calculus summer pkt
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8/10/2019 Calculus Summer Pkt
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Coral Springs Charter
SUMMER REVIEW PACKET
For students in entering CALCULUS AP
Name: __________________________________________________________
1. This packet is to be handed in to your Calculus teacher on the first day of the school year.
. All !ork must be sho!n in the packet "# on separate paper attached to the packet.$. Completion of this packet is !orth one%half of a ma&or test grade and !ill be counted in your first
marking period grade.
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Summer Review Packet for Students Entering Calculus (all levels)
Complex Fractions
Simplif each of the follo!ing"
1.
'
a− a
'+ a
.
−(
x +
'+1)
x +
$.
( −1
x − $
'+1'
x − $
(.
x
x +1−1
x x
x +1+1
x
'.
1− x
$ x − (
x +$
$ x − (
F#nctions
$et f * x+ = x +1 and g * x+ = x −1 " Fin% each"
,. f *+ = ____________ -. g *−$+ = _____________ . f *t +1+ = __________
/. f g *−+ = __________ 1). g f *m + + = ___________ 11. f * x + h+ − f * x+
h= ______
$et f ( x) = sin x Fin% each exactl"
1. f π
2
= ___________ 1$. f
2π
3
= ______________
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$et f * x+ = x 0 g * x+ = x +'0 and h* x+ = x −1 " Fin% each"
1(. h f *−+ = _______ 1'. f g * x −1+ = _______ 1,. g h* x$+ = _______
Fin% f * x + h+ − f * x+
h for the gi&en f#nction f.
1-. f * x+ = / x + $ 1. f * x+ = '− x
Intercepts an% Points of Intersection
To find the %intercepts0 let y 2 ) in your e3uation and sol4e.
To find the y%intercepts0 let 2 ) in your e3uation and sol4e
Fin% the x an% intercepts for each"
1/. y = 2 x − 5 ). y = x2 + x − 2
1. y = x 16 − x2 . y
2 = x3 − 4 x
$
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Use s#'stit#tion or elimination metho% to sol&e the sstem of e(#ations"
Fin% the point)s* of intersection of the graphs for the gi&en e(#ations"
$. x + y = 8
4 x − y = 7(.
x2 + y = 6
x + y = 4'.
x2 − 4 y
2 − 20 x − 64 y − 172 = 0
16 x2 + 4 y2 − 320 x + 64 y +1600 = 0
Inter&al +otation
,. Complete the table !ith the appropriate notation or graph.
Sol#tion Inter&al +otation ,raph
−2 < x ≤ 4
−1, 7)
-
5ol4e each e3uation. 5tate your ans!er in 6"T7 inter4al notation and graphically.
-. 2 x − 1 ≥ 0 . −4 ≤ 2 x − 3 < 4 /. x
2−
x
3> 5
.omain an% Range
Fin% the %omain an% range of each f#nction" Write o#r ans!er in I+TERVA$ notation"
$). f ( x) = x2 − 5 $1. f ( x) = − x + 3 $. f ( x) = 3sin x $$. f ( x) = 2
x −1
(
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In&erses
Fin% the in&erse for each f#nction"
/0" f ( x) = 2 x + 1 /1" f ( x) = x2
3
Also0 recall that to P#"89 one function is an in4erse of another function0 you need to sho! that: f (g( x)) = g( f ( x)) = x
Example2
If2 f ( x) = x − 9
4and g( x) = 4 x + 9 sho! f(x) and g(x) are in&erses of each other"
f (g( x)) = 4 x − 9
4
+ 9 g( f ( x)) =
4 x + 9( )− 9
4
= x − 9 + 9 = 4 x + 9 − 9
4
= x = 4 x
4
= x
f (g( x)) = g( f ( x)) = x therefore they are inverses
of each other.
Pro&e f and g are in&erses of each other"
/3" f ( x) = x
3
2g( x) = 2 x3 /4" f ( x) = 9 − x
2 , x ≥ 0 g( x) = 9 − x
'
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E(#ation of a line
Slope intercept form2 y = mx + b Vertical line2 2 c *slope is undefined+
Point5slope form2 y − y1 = m( x − x1
) 6ori7ontal line2 y 2 c *slope is )+
$. se slope%intercept form to find the e3uation of the line ha4ing a slope of $ and a y%intercept of '.
$/. ;etermine the e3uation of a line passing through the point *'0 %$+ !ith an undefined slope.
(). ;etermine the e3uation of a line passing through the point *%(0 + !ith a slope of ).
(1. se point%slope form to find the e3uation of the line passing through the point *)0 '+ !ith a slope of <$.
(. Find the e3uation of a line passing through the point *0 + and parallel to the line y = 5
6 x − 1 .
($. Find the e3uation of a line perpendicular to the y% ais passing through the point *(0 -+.
((. Find the e3uation of a line passing through the points *%$0 ,+ and *10 +.
('. Find the e3uation of a line !ith an %intercept *0 )+ and a y%intercept *)0 $+.
,
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2
-2
(-1,0)
(0,-1)
(0,1)
(1,0)
Ra%ian an% .egree Meas#re
(,. Con4ert to degrees: a.5π
6 b.
