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Essential Tools for Understanding Calculus - Rules, Concepts, Variables, Equations, Examples, j) Helpful Hints & Lh Common Pitfalls STRATEGY FOR SOLVING PROBLEMS EFFECTIVELY I. Understand the principle (business or scientific) required. II.j) Develop a mathematical strategy. A. There are eight useful steps that will help you develop the correct strategy. I. Sketch, diagram or chart the relationships and information that is subject of the problem. 2. Identify all relevant variables, concepts and constants. 3. Describe the problem situations using appropriate mathematical relationships, functions, formulas, equations or graphs. 4. Collect all essential information and data. S. Lh ~ extra and unnecessarY information and data. 6. Derive a mathematical expression or statement for the problem, making sure all measurements are in' the correct unit. 7. Complete the appropriate mathematical manipulations and solution techniques. 8. Check the final answer by using the original problem and information to make certain that the answers, units, signs, magnitudes, etc., all make sense and are correct! FUNCTIONS I. Definitions A. A relation is a set oforder pairs; written (x,y) or (x, fix»~. B. A function is a relation that has x-values that are all different for differenty-values . A vertical line test can be used to determine a function; every vertical line intersects the graph, at most, once. C. A one-to-one function is a function that has y values that are all different for different x-values. A horizontal line test can be used to determine a one-to-one function; every horizontal line intersects the graph, at most, once. D. Domain is the set of all x-values of a relation. E. Range is the set of all y-values of a relation. F. A function is an even function iff(- x) = fix). G.A function is an odd function iff(- x) = -f(x). H. The one-to-one functions f(x) and g(x) are inverse functions iff(g(x» = g(f(x» =x;f '(x) and g-'(x) indicate the inverse functions of fix) and g(x), respectively. Inverse functions are reflections over the line graph ofy = x . I. Dependent variable is the output variable in an equation and depends on or is determined by the input variable. 1. Independent variable is the input variable in an equation. II. Common Function Summary A. Linear:f(x) = mx + b I. m is the slope; m =Y2 - y, = y, - Y2 = ~y = rise x 2 -x, x, -x2 ~ run 2. b is the y-intercept. 3.lt is a constant function when m = 0; it is a horizontal line. B. Absolute value:f(x) = a~ - hi + k I. (h, k) is the vertex. 2. If a> 0, the graph opens up. 3. If a < 0, the graph opens down. 4.±aare the slopes of the two sides of the graph. C. Square root: f(xl=a.Jx-h +k I. (h, k) is the endpoint. 2. f2 If a> 0, the graph goes to the right. 3. p If a < 0, the graph goes to the left. D. Polynomial: f(x) = a"x' + a~,x~' +...+ a,x + ao 1. ao is the y-intercept. 2. There are, at most, n zeros or x-intercepts where f(x) = o. 3. There are, at most, n - 1 points ofchange or turns in the graph. 4. j) The extreme left and right sections of the graph both go up or both go down if n is even, and go in opposite directions if n is odd. E. Quadratic:f(x) = a(x - h)2 + k I. This is a special case of a polynomial. 2. (h, k) is the vertex. 3. If a> 0, the graph opens up. 4. If a < 0, the graph opens down. S. Use quadratic formula to solve for the zeros or x-intercepts: X= -b+~ . () 2a F. Rational: f(x) = p x qCx) l.p(x) and q(x) are polynomials, and q(x) f. o. 2. If degree p(x) < degree q(x), the asymptote is the x-axis wherey = O. 3. If degree p(x) = degree q(x), the asymptote is y = (lead coefficient ofp(x»/(lead coefficient ofq(x» . 4. If degree p(x) > degree q(x), the asymptote is y = (the quotient ofp(x).,. q(x»; a diagonal asymptote. G. Exponential:f(x) = ax I a>Oandaf.]. 2. If a > 1, the function is increasing. 3. If a < 1, the function is decreasing. 