KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a
|
|
- Vincent Higgins
- 5 years ago
- Views:
Transcription
1 KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider the mgnitude only i.e. A = y d in this cse.. Are etween the curves y = f () & y = g () etween the ordintes t = & = is given y, A = f () d g () d = [ f () g () ] d.. Averge vlue of function y = f () w.r.t. over n intervl is defined s : y (v) = f () d. 5. The re functiona stisfies the differentil eqution da = f () with initil conditiona d =. Note : If F () is ny integrl of f () then, A = f () d = F () + c A = = F () + c c = F () hence A = F () F (). Finlly y tking = we get, A = F () F (). 6. CURVE TRACING : The following outline procedure is to e pplied in Sketching the grph of function y = f () which in turn will e etremely useful to quickly nd correctly evlute the re under the curves. () Symmetry : The symmetry of the curve is judged s follows : (i) If ll the powers of y in the eqution re even then the curve is symmetricl out the is of. (ii) If ll the powers of re even, the curve is symmetricl out the is of y. (iii) If powers of & y oth re even, the curve is symmetricl out the is of s well s y. (iv) If the eqution of the curve remins unchnged on interchnging nd y, then the curve is symmetricl out y =. (v) If on interchnging the signs of & y oth the eqution of the curve is unltered then there is symmetry in opposite qudrnts. () Find dy/d & equte it to zero to find the points on the curve where you hve horizontl tngents. (c) Find the points where the curve crosses the is & lso the yis. (d) Emine if possile the intervls when f () is incresing or decresing. Emine wht hppens to y when or. 7. USEFUL RESULTS : (i) Whole re of the ellipse, / + y / = is π. (ii) Are enclosed etween the prols y = & = y is 6/. (iii) Are included etween the prol y = & the line y = m is 8 / m. EXERCISE I Q. Find the re ounded on the right y the line + y =, on the left y the prol y = nd elow y the is. Q. Find the re of the region ounded y the curves, y = ² + ; y = ; = & =. Pge 9 of Are Under Curve
2 Q. Find the re of the region {(, y) : y ² +, y +, }. Q. Find the vlue of c for which the re of the figure ounded y the curves y = sin, the stright lines = π/6, = c & the sciss is is equl to /. Q.5 The tngent to the prol y = hs een drwn so tht the sciss of the point of tngency elongs to the intervl [, ]. Find for which the tringle ounded y the tngent, the is of ordintes & the stright line y = hs the gretest re. Q.6 Compute the re of the region ounded y the curves y = e.. ln & y = ln /(e. ) where ln e=. Q.7 A figure is ounded y the curves y = sin π, y =, = & =. At wht ngles to the positive is stright lines must e drwn through (, ) so tht these lines prtition the figure into three prts of the sme size. Q.8 Find the re of the region ounded y the curves, y = log e, y = sin π & =. Q.9 Find the re ounded y the curves y = nd y =. Also find the rtio in which the y-is divided this re. Q. If the re enclosed y the prols y = nd y = is 8 sq. units. Find the vlue of ''. Q. The line + y = divides the re enclosed y the curve, 9 + y 8 6y = into two prts. Find the rtio of the lrger re to the smller re. Q. Find the re of the region enclosed etween the two circles ² + y² = & ( )² + y² = Q. Find the vlues of m (m > ) for which the re ounded y the line y = m + nd = y y is, (i) 9/ squre units & (ii) minimum. Also find the minimum re. Q. Find the rtio in which the re enclosed y the curve y = cos ( π/) in the first qudrnt is divided y the curve y = sin. Q.5 Find the re enclosed etween the curves : y = log e ( + e), = log e (/y) & the