4π
5c. .,$ radians
(-. Con4ert to radians: a. 45o b. −17
o c. $- o
Unit Circle
(. a.) sin180o
b.) cos270
o
c.) sin(−90o) d .) sinπ
e.) cos360
o f .) cos(−π )
,raphing Trig F#nctions
,raph t!o complete perio%s of the f#nction"
(/. f ( x) = 5sin x '). f ( x) = cos x − 3
Trigonometric E(#ations2
5ol4e each of the e3uations for 0 ≤ x < 2π . =solate the 4ariable0 sketch a reference triangle0 find all the
solutions !ithin the gi4en domain0 0 ≤ x < 2π . #emember to double the domain !hen sol4ing for a double
angle. se trig identities0 if needed0 to re!rite the trig functions. *5ee formula sheet at the end of the packet.+
'1. sin x = − 1
2'. 2cos x = 3
'$. cos2 x = 1
2'(. sin2
x = 1
2
-
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Minor Axis
Major Axis
ab
c FOCUS (h + c, k)FOCUS (h - c, k)
CENTER (h, k)
( x − h)2
a2 + ( y − k )2
b2 = 1
''. sin2 x = − 3
2',. 2cos2
x − 1 − cos x = 0
'-. 4cos2 x − 3 = 0 '. sin2 x + cos 2 x − cos x = 0
In&erse Trigonometric F#nctions2
For each of the follo!ing8 express the &al#e for 9: in ra%ians"
'/. y = arcsin − $
,). y = arccos −1( ) ,1. y = arctan*−1+
For each of the follo!ing gi&e the &al#e !itho#t a calc#lator"
,. tan arccos
$
,$. sec sin
−11
1$
,(. sin arctan1
'
,'. sin sin−1 -
.
Circles an% Ellipses
r 2 = ( x − h)2 + ( y− k )2
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4
2
-2
-4
-5 5
4
2
-2
-4
-5 5
4
2
-2
-4
-5 5
4
2
-2
-4
-5 5
For a circle centered at the origin0 the e3uation is x2 + y
2 = r 2
0 !here r is the radius of the circle.
For an ellipse centered at the origin0 the e3uation is x
2
a2 +
y2
b2 = 1 0 !here a is the distance from the center to the
ellipse along the %ais and ' is the distance from the center to the ellipse along the y%ais. =f the largernumber is under the y2 term0 the ellipse is elongated along the y%ais. For our purposes in Calculus0 you !ill not
need to locate the foci.
,raph the circles an% ellipses 'elo!2
,,. x2 + y
2 = 16 ,-. x2 + y
2 = 5
,. x
2
1+
y2
9= 1 ,/.
x2
16+
y2
4= 1
-)%/ *#epresented by net set of problems+
/
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Vertical Asmptotes
;etermine the 4ertical asymptotes for the function. 5et the denominator e3ual to >ero to find the %4alue for
!hich the function is undefined. That !ill be the 4ertical asymptote.
/). f ( x) = 1
x2
/1. f ( x) = x
2
x2 − 4
/. f ( x) = 2 + x
x2 (1 − x)
6ori7ontal Asmptotes
;etermine the hori>ontal asymptotes using the three cases belo!.
Case I. ;egree of the numerator is less than the degree of the denominator. The asymptote is y 2 ).
Case II" ;egree of the numerator is the same as the degree of the denominator. The asymptote is the ratio ofthe lead coefficients.
Case III. ;egree of the numerator is greater than the degree of the denominator. There is no hori>ontalasymptote. The function increases !ithout bound. *=f the degree of the numerator is eactly 1 more than the
degree of the denominator0 then there eists a slant asymptote0 !hich is determined by long di4ision.+
.etermine all 6ori7ontal Asmptotes"
/$. f ( x) =
x2 − 2 x + 1
x
3
+ x − 7/(. f ( x)
=
5 x3 − 2 x
2 + 8
4 x − 3 x 3 + 5/'. f ( x)
=
4 x5
x2 − 7
?iscellaneous @no!ledge
/,. #ationali>e the denominator: *a+ *b+ *c+
1)
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/-. 5ol4e for x *do not use a calculator+:
*a+ '* x 1+ 2 ' *b+ *c+ log x 2 $ *d+ log$ x 2 log$( % ( log$'
/. 5implify: *a+ log' log* x % 1+ % log* x % 1+ *b+ log(/ % log$
//. Factor completely: *a+ x, % 1, x( *b+ ( x$ % x % ' x ') *c+ x$ - *d+ x( %1
1)). Find all real solutions to: *a+ x, % 1, x( 2 ) *b+ ( x$ % x % ' x ') 2 ) *c+ x$ - 2 )
1)1. Find the remainders on di4ision of
*a+ x' % ( x( x$ % - x 1 by x *b+ x' % x( x$ x % x ( by x$ 1
1). Find the domain of the function
1)$. Find the domain and range of the functions:
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cos2 x = cos2 x − sin
2 x
= 1− 2sin2 x
= 2cos2 x − 1
Form#la Sheet
#eciprocal =dentities: csc x = 1
sin xsec x =
1
cos xcot x =
1
tan x
Buotient =dentities: tan x = sin x
cos x cot x = cos x
sin x
Pythagorean =dentities: sin2 x + cos2
x = 1 tan2 x + 1 = sec2
x 1 + cot2 x = csc2
x
;ouble Angle =dentities: sin2 x = 2sin x cos x
tan2 x = 2tan x
1 − tan2 x
ogarithms: y = loga x is e3ui4alent to x = a y
Product property: logb mn = logb m + logb n
Buotient property: logb
m
n= log
b m − logb n
Po!er property: logb m p = p logb m
Property of e3uality: =f logb m = logb n 0 then m 2 n
Change of base formula: loga n =
logb n
logb a
5lope%intercept form: y = mx + b
Point%slope form: y − y1 = m( x − x1)
5tandard form: A 6y C 2 )
1