4. Rules for exponents: a. x'" • x" = x"'" h. XIII =x",-n xn c. (x'" r =x",n d - ", __1_ .X - x'" e. _1_= x"' x - ", f. (xy)"' =x"'y"' g'(;T=~: h.x~ =~=(!!/Xr i . ~=!!/X.~ j . ~=~ k.~="'!!/X I. Ifx' =x', then a = b. m. Ifax = bx , then a = b if a f. O. H. Logarithmic:f(x) = log.x I.x>O 2. a> 0 and a f. 1. 3. Lhf(x) = log.x, IF and only IF. aftx )= x. 4. a is the base. S. Logarithms are exponents. 6. Rules for logarithms: a. log.I = 0 b.log.a = I c.log.ax=x d. a'og.x = x e. Iflog.x = log.y, then x =y . f. f(x) = log, x = In x; this is the natural log. log. x logx Inx g.log. X= 10gb a = loga =III/l; r-----------------~ this is the change-of-base rule. h. log.xy = log.x + log.y i.log.y=log.x-log.y j. log.xY = ylog.x I.Trigonometric 1. Basics a. j) Angles can be measured in degrees and radians. i. I radian = (] !O) degrees ii . I degree = (] ;0) radians iii.proportion conversion of angle measurements: angle in degrees angle in radians 180 0 It radians iv. unit circle (a) center at (0, 0) (b) radius = one unit (c) points on the circle = p(x,y) (d) j) Positive angles move counterclockwise ~ from P(I, 0). (e) j) Negative angles move clockwise from 0 P(I,O). (f) j)Angles rotating one or more full times ~ require adding ±21t for each rotation. ~ v. function definitions III (a)sinS=y Z (b) cos S =x (c) tanS=~ ~ (d) cscS=.!. y (e) secS=.!. x (f) cotS=,,!. y vi . j) useful values 8 = degrees; t = radians',{) = undefined 8 0 30 4S 60 90 180 t 0 .n. 6 .n. 4 .n. 3 .n. 2 It sin 0 l 2 J2 2 .J3 2 1 0 cos I .J3 2 J2 2 l 2 0 - ] .J3tan 0 1 .J3 {) 0 3 2. Graphing properties a. Amplitude of sine and cosine is half the difference between the maximum and tht! minimum values, or lal. ~ b. Period is the radians needed to complete one full cycle of the curve, or 2: . c. Horizontal shift or phase shift is c. " d. Vertical shift or average value is d. W e. Sine:f(x) =aslnb(x- c) + d m f. Cosine:f(x) = acosb(x - c) + d g. Tangent:f(x) = atanb(x - c) + d ZLh [CAUTION! Tangent has no amplitude. so a affects the vertical stretch and shrink only.] ~ h. Cosine is even; sine & tangent are odd. 3. Important identities & formulas 1 0 Functions {Sontinued) a. Pythagorean identities i. sin' u + cos' U = 1 ii.l + tan'u = sec'u iii.l + cot'u = csc'u b. Sum/difference formulas i. sin(u ± v) = sinu cosv ± cosu sinv ii. cos(u ± v) = cosu COSV 'f sinu sinv iii tan(u+v)= tanu+tanv . - 1 'f tan u tanv c. Half-angle formulas 1. sini=±p-~osu ii . cosi=±~I+~OSU iii. tan.!!. = I-cosu =~ 2 sinu l+cosu d. Double-angle formulas i. sin2u = 2sinu cosu ii. cos2u = cos' u - sin'u = 1-2sin'u = 2cos' u - 1 iii.tan2u= 2tanu I-tan' u e. Power-reducing formulas 1. sin' u l-cos2u 2 ii. cos' U= l+cos2u 2 iii. tan' U= l-cos2u 1+cos2u Ill. Basic Common Function Graphs A.Cons*<lnt:j(x) = c F. Cubic:f(x) =x' B. Linear identity:j(x) =x G.Rational: j(x)=~ C. Absolute value: H. Exponential: j(x) = Ix! j(x) =eX D.Square root: I. Logarithmic: j(xb-.JX j(x)=lnx E. Quadratic function: J. Sine: f(x) =x' j(x) = sinx y f-- 2+--+-+-f-IH 1'\-- I t-:r ...... "+-t---+:v.:,,,x K.Cosine:f(x) = cosx yI I I I I I I r If j x " j 2" - -I =+ "'~ - -2 I l I I M.General Transformations 1. When given the function f(x) and the number a, then the function: f(x) ± a has a vertical shift up for +a; down for-a. j(x ± a) has a horizontal shift right for -a; left for +a. aj(x) has verticalstretch if a> 1; vertical shrink if 0< a < 1; x-axis reflection if a is negative. j(ax) has horizontal shrink ifa> 1; horizontal stretch ifO < a < 1; y-axis reflection if a is negative. LIMITS & CONTINUITY I. Definitions A. A is the limit of f(x) as x approaches a, written ~!T. j(x) = A; means that for each neighborhood ~A (neighborhood of A with radius p), there exists a punctured neighborhood N,a such that f(N,a) is a subset of~A when N,a is a subset ofthe domain off. B. ~!T. j(x) = A if for every E > 0, there exists Ii > ° such that If(x) - AI < E when Ix-aI<Ii, assumingj(x) is defined for all x in an open interval containing a. C. Geometrically, ~i,"! j (x) is the y-value of the point in which the graph ofj should intersect the line x = a. D. (l Generally, lim j(x) = A implies thatf(x) comes /-' X-HI infinitely close to A as x gets infinitely close to a. E. Limits can equal +00 or -00, usually as j(x) approaches an asymptote. F. (l Informally consider that the number limj(x)~ x~a is the number that is approximated by j(x) when x is close to, but not equal to, a. G.One-sided limits 1. A left-sided limit equals L, written lim j(x) = lim j(xl= L, ifj(x) gets close to L X~fI- xi" as x gets close to a from the left; that is, x gets close to a but remains less than a. 2. A right-sided limit equals R, written lim j(x)=limj(x)=R, ifj(x) gets close to R X-)Q+ xJ,Q as x gets close to a from the right; that is, x gets close to a but remains greater than a. 3. l' limj(x) exists and limj(x)=A, lE..AWl ~ x-). X-)(I H. The functionjis continuous at the point a ifj(a) is defined and t!T. j(xl= j(a). 1. The functionjis continuous iffor every E > 0, there exists Ii > °such that for x andy in the domain ofj when Ix - yl < Ii, then If(x) - j(Y)1 < E. II. Theorems A.lim[j(xl±g(x)]=limj(xl±limg(x) X-HI X-)Q X-)Q B. If function g is continuous at point A and limj(xl=A, then limg{J(x») = g(limj(x»). X-)(I X-)Q X-)Q C. lim[j(x). g(x)]=(limj(x»)(limg(x») X_CHI X----)(I X-HI j(x) lim j(x) D lim--=~ provided g(x "# 0) and .X-->. g(x) limg(x) ' X-->. limg(x) "" 0. E. ;:{J(x»)" =(limj(x)"), provided n is a positive X----)Q X-)Q integer. F. Iimj(x)=A is equivalent to lim[j(x)-A]=O. x----). X----)Q G.lf j(x) < g(x) < hex) for every x in a punctured neighborhood of a (that is, x near a), and limj(x)= IimhCxl= A, then limg(x)=A. X-HI x--+a X.....,IIf 2 H.~Forexample, when finding lim x 3 -28 such thaI 0/' x-+l x x 3 , x-2 x-2' x"# 2 - 8 becomes (x - 2)(x2 + 2x + 4) and then (x' + 2x + 4); consequently, when x is close to 2, (xl + 2x + 4) is close to 12; therefore, lim xl-: = 12. III. Rules x----)2 x- A. For polynomial p(x) to the n'h power with the lead term of ax" and polynomial Q(x) to the m'h power with the lead term of bX"', if p(x) x =Q(x) and Q(x) "# 0, then when: j() l.n=m, the limj(xl=![ x----)oo b 2. n > m, the lim j(x) = lim P «x» X--,)ooo X----) <XI Q x 3.n<m,the limj(x)=O X-->~ B. lim c = c, when c is a constant. x-.. cr eX_I=1 ·x~ x D.lim x: =0 X----)<><> e E. lim(X+l)X=e X----) OO X F lim aX -I = Ina • X-+O X G. lim sinx = I x--+o x H.limcosx-l =0 x----)o x I. lim tanx =1 X~O x DERIVATIVES 1. Definitions A. If x is a number in the domain of a function, /. j(x+h)- j(x)( ) then the derivative f' x = lim , provided the limit exists. h-.O h I. If the limit exists at x; thenjis differentiable at x; 2. j) Not all continuous functions are differentiable. B. If a function,/. is differentiable at point a, then the tangent line to the graph ofj at the point (a, /{a» j '() 'th j'() I' j(a + h)- j(a)has sIope a WI II = 1m . X~. h I. This tangent line is unique. 2. This derivative is the instantaneous rate of change of j; it tells how fast j is increasing or decreasing with respect to x as x approaches a; that is, near x =a. 3. rP For example, when y==.J4-x2 , the slope of the tangent line to the curve at point (x,y) is -;; therefore, the derivative of y is y'=f'Cx)=-~=~ . y .J4-x' y -2 -\ C. The first derivative function notations include f'(x), y', dy, DxY, and ..!L j(x); second dx . fix. • d1y derivative notations mclude j (x), y, dX' and D;y. D. The derivative atx = a is usually written as: /'(11), D(f)(II), or 1;L. 2 II. RuleslFormulas A. Assume fix) and g(x) are differentiable functions, D,f(x) = :x lex) = f'(x), and D.g(x) = ! g(x) = g'(x) for the following statements: 1. L'Hopital's Rule: If/and g are differentiable for x near a and Iim/(x)=limg(x)=O; or, X~1l X--+Q lim/(x)=limg(x):±oo and g'(x) t- 0, then X--+d' x--+« lim lex) = lim f'(x) . x~. g(x) x~. g'(x) 2. Chain Rule: Ifll =g(x), thenD,f(II) =DJ(II)D.II; dy dy dll or, D.y=D.yD.u; or, dx = dll dx wheny=f(II). 3. RoUe's Theorem: If/is continuous in the closed interval [a, b] and iflea) = feb), then there is at least one point m in the open interval (a, b) such thatf'(m) = O. 4. Mean Value Theorem: If/is continuous in the closed interval [a, b], then there is a point m in the . /(b)- lea) () open mterval (a, b) such that b-a f' m . 5. Lmg(x)=mg'(x) for all real numbers m. dx 6. L{J(xhg(x») = f'(xhg'(x) dx 7. L[/(x)g(x)j =lex) g'(x)+g(x) f'(x) dx 8.L[/(X)]= g(x) f'(x)- /(x)g'(x) for g(x}t-O. dx g(x) [g(x)jl 9. Lc=O, when c is a constant. dx . 10.Lx=1 dx II. L(l)=_...Ldx x Xl 12.L(mx+b)=m' for all real numbers m. dx 13. L(xt'= nx"-I, when n is a real number; n t- 0, dx x..-1 is defined. 14. L £ = I, dx 2vx 15. L JI (x) I ; derivative of an dx f'{j-I (x») inverse function whenf'if-l(x» t- o. 16. L r=r dx 17.L~ =~Ina dx 18. L lnx=ldx x 19.A... log x=_I_ dx • xlna 20. L(sinx) = cosx dx 21. L(cosx)=-sinx dx 22. L (tanx)=secl x dx 23. L (cotx)=-cscl x dx 24. L(secx) = secx' tanx dx 25. L(cscx)= -cscX'cotx dx 26. L(arcsinx) = ,....!-., dx vI_Xl 27.L(arccosx)=- ,....!-., dx vI-xl 28. L(arctanx) = _1_ dx l+xl 29. L (arccotx)= __I- dx I+x 30. L (arcsecx)= ~ dx x xl-l 31. L(arccscx) 1 dx x.Jxl-l III. Applications A.lmplicit Differentiation 1. Used when it is difficult or undesirable to solve an equation for y, such as x' + y' = 1. 2. Differentiate both sides of the equation with respect to x. 3. Apply the Chain Rule. 4. Substitute y' for : and 1 for ::; . 5. Solve for y '. 6. ~ For example, when finding the derivative of y= Jx~\ f(;)[;(X] - 1)with and g(x) .Jx+l' dx g(x) becomes L[ (x-I)]= .Jx+IDx (x-l)-(x-I) Dx .Jx+I ; dx .Jx+I [.Jx+lf then, using the powerformula to find Dx .Jx + I , .Jx+l (1)- (x-I) 2.Jx+lthe statement becomes x+1 finally, using algebra to simplify the expression, the derivative becomes x+3 or (x+3),Jx+t B. Graphs 2(x+ If. 2(x+ 1)1 . I. Increasing/decreasing a. A function/ is increasing in an interval (a, b) iff(a) <feb) whenever a < b. b. A function/ is decreasing in an interval (a, b) if/Cal >feb) whenever a < b. c. If/ is continuous and/'(x) > 0 at every point of an open interval (a, b), then/ is increasing in this interval. d. If/ is continuous andf' (x) < 0 at every point of an open interval (a, b), then/ is decreasing in this interval. e. Considering a point traveling left to right along a curve off, ifthe point goes up in any interval of the curve, then/ is increasing in that interval; if the point goes down in any interval ofthe curve, then / is decreasing in that interval. 2. Concavity a. A curve or part of a curve is concave up if the curve lies above the lines thatare tangent to the points on the curve. b. A curve or part of a curve is concave down if the curve lies below the lines that are tangent to the points on the curve. c. Ifj" (x) > 0 at every point in an interval, then the graph off{x) is concave up in this interval. d.lfj"(x) < 0 at every point in an interval, then the graph of/ex) is concave down in this interval. 3. Inflection point a. If / is differentiable in a right and in a left interval or neighborhood ofany point a at which the graph of/is continuous, and ifj" is positive for all values in one ofthe intervals but negative for all values in the other interval, then (a,f(a» is a point of inflection of the graph off. 4. Maximum/minimum a. Point (a,f{a» is a relative or local minimum pOint of any interval of the graph of/if/Cal < fix) for any x in this interval; the number lea) is the minimum value. b. Point (a,f(a» is a relative or local maximum point of any interval of the graph of/if/Cal > lex) for any x in this interval; the number lea) is the maximum value. c. The global or absolute minimum is the point that has the leastf{x) value in the domain. d. The global or absolute maximum is the point that has the greatest/ex) value in the domain. e. Extreme Value Theorem: If/is a continuous function on a closed interval [a, b], then/ has a maximum and a minimum; and, the global or absolute maximum and minimum occur only at critical points or endpoints. f. A critical point, (x,f(x», is a point ofthe graph of/ that satisfies one of these conditions: i. f'(x) = 0 ii.f'(x) does not exist; OR iii.(x,f{x» is an endpoint of the graph. iv.~YFor example, on the graph ofy = lex), at the relative maximum point P and the relative minimum point Q, the curve has a horizontal tangent as it also does at point R, which is neither a maximum nor a minimum point; additionally, if the search for maximum and minimum points is limited to those points whose x-coordinates satisfy rex) = 0, then the maximum point S and the minimum point T. which is an endpoint, will be missed, and these are all critical poipts. y s -L ( I p "\. R ./f" ./ / - I- - Q \ N \ T x I g. P A maximum point or a minimum point must be a critical point, but critical points need not be maximum points or minimum points. h. If / is differentiable in an open interval that contains point a. such thatr(a) = 0, then: i. f{a) is a maximum value off, ifj"(a) < 0; AND ii.f(a) is a minimum value of/ ifj"(a) > O. 6 [CAUTION! This test does not apply if f"(a) = 0.] C. P Helpful Hints for Sketching a Curve I . Determine the domain for the function,f{x). 2. Analyze all points where lex) is not continuous. 3. Sketch all vertical, horizontal and oblique asymptotes, if there are any. 4. Evaluater(x) andj"(x). 5. Find and plot all critical points, a, where f'(a) does not exist or wherer(a) = O. 6. Find and plot all relative maximum and all relative minimum points. 7. Find and plot all possible inflection points, b, wherej"(b) does not exist or wherej"(b) = O. 8. Find and plot the x-intercepts and the y-intercepts, if there are any. 9. Complete the sketch of the curve. D. Rate ofChange I. Average rate of change of/over the interval [a,x]: /(x)- lea)Ia. s x a . b. As x approaches a. the average rate of change approachesr(a). c. It is the slope of the line containing the endpoints of the interval. 2. Instantaneous rate of change off; a.lsr(a)whenx=a. b. It is the slope of the unique line tangent to the graph of/ at point a. c. It measures how fast/ increases or decreases at point a. d. Instantaneous velocity is 1'(1), where s is the position, s = let), and I is time. e. Instantaneous acceleration isr(I), where" is velocity, " = f(1), and I is time. 3 INTEGRATION I. Area Under a Curve A. If a function, f(x), is a curve graphed in the interval [a, b], then the area bounded by the curve, the x-axis, and the vertical lines containing the endpoints of the interval [a, b] may be approximated through the following: I. Rectangular methods a. Divide the interval into rectangles with equal width of b;; a . b. This results in n + 1 points on the x-axis. . b-a c. These pomts are Xo = a, XI = a+ --;;- , x 2 =a+2(b~a), .•• ,xo =a+n( b~a). d. Find the sum of these n rectangles of equal width in the bounded region. e. Left-endpolnt method i. The height of each rectangle is the vertical left side. ii. The height of each rectangle is f(x,) for o:s i:S (n - 1). iii .The sum of these rectangle areas is b;;a [f(Xt) + f(x.) + f(X2) +,..+ f(x_.)] = b n-I ;a Lf(xJ ;=0 f. Right-endpolnt method i. The height of each rectangle is the vertical right side. ii. The height of each rectangle is f(xj) for 1 :s i:S n. g. The sum of these rectangle areas is b-a o n [f(x.) + f(X2) +,..+ f(x.)] = b~a L f(xj) . h. Midpoint method ,~I i. The height of each rectangle is the 'vertical line segment from the midpoint of the rectangle base to the midpoint of the opposite side. ii . The height of each rectangle is f( Xj +2Xj+1 ) for O:S i:S (n -I). iii.The sum of these rectangle areas is b:a [f( Xo ;XI )+f( