R g ii no -intercepts, -intercept iii is function -\We_ - 6 =, = 7 8 f () = f () = 9 (f ± h )() =(h ± f) () = REVIEW SET B Domin = f j R g, Rnge = f j > g Domin = f j 6= 0, g, Rnge = f j 6 or >0g + + =0 nd f() = 8 ) f = f ) f is self-inverse function 9 f () = + i (f ± f )() = ii (f ± f)() = 0 0 = i ii 6 = (f ± g )() = + nd (g ± f) () = Is not Is c Is d Is e Is i is the onl one c Domin = f j 6= 0g, Rnge = f j >0g
16 698 ANSWERS =, = Domin = f j 6= g, Rnge = f j 6= g 6 verticl smptote =, horizontl smptote = s!,! s!,! s! +,! s!,! + c -intercept, -intercept d 7 (g ± f)() = = + c i verticl smptote =, horizontl smptote =0 ii -\Qe_ iii Domin = f j 6= g, Rnge = f j 6= 0g 8 f () = +7 \Qw_ 9 Rnge = f j g c i ¼ :8 ii = REVIEW SET C \Qw_ 0 6 = p p =, = 6, c = () (, ) (, ) Qw_ -\Qr_ ( ) (, ) f ( ) (, ) f ( ) Domin = f j > g, Rnge = f j 6 <g Domin = f R g, Rnge = f j =, or g 6 f () = 7 f () = 7 (f ± h )() =(h ± f) +6 () = 8 6 EXERCISE A,,,, 9,, 7 c, 6, 8, d 96, 8,, Strts t 8 nd ech term is 8 more thn the previous term. Net two terms 0, 8. Strts t, ech term is more thn the previous term;, 7. c Strts t 6, ech term is less thn the previous term; 6,. d Strts t 96, ech term is 7 less thn the previous term; 68, 6. e Strts t, ech term is times the previous term; 6, 0. f Strts t, ech term is times the previous term; 6, 86. g Strts t 80, ech term is hlf the previous term; 0,. h Strts t, ech term is of the previous term;,. i Strts t 0 000, ech term is of the previous term; 80, 6. Ech term is the squre of the term numer;, 6, 9. Ech term is the cue of the term numer;, 6,. c Ech term is n(n + ) where n is the term numer; 0,, 6. 79, 7 80, 0 c 6, 96 d, 7 e 6, f, 8 EXERCISE B,, 6, 8, 0, 6, 8, 0, c,,, 7, 9 d,,,, 7 e, 7, 9,, f,, 7, 9, g, 7, 0,, 6 h,, 9,, 7,, 8, 6, 6,,, 8, 96 c, d,, 8, 6,,, 8, 6 7,,,, 7 EXERCISE C 7 6 c 0 07 c + d u =6, d = u n = n c d es, u 0 e no u = 87, d =, u n = 9 n c 69 d u 97 u =, d = c 69 d u = 6 u =, d = 7 c 7 d n > 68 7 k = 7 k = c k = d k =0 e k = or f k = or
17 ANSWERS 709 iii g( ) g( ) g( ) g( ) g( ) = f( ) is the reflection of = f() in the -is. i (, 0) ii (, ) iii (, ) i (7, ) ii (, 0) iii (, ) 6 i (, ) ii (0, ) iii (, ) i (, ) ii (0, ) iii (, ) 7 A rottion out the origin through 80 o. (, 7) c (, ) 6 h( ) h( ) h( ) h( ) h( ) EXERCISE B. ( ) ( ) ( ) ( ) REVIEW SET A 8 c d + e 6 c + + d e + i ii iii c i = : ii =0:9 c ( ) ( ) g() = + 6 ( ) ( ) ( ) c ( ) ( ) ( ) 6 c c c ( ) ( c) f ( ) f ( ) f ( ) f ( ) f ( ) A B c D d C 7 g() ( ) f ( ) f ( ) f () f ( ) f ( ) f ( ) A' (, ) = c A 0 (, ) A (, )
18 70 ANSWERS 8 i = ii REVIEW SET B c V i nd ii c V(, ) i true iii flse ii flse iv true 6 i = ii c f ( ) f ( ) f ( ) f ( ) iii For =, VA is =0, HA is =0 For =, VA is =, HA is = iv For =, domin is f j 6= 0g, rnge is f j 6= 0g For =, domin is f j 6= g, rnge is f j 6= g f ( ) f ( ) f ( ) f ( ) g() =( ) +8 f j > g f j > 8g iii For =, HA is =0, no VA For =, HA is =, no VA iv For =, domin is f j R g, rnge is f j >0g For =, domin is f j R g, rnge is f j > g c i = log ( ) ii iii For = log, VA is =0, no HA For = log ( ), VA is =, no HA iv For = log, domin is f j >0g, rnge is f R g For = log ( ), domin is f j >g, rnge is f R g 7 -intercepts nd, -intercept c (, ) 8 REVIEW SET C log ( ) g ( ) ( )X c 8 log f ( ) f ( ) f () f ( ) d 0 + f ( ) f ( ) f ( ) f () f ( )
19 ANSWERS 7 (, ) nd ( ) (, 0) g() = g() = i = +8 ii = +8 f( + k) =( + k)+ = + + k = f()+k EXERCISE 6A. =0, 7 =0, c =0, 7 d =0, e =0, 8 f =0, g =, h =, i =, 7 j = k =, l =, = =, 7 c =, 6 d =, e =, f =, g =, h =, i =, j =, k = 8, l =, 7 =, =, c =0, d =, e =, f = EXERCISE 6A. = p no rel solns. c = p d =8 p 7 e = p f = p 6 g = p 0 h = p p i = 7 = p = p 7 c =7 p d = p 7 e = p f = p 7 g = p h = p 6 i no rel solns. = p = p 9 c = p 7 p d = 7 p e = 7 f = p 0 6 EXERCISE 6A. (, ) (, ) (, ) (, ) = p 7 = p c = p d = p e = p f = p 7 g = p 9 7 h = 7 p 9 97 = p = p 8 7 c = p 8 d = p p 7 e = f = p 7 g() ('\\Ow_) f ( ) f ( ) f ( ) f ( ) EXERCISE 6B rel distinct roots rel distinct roots c rel distinct roots d no rel roots e repeted root, c, d, f = 6 m i m = ii m< iii m> =9 8m i m = 9 8 ii m< 9 8 iii m> 9 8 c =9 m i m = 9 ii m< 9 iii m> 9 =k +8k i k< 9 iii k = 9 f = k k EXERCISE 6C. or k> ii k 6 9 or k > or iv 9 <k< i <k<0 ii 6 k 6 0 iii k = or 0 iv k< or k>0 =( )( + ) = ( )( + ) c = ( + )( + ) d = ( + ) 0 8 Oi_ Or_ 0 i k< 8 or k>0 ii k 6 8 or k > 0 iii k = 8 or 0 iv 8 <k<0 = k i <k< ii 6 k 6 iii k = iv k< or k> c =k +k i k< 6 or k> ii k 6 6 or k > iii k = 6 or iv 6 <k< d =k k 6 i k< or k>6 ii k 6 or k > 6 iii k =6or iv <k<6 e =9k k 9 -\Ql_E_ -\Re_ 0 m m m k k k k k k
23 7. h( ) + + = + HA: none VA: none OA: = + Hole: none -int(s): (-.9,0) -int: (0,) End ehvior: s, s, 8. i( ) = HA: none VA: = OA: none Hole:, 6 -int(s): (0,0), (,0) -int: (0,0) End ehvior: s, s, Fun Fct: This grph ctull hs prolic smptote, ut ou do not hve to know how to find it.
24 7 ANSWERS EXERCISE 6G min., when = m. 8, when = c m. 8, when = d min. 8, when = e min. 6 when = 8 f m. 6 8, when = 7 0 refrigertors, $ m 0 m c 00 m : m 6 m m 0 m m 7 8 units 8 =6 cm cm REVIEW SET 6A, e = c d (, 9 ) =0or = = or or c = p = p 6 = 7 p 6 = + 8 = = which is < 0 ) m. m. =when = 8 (, ) nd (, 8) 9 k< 8 0 m = 9 8 m< 9 8 c m> or 6 m =, n = k =7 c (, ) d f() hs domin f j R g, rnge f j > g g() hs domin f j R g, rnge f j > g REVIEW SET 6B ('-) = + d (, ) c ¼ 0:86 or : ¼ 0:86 or :686 V( ) =, V(, ) two distinct rtionl roots repeted root - &-\Qw_\' - &-\Ew_\'-\Qs_T_* - 6 :9 cm 7 c> 6 emple: c =, (, ) nd (, 7) 8 = d (, ) c -intercept, -ints. p 6 9 : cm : cm 0 touch t (, 9) min. = when = m. = 8 when = ³ A = c 7 9 m m k = or (0, ) c horizontl trnsltion of units REVIEW SET 6C = d (, ) c = p 7 = 7 p 7 = 7 p 7 no rel roots = 0 9 ( ) 0 = ( )( 7) 7 c = ( + ) 9 grph cuts -is twice grph cuts -is twice 6 neither positive definite d 0 m 7 = ( )( + ) = 6( ) cm 9 0 k< = + + m = or 9 i A( m, 0), B( n, 0) m n ii = i positive ii negtive EXERCISE p +p q +pq + q c d e f g h p p q +6p q pq + q c d e f g h \Qw_\~`6 -+\Qw_\~`6 - ('-) (-'-)
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PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
ALGEBRA B Semester Em Review The semester B emintion for Algebr will consist of two prts. Prt will be selected response. Prt will be short nswer. Students m use clcultor. If clcultor is used to find points
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
MCR U MCR U Em Review Introduction to Functions. Determine which of the following equtions represent functions. Eplin. Include grph. ) b) c) d) 0. Stte the domin nd rnge for ech reltion in question.. If
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
NAME: M. WAIN FUNCTIONS evision o Solving Polnomil Equtions i one term in Emples Solve: 7 7 7 0 0 7 b.9 c 7 7 7 7 ii more thn one term in Method: Get the right hnd side to equl zero = 0 Eliminte ll denomintors
6000_000.qd //0 :6 AM Pge 0-0- CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
Ch. 7.-7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,
Alger 00 Midterm Review Nme: Dte: Directions: For the following prolems, on SEPARATE PIECE OF PAPER; Define the unknown vrile Set up n eqution (Include sketch/chrt if necessr) Solve nd show work Answer
MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled
Prerequisite Knowledge Required from O Level Add Mth ) Surds, Indices & Logrithms Rules for Surds. b= b =. 3. 4. b = b = ( ) = = = 5. + b n = c+ d n = c nd b = d Cution: + +, - Rtionlising the Denomintor
ME rctice ook ES3 3 ngle Geometr 3.3 ngle Geometr 1. lculte the size of the ngles mrked with letter in ech digrm. None to scle () 70 () 20 54 65 25 c 36 (d) (e) (f) 56 62 d e 60 40 70 70 f 30 g (g) (h)
. If A, nd B 8, REVIEW SHEET FOR PRE-CALCULUS MIDTERM. For the following figure, wht is the eqution of the line?, write n eqution of the line tht psses through these points.. Given the following lines,
.1 Understnd nd use the lws of indices for ll rtionl eponents.. Use nd mnipulte surds, including rtionlising the denomintor..3 Work with qudrtic nd their grphs. The discriminnt of qudrtic function, including
THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used
C. Rel Numers nd the Rel Numer Line C C Preclculus Review C. Rel Numers nd the Rel Numer Line Represent nd clssif rel numers. Order rel numers nd use inequlities. Find the solute vlues of rel numers nd
4.1 One-to-One Functions; Inverse Functions Finding Inverses of Functions To find the inverse of function simply switch nd y vlues. Input becomes Output nd Output becomes Input. EX) Find the inverse of
Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q
In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
Differentition pplictions 3: The Men Vlue Theorem 169 D 3: The Men Vlue Theorem Model 1: Pennslvni Turnpike You re trveling est on the Pennslvni Turnpike You note the time s ou pss the Lenon/Lncster Eit
P g e 30 4.2 Eponentil Functions 1. Properties of Eponents: (i) (iii) (iv) (v) (vi) 1 If 1, 0 1, nd 1, then E1. Solve the following eqution 4 3. 1 2 89 8(2 ) 7 Definition: The eponentil function with se
Mth 0- Nme: Sequences nd Series RETest Worksheet Short Answer Use n infinite geometric series to express 353 s frction [ mrk, ll steps must be shown] The popultion of community ws 3 000 t the beginning
_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
TEKS 8. A..A, A.0.F Add nd Subtrct Rtionl Epressions Before Now You multiplied nd divided rtionl epressions. You will dd nd subtrct rtionl epressions. Why? So you cn determine monthly cr lon pyments, s
CET MATHEMATICS VERSION CODE: C. If sin is the cute ngle between the curves + nd + 8 t (, ), then () () () Ans: () Slope of first curve m ; slope of second curve m - therefore ngle is o A sin o (). The
BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht
First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.
Stright line grphs, Mied Eercise Grdient m ( y ),,, The eqution of the line is: y m( ) ( ) + y + Sustitute (k, ) into y + k + k k Multiply ech side y : k k The grdient of AB is: y y So: ( k ) 8 k k 8 k
Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!
A Trigonometric Fnctions (pp 8 ) Rtios of the sides of right tringle re sed to define the si trigonometric fnctions These trigonometric fnctions, in trn, re sed to help find nknown side lengths nd ngle
8.5 The Ellipse Kidne stones re crstl-like ojects tht cn form in the kidnes. Trditionll, people hve undergone surger to remove them. In process clled lithotrips, kidne stones cn now e removed without surger.
Anti-differentition nd introduction to integrl clculus. Kick off with CAS. Anti-derivtives. Anti-derivtive functions nd grphs. Applictions of nti-differentition.5 The definite integrl.6 Review . Kick off
Lesson-5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).
Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble
Answers to Eercises CHAPTER 9 CHAPTER LESSON 9. CHAPTER 9 CHAPTER. c 9. cm. cm. b 5. cm. d 0 cm 5. s cm. c 8.5 cm 7. b cm 8.. cm 9. 0 cm 0. s.5 cm. r cm. 7 ft. 5 m.. cm 5.,, 5. 8 m 7. The re of the lrge
Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution
. A. A. C. B. C 6. A 7. A 8. B 9. C. D. A. B. A. B. C 6. D 7. C 8. B 9. C. D. C. A. B. A. A 6. A 7. A 8. D 9. B. C. B. D. D. D. D 6. D 7. B 8. C 9. C. D. B. B. A. D. C Section A. A (68 ) [ ( ) n ( n 6n
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
SAINT IGNATIUS COLLEGE Directions to Students Tril Higher School Certificte 0 MATHEMATICS Reding Time : 5 minutes Totl Mrks 00 Working Time : hours Write using blue or blck pen. (sketches in pencil). This
654 CHAPTER 1 PARAETRIC EQUATIONS AND POLAR COORDINATES ; 43. The points of intersection of the crdioid r 1 sin nd the spirl loop r,, cn t be found ectl. Use grphing device to find the pproimte vlues of
Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the -coordinte of ech criticl vlue of g. Show
1. Find the distnce between the pir of points. Give n ect, simplest form nswer nd deciml pproimtion to three plces., nd, MAC 110 Finl Em Review, nd,0. The points (, -) nd (, ) re endpoints of the dimeter
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
Rtionl Prents (pp of 4) Unit: 08 Lesson: 0 The grphs below describe two prent functions, ech of which is referred to s rtionl function Why do you think they re clled rtionl functions? From the grphs, provide
1 Prctice A In Eercises 1 6, tell whether nd show direct vrition, inverse vrition, or neither.. 7. 6. 10. 8 6. In Eercises 7 10, tell whether nd show direct vrition, inverse vrition, or neither. 8 10 8.
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right