is. Q.6 Find the re of the figure enclosed y the curve (y rc sin ) =. Q.7 For wht vlue of '' is the re ounded y the curve y = + + nd the stright line y =, = & = the lest? Q.8 Find the positive vlue of '' for which the prol y = + isects the re of the rectngle with vertices (, ), (, ), (, + ) nd (, + ). Q.9 Compute the re of the curviliner tringle ounded y the yis & the curve, y = tn & y = (/) cos. Q. Consider the curve C : y = sin sin, C cuts the is t (, ), ( π, π). A : The re ounded y the curve C & the positive is etween the origin & the ordinte t =. A : The re ounded y the curve C & the negtive is etween the ordinte = & the origin. Prove tht A + A + 8 A A =. Q. Find the re ounded y the curve y = e ; y = nd = c where c is the -coordinte of the curve's inflection point. Q. Find the vlue of 'c' for which the re of the figure ounded y the curve, y = 8 5, the stright lines = & = c & the sciss is is equl to 6/. Q. Find the re ounded y the curve y² = & = y. Q. Find the re ounded y the curve y = e, the -is, nd the line = c where y (c) is mimum. Q.5 Find the re of the region ounded y the is & the curves defined y, y = tn, π / π / y = cot, π / 6 π / Pge of Are Under Curve
3 EXERCISE II Q. In wht rtio does the -is divide the re of the region ounded y the prols y = ² & y = ²? Q. Find the re ounded y the curves y = & y =. Q. Sketch the region ounded y the curves y = 5 & y = & find its re. Q. Find the eqution of the line pssing through the origin nd dividing the curviliner tringle with verte t the origin, ounded y the curves y =, y = nd = into two prts of equl re. Q.5 Consider the curve y = n where n > in the st qudrnt. If the re ounded y the curve, the -is nd the tngent line to the grph of y = n t the point (, ) is mimum then find the vlue of n. Q.6 Consider the collection of ll curve of the form y = tht pss through the the point (, ), where nd re positive constnts. Determine the vlue of nd tht will minimise the re of the region ounded y y = nd -is. Also find the minimum re. Q.7 In the djcent grphs of two functions y = f() nd y = sin re given. y = sin intersects, y = f() t A (, f()); B(π, ) nd C(π, ). A i (i =,,,) is the re ounded y the curves y = f () nd y = sin etween = nd = ; i =, etween = nd = π; i =, etween = π nd = π; i =. If A = sin + ( )cos, determine the function f(). Hence determine nd A. Also clculte A nd A. Q.8 Consider the two curves y = /² & y = /[ ( )]. (i) At wht vlue of ( > ) is the reciprocl of the re of the fig. ounded y the curves, the lines = & = equl to itself? (ii) At wht vlue of ( < < ) the re of the figure ounded y these curves, the lines = & = equl to /. Q.9 ln c Show tht the re ounded y the curve y =, the -is nd the verticl line through the mimum point of the curve is independent of the constnt c. Q. For wht vlue of '' is the re of the figure ounded y the lines, y =, y = 5? Q. Compute the re of the loop of the curve y² = ² [( + )/( )]. Q. Find the vlue of K for which the re ounded y the prol y = + nd the line y = K + is lest. Also find the lest re. Q. Let A n e the re ounded y the curve y = (tn ) n & the lines =, y = & = π/. Prove tht for n >, A n + A n = /(n ) & deduce tht /(n + ) < A n < /(n ). Q. If f () is monotonic in (, ) then prove tht the re ounded y the ordintes t = ; = ; y = f () + nd y = f (c), c (, ) is minimum when c =. Hence if the re ounded y the grph of f () = +, the stright lines =, = nd the -is is minimum then find the vlue of ''. Q.5 Consider the two curves C : y = + cos & C : y = + cos ( α) for α, π ; [, π]. Find the vlue of α, for which the re of the figure ounded y the curves C, C & = is sme s tht of the figure ounded y C, y = & = π. For this vlue of α, find the rtio in which the line y = divides the re of the figure y the curves C, C & = π. Q.6 Find the re ounded y y² = ( + ), y² = ( ) & y = ove is of. Q.7 Compute the re of the figure which lies in the first qudrnt inside the curve Pge of Are Under Curve
4 ² + y² = ² & is ounded y the prol ² = y & y² = ( > ). Q.8 Consider squre with vertices t (, ), (, ), (, ) & (, ). Let S e the region consisting of ll points inside the squre which re nerer to the origin thn to ny edge. Sketch the region S & find its re. Q.9 Find the whole re included etween the curve ² y² = ² (y² ²) & its symptotes (symptotes re the lines which meet the curve t infinity). Q. For wht vlues of [, ] does the re of the figure ounded y the grph of the function y = f () nd the stright lines =, = & y = f() is t minimum & for wht vlues it is t mimum if f () =. Find lso the mimum & the minimum res. Q. Find the re enclosed etween the smller rc of the circle ² + y² + y = & the prol y = ² + +. Q. D r w n e t n d c l e n g r p h o f t h e f u n c t i o n f D r w n e t n d c l e n g r p h o f t h e f u n c t i o n f () = cos ( ), [, ] nd find the re enclosed etween the grph of the function nd the is s vries from to. Q. Let C & C e two curves pssing through the origin s shown in the figure. A curve C is sid to "isect the re" the region etween C & C, if for ech point P of C, the two shded regions A & B shown in the figure hve equl res. Determine the upper curve C, given tht the isecting curve C hs the eqution y = & tht the lower curve C hs the eqution y = /. Q. For wht vlues of [, ] does the re of the figure ounded y the grph of the function y = f () & the stright lines =, =, y = f() hve the gretest vlue nd for wht vlues does it hve the lest vlue, if, f() = α + β, α, β R with α >, β >. Q.5 Given f () = t e (logsec t sec y = f () nd y = g () etween the ordintes = nd = π. t)dt ; g () = e tn. Find the re ounded y the curves EXERCISE III Q. Let f () = Mimum {, ( ), ( )}, where. Determine the re of the region ounded y the curves y = f (), is, = & =. [ JEE '97, 5 ] Q. Indicte the region ounded y the curves = y, y = + nd is nd otin the re enclosed y them. [ REE '97, 6 ] Q. Let C & C e the grphs of the functions y = & y =, respectively. Let C e the grph of function y = f (),, f() =. For point P on C, let the lines through P, prllel to the es, meet C & C t Q & R respectively (see figure). If for every position of P (on C ), the res of the shded regions OPQ & ORP re equl, determine the function f(). [JEE '98, 8] Q. Indicte the region ounded y the curves y = ln & y = nd otin the re enclosed y them. [ REE '98, 6 ] Q.5 () For which of the following vlues of m, is the re of the region ounded y the curve y = nd the line y = m equls 9/? (A) (B) (C) (D) for () Let f() e continuous function given y f() = + + for > Find the re of the region in the third qudrnt ounded y the curves, = y nd Pge of Are Under Curve
5 y = f() lying on the left of the line 8 + =. [ JEE '99, + (out of ) ] Q.6 Find the re of the region lying inside + (y ) = nd outside c + y = c where c =. [REE '99, 6] Q.7 Find the re enclosed y the prol (y ) =, the tngent to the prol t (, ) nd the -is. [REE,] Q.8 Let nd for j =,,,...n, let S j e the re of the region ounded y the y is nd the curve jπ ( j+ ) π e y = siny, y. Show tht S, S, S,...S n re in geometric progression. Also, find their sum for = nd = π. [JEE', 5] Q.9 The re ounded y the curves y = nd y = + is (A) (B) (C) (D) [JEE', (Scr)] Q. Find the re of the region ounded y the curves y =, y = nd y =, which lies to the right of the line =. [JEE ', (Mins)] Q. If the re ounded y y = nd = y, >, is, then = (A) (B) (C) (D) [JEE ', (Scr)] Q.() The re ounded y the prols y = ( + ) nd y = ( ) nd the line y = / is (A) sq. units (B) /6 sq. units (C) / sq. units (D) / sq. units [JEE '5 (Screening)] () Find the re ounded y the curves = y, = y nd y =. f ( ) + (c) If f () = +, f () is qudrtic function nd its mimum vlue occurs t c c f () c + c point V. A is point of intersection of y = f () with -is nd point B is such tht chord AB sutends right ngle t V. Find the re enclosed y f () nd chord AB. [JEE '5 (Mins), + 6] Q. Mtch the following π cos sin (i) (sin ) (cos cot log(sin ) )d (A) (ii) Are ounded y y = nd = 5y (B) (iii) Cosine of the ngle of intersection of curves y = log nd y = is (C) 6 ln (D) / [JEE 6, 6] ANSWER EXERCISE I Q. 5/6 sq. units Q. / sq. units Q. /6 sq. units Q. c = π 6 or π Q 7. π tn π Q 9. π ; π π + ; π tn π Q 5. =, A( ) = 8 Q 8. 8 Q 6. (e 5)/ e sq. units sq. units Q. = 9 Q. π + π Pge of Are Under Curve
6 Q. π sq. units Q. (i) m =, (ii) m = ; Amin = / Q. Q 5. sq. units Q 6. π/ Q 7. = / Q 8. Q 9. + l n sq. units 8 7 / Q. e Q. C = or ( ) Q. / Q. ( e / ) Q 5. ln EXERCISE II Q. : Q. 8/5 sq. units Q. (5 π )/ sq. units Q. y = / Q 5. + Q 6. = /8, A minimum = sq. units Q 7. f() = sin, = ; A = sin; A = π sin; A = (π ) sq. units Q 8. = + e, = + e Q.9 / Q. = 8 or ( 6 ) 5 Q. (π/) sq. units Q. K =, A = / Q. = 8 8 / Q 5. α = π/, rtio = : Q 6. ( ) ( ) Q sin rc sq. units Q 8. ( 6 ) Q 9. Q. = / gives minim, A = π π ; = gives locl mim A() = ; = gives mimum vlue, A() = π/ Q. 8 + π Q. ( ) sq. units Q. (6/9) Q. for =, re is gretest, for = /, re is lest Q5. e π log sq. units EXERCISE III Q. 7/7 Q. 5/6 sq. units Q. f() = Q. 7/ π Q.5 () B, D () 57/9 ; = ; = Q.6 π sq. units π Sj Q.7 9 sq. units Q.8 = e ; S S j+ e = π + + for =, = π, S = π ( e + ) nd r = π π + Q.9 B Q. sq. units Q. B 5 Q. () D ; () sq. units ; (c) sq. units Q. (i) A, (ii) D, (iii) A Pge of Are Under Curve
7 EXERCISE IV. The re ounded y the curve = y, -is nd the line = is (A) (B) (C). The re ounded y the -is nd the curve y = is (A) (B) (C) (D) (D) 8 Pge 5 of Are Under Curve. The re ounded y the curve y = sin with -is in one rc of the curve is (A) (B) (C) (D). The re contined etween the curve y =, the verticl line =, = ( > ) nd -is is (A) log (B) log (C) log (D) log 5. The re of the closed figure ounded y the curves y =, y = & y = is: (A) 9 (B) 8 9 (C) 6 9 (D) none 6. The re of the closed figure ounded y the curves y = cos ; y = + π & = π is (A) π + (B) π (C) π + 7. The re included etween the curve y = ( ) & its symptote is: (A) π 8. The re ounded y ² + y² = & y = sin π (A) π π (B) π π (D) π (B) π (C) π (D) none (C) π in the upper hlf of the circle is: 8 π (D) none 9. The re of the region enclosed etween the curves 7 + 9y + 9 = nd y + 7 = is: (A) (B) (C) 8 (D) 6. The re ounded y the curves y = ( ln ); = e nd positive Xis etween = e nd = e is : e (A) 5 e e (B) 5 e e e (C) 5 5e (D). The re enclosed etween the curves y = log e ( + e), = log e y nd the -is is (A) (B) (C) (D) none of these e
8 . The re ounded y the curves + y = nd + y = is (A) (B) 6 (C) (D) none of these. The re ounded y -is, curve y = f(), nd lines =, = is equl to ( + ) for ll >, then f() is (A) ( ) (B) ( + ) (C) ( + ) (D) / (+ ). The re of the region for which < y < nd > is (A) ( ) d (B) ( ) d (C) ( ) d (D) ( 5. The re ounded y y =, y = [ + ], nd the y-is is (A) / (B) / (C) (D) 7/ 6. The re ounded y the curve = cos t, y = sin t is (A) π 8 (B) π 6 (C) π (D) π 7. If A is the re enclosed y the curve y =, -is nd the ordintes =, = ; nd A is the re enclosed y the curve y =, -is nd the ordintes =, =, then (A) A = A (B) A = A (C) A = A (D) A = A 8. The re ounded y the curv e y = f(), -is nd the ordintes = nd = is ( ) sin ( + ), R, then f() = (A) ( ) cos ( + ) (B) sin ( + ) (C) sin ( + ) + ( ) cos ( + ) (D) none of these 9. Find the re of the region ounded y the curves y = +, y =, = nd =. (A) sq. unit (B) sq. unit (C) sq. unit (D) none of these. The res of the figure into which curve y = 6 divides the circle + y = 6 re in the rtio (A) (B) π 8π + (C) π + 8π (D) none of these. The tringle formed y the tngent to the curve f() = + t the point (, ) nd the coordinte es, lies in the first qudrnt. If its re is, then the vlue of is [IIT - ] (A) (B) (C) (D) EXERCISE V. Find the re of the region ounded y the curve y = y nd the y-is.. Find the vlue of c for which the re of the figure ounded y the curves y = sin, the stright lines = π/6, = c & the sciss is is equl to /.. For wht vlue of '' is the re ounded y the curve y = + + nd the stright line y =, = & = the lest?. Find the re of the region ounded in the first qudrnt y the curve C: y = tn, tngent drwn to ) d Pge 6 of Are Under Curve
9 C t = π nd the is. 5. Find the vlues of m (m > ) for which the re ounded y the line y = m + nd = y y is, (i) 9/ squre units & (ii) minimum. Also find the minimum re. 6. Consider the two curves y = /² & y = /[ ( )]. (i) At wht vlue of ( > ) is the reciprocl of the re of the figure ounded y the curves, the lines = & = equl to itself? (ii) At wht vlue of ( < < ) the re of the figure ounded y these curves, the lines = & = equl to /. 7. A norml to the curve, + α y + = t the point whose sciss is, is prllel to the line y =. Find the re in the first qudrnt ounded y the curve, this norml nd the is of ' '. 8. Find the re etween the curve y ( ) = & its symptotes. 9. Drw net & clen grph of the function f () = cos ( ), [, ] & find the re enclosed etween the grph of the function & the is s vries from to.. Find the re of the loop of the curve, y = ( ).. Let nd for j =,,,..., n, let S j e the re of the region ounded y the yis nd the curve e y = sin y, j π ( j +) π y. Show tht S, S, S,..., S n re in geometric progression. Also, find their sum for = nd = π. [IIT -, 5]. Find the re of the region ounded y the curves, y =, y = & y = which lies to the right of the line =. [IIT -, 5]. If c c f( ) f() = f() c + +, f() is qudrtic function nd its mimum vlue occurs t + c point V. A is point of intersection of y = f() with -is nd point B is such tht chord AB sutends right ngle t V. Find the re enclosed y f() nd cheord AB. [IIT - 5, 6] ANSWER EXERCISE IV. B. C. B. B 5. B 6. D 7. C 8. A 9. C. B. A. A. D. C 5. B 6. A 7. D 8. C 9. A. C. C EXERCISE V. / sq. units. c = π 6 or π.. = ln 5. (i) m =, (ii) m = ; A min = / 6. = + e, = + e ( ). 7 6 sq. units π sq. units. 5 squre units. Pge 7 of Are Under Curve
Mathematics. Area under Curve.
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
More informationLesson-5 ELLIPSE 2 1 = 0
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).
More informationJEE Advnced Mths Assignment Onl One Correct Answer Tpe. The locus of the orthocenter of the tringle formed the lines (+P) P + P(+P) = 0, (+q) q+q(+q) = 0 nd = 0, where p q, is () hperol prol n ellipse
More informationR(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of
Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of
More information/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2
SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the
More informationCalculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION
lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..
More informationCONIC SECTIONS. Chapter 11
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
More informationFINALTERM EXAMINATION 9 (Session - ) Clculus & Anlyticl Geometry-I Question No: ( Mrs: ) - Plese choose one f ( x) x According to Power-Rule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationEigen Values and Eigen Vectors of a given matrix
Engineering Mthemtics 0 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE : Engineering Mthemtics I : 80/MA : Prolem Mteril : JM08AM00 (Scn the ove QR code for the direct downlod of this mteril) Nme
More information, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF
DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs
More informationDrill Exercise Find the coordinates of the vertices, foci, eccentricity and the equations of the directrix of the hyperbola 4x 2 25y 2 = 100.
Drill Exercise - 1 1 Find the coordintes of the vertices, foci, eccentricit nd the equtions of the directrix of the hperol 4x 5 = 100 Find the eccentricit of the hperol whose ltus-rectum is 8 nd conjugte
More informationk ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.
Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.
More informationx 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx
. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute
More informationCh AP Problems
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,
More informationReview Exercises for Chapter 4
_R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationTO: Next Year s AP Calculus Students
TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC
More informationLinear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically.
Liner Inequlities: Ech of the following crries five mrks ech:. Solve the system of equtions grphiclly. x + 2y 8, 2x + y 8, x 0, y 0 Solution: Considerx + 2y 8.. () Drw the grph for x + 2y = 8 by line.it
More informationPrerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,
R rern Tower, Rod No, Contrctors Are, Bistupur, Jmshedpur 800, Tel 065789, www.prernclsses.com IIT JEE 0 Mthemtics per I ART III SECTION I Single Correct Answer Type This section contins 0 multiple choice
More informationAPPM 1360 Exam 2 Spring 2016
APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the
More informationMATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A,B and C. SECTION A
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. TIME : 3hrs M. Mrks.75 Note: This question pper consists of three sections A,B nd C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. X = ) Find the eqution
More informationMath 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
More informationTime : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A
Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 1 Instructions : 1. Answer ll questions.. Write your nswers ccording to the instructions given below with the questions.. Begin ech section on new
More information15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )
- TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the
More informationSection 7.1 Area of a Region Between Two Curves
Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region
More informationNORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y
LOCUS 50 Section - 4 NORMALS Consider n ellipse. We need to find the eqution of the norml to this ellipse t given point P on it. In generl, we lso need to find wht condition must e stisfied if m c is to
More informationMTH 4-16a Trigonometry
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
More informationP 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
More informationEdexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1
More informationA LEVEL TOPIC REVIEW. factor and remainder theorems
A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division
More informationFirst Semester Review Calculus BC
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.
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More information1. If * is the operation defined by a*b = a b for a, b N, then (2 * 3) * 2 is equal to (A) 81 (B) 512 (C) 216 (D) 64 (E) 243 ANSWER : D
. If * is the opertion defined by *b = b for, b N, then ( * ) * is equl to (A) 8 (B) 5 (C) 6 (D) 64 (E) 4. The domin of the function ( 9)/( ),if f( ) = is 6, if = (A) (0, ) (B) (-, ) (C) (-, ) (D) (, )
More informationPARABOLA EXERCISE 3(B)
PARABOLA EXERCISE (B). Find eqution of the tngent nd norml to the prbol y = 6x t the positive end of the ltus rectum. Eqution of prbol y = 6x 4 = 6 = / Positive end of the Ltus rectum is(, ) =, Eqution
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationEllipse. 1. Defini t ions. FREE Download Study Package from website: 11 of 91CONIC SECTION
FREE Downlod Stud Pckge from wesite: www.tekoclsses.com. Defini t ions Ellipse It is locus of point which moves in such w tht the rtio of its distnce from fied point nd fied line (not psses through fied
More informationSuppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.
Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot
More informationMath 1431 Section 6.1. f x dx, find f. Question 22: If. a. 5 b. π c. π-5 d. 0 e. -5. Question 33: Choose the correct statement given that
Mth 43 Section 6 Question : If f d nd f d, find f 4 d π c π- d e - Question 33: Choose the correct sttement given tht 7 f d 8 nd 7 f d3 7 c d f d3 f d f d f d e None of these Mth 43 Section 6 Are Under
More informationy = f(x) This means that there must be a point, c, where the Figure 1
Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile
More informationThomas Whitham Sixth Form
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
More informationCET MATHEMATICS 2013
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
More informationFORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81
FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first
More informationMathematics Extension 1
04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
More informationMH CET 2018 (QUESTION WITH ANSWER)
( P C M ) MH CET 8 (QUESTION WITH ANSWER). /.sec () + log () - log (3) + log () Ans. () - log MATHS () 3 c + c C C A cos + cos c + cosc + + cosa ( + cosc ) + + cosa c c ( + + ) c / / I tn - in sec - in
More informationMath 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions
Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationPrecalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.
Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More informationES.182A Topic 32 Notes Jeremy Orloff
ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationCHAPTER : INTEGRATION Content pge Concept Mp 4. Integrtion of Algeric Functions 4 Eercise A 5 4. The Eqution of Curve from Functions of Grdients. 6 Ee
ADDITIONAL MATHEMATICS FORM 5 MODULE 4 INTEGRATION CHAPTER : INTEGRATION Content pge Concept Mp 4. Integrtion of Algeric Functions 4 Eercise A 5 4. The Eqution of Curve from Functions of Grdients. 6 Eercise
More informationEXERCISE I. 1 at the point x = 2 and is bisected by that point. Find 'a'. Q.13 If the tangent at the point (x ax 4 touches the curve y =
TANGENT & NORMAL EXERCISE I Q. Find the equtions of the tngents drwn to the curve y 4y + 8 = 0 from the point (, ). Q. Find the point of intersection of the tngents drwn to the curve y = y t the points
More informationCalculus AB. For a function f(x), the derivative would be f '(
lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:
More informationMAT137 Calculus! Lecture 20
officil website http://uoft.me/mat137 MAT137 Clculus! Lecture 20 Tody: 4.6 Concvity 4.7 Asypmtotes Net: 4.8 Curve Sketching 4.5 More Optimiztion Problems MVT Applictions Emple 1 Let f () = 3 27 20. 1 Find
More informationAlgebra II Notes Unit Ten: Conic Sections
Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting
More informationIMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB
` K UKATP ALLY CE NTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB 7-8 FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd - 5 7 Ph: -66 Regd
More informationOn the diagram below the displacement is represented by the directed line segment OA.
Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
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.
More information7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?
7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationAlg. Sheet (1) Department : Math Form : 3 rd prep. Sheet
Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,
More informationTrigonometric Functions
Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd
More informationELLIPSE. Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is. ( a,0) A'
J-Mthemtics LLIPS. STANDARD QUATION & DFINITION : Stndrd eqution of n ellipse referred to its principl es long the co-ordinte es is > & = ( e ) = e. Y + =. where where e = eccentricit (0 < e < ). FOCI
More informationChapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1
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
More information10 If 3, a, b, c, 23 are in A.S., then a + b + c = 15 Find the perimeter of the sector in the figure. A. 1:3. A. 2.25cm B. 3cm
HK MTHS Pper II P. If f ( x ) = 0 x, then f ( y ) = 6 0 y 0 + y 0 y 0 8 y 0 y If s = ind the gretest vlue of x + y if ( x, y ) is point lying in the region O (including the boundry). n [ + (n )d ], then
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Extension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors m be used A tble of stndrd
More informationLoudoun Valley High School Calculus Summertime Fun Packet
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!
More informationFUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y
FUNCTIONS: Grde 11 The prbol: ( p) q or = +b + c or = (- 1)(- ) The hperbol: p q The eponentil function: b p q Importnt fetures: -intercept : Let = 0 -intercept : Let = 0 Turning points (Where pplicble)
More informationSpace Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
More informationHYPERBOLA. AIEEE Syllabus. Total No. of questions in Ellipse are: Solved examples Level # Level # Level # 3..
HYPERBOLA AIEEE Sllus. Stndrd eqution nd definitions. Conjugte Hperol. Prmetric eqution of te Hperol. Position of point P(, ) wit respect to Hperol 5. Line nd Hperol 6. Eqution of te Tngent Totl No. of
More information6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS
6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.
More informationChapter 7: Applications of Integrals
Chpter 7: Applictions of Integrls 78 Chpter 7 Overview: Applictions of Integrls Clculus, like most mthemticl fields, egn with tring to solve everd prolems. The theor nd opertions were formlized lter. As
More informationUnit Six AP Calculus Unit 6 Review Definite Integrals. Name Period Date NON-CALCULATOR SECTION
Unit Six AP Clculus Unit 6 Review Definite Integrls Nme Period Dte NON-CALCULATOR SECTION Voculry: Directions Define ech word nd give n exmple. 1. Definite Integrl. Men Vlue Theorem (for definite integrls)
More information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More informationr = cos θ + 1. dt ) dt. (1)
MTHE 7 Proble Set 5 Solutions (A Crdioid). Let C be the closed curve in R whose polr coordintes (r, θ) stisfy () Sketch the curve C. r = cos θ +. (b) Find pretriztion t (r(t), θ(t)), t [, b], of C in polr
More informationFinal Exam - Review MATH Spring 2017
Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.
More informationCoimisiún na Scrúduithe Stáit State Examinations Commission
M 30 Coimisiún n Scrúduithe Stáit Stte Exmintions Commission LEAVING CERTIFICATE EXAMINATION, 005 MATHEMATICS HIGHER LEVEL PAPER ( 300 mrks ) MONDAY, 3 JUNE MORNING, 9:30 to :00 Attempt FIVE questions
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationI. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=
11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel
More informationS56 (5.3) Vectors.notebook January 29, 2016
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
More informationOptimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.
Optimiztion Lecture 1 Review of Differentil Clculus for Functions of Single Vrible http://users.encs.concordi.c/~luisrod, Jnury 14 Outline Optimiztion Problems Rel Numbers nd Rel Vectors Open, Closed nd
More informationMATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC
FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot
More informationPractice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator.
Nme: MATH 2250 Clculus Eric Perkerson Dte: December 11, 2015 Prctice Finl Show ll of your work, lbel your nswers clerly, nd do not use clcultor. Problem 1 Compute the following limits, showing pproprite
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More informationDate Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )
UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4
More informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More informationDA 3: The Mean Value Theorem
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
More informationForm 5 HKCEE 1990 Mathematics II (a 2n ) 3 = A. f(1) B. f(n) A. a 6n B. a 8n C. D. E. 2 D. 1 E. n. 1 in. If 2 = 10 p, 3 = 10 q, express log 6
Form HK 9 Mthemtics II.. ( n ) =. 6n. 8n. n 6n 8n... +. 6.. f(). f(n). n n If = 0 p, = 0 q, epress log 6 in terms of p nd q.. p q. pq. p q pq p + q Let > b > 0. If nd b re respectivel the st nd nd terms
More information( β ) touches the x-axis if = 1
Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I - Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without
More informationcos 3 (x) sin(x) dx 3y + 4 dy Math 1206 Calculus Sec. 5.6: Substitution and Area Between Curves
Mth 126 Clculus Sec. 5.6: Substitution nd Are Between Curves I. U-Substitution for Definite Integrls A. Th m 6-Substitution in Definite Integrls: If g (x) is continuous on [,b] nd f is continuous on the
More information10.2 The Ellipse and the Hyperbola
CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point
More informationCBSE-XII-2015 EXAMINATION. Section A. 1. Find the sum of the order and the degree of the following differential equation : = 0
CBSE-XII- EXMINTION MTHEMTICS Pper & Solution Time : Hrs. M. Mrks : Generl Instruction : (i) ll questions re compulsory. There re questions in ll. (ii) This question pper hs three sections : Section, Section
More informationLog1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?
008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing
More informationInstantaneous Rate of Change of at a :
AP Clculus AB Formuls & Justiictions Averge Rte o Chnge o on [, ]:.r.c. = ( ) ( ) (lger slope o Deinition o the Derivtive: y ) (slope o secnt line) ( h) ( ) ( ) ( ) '( ) lim lim h0 h 0 3 ( ) ( ) '( ) lim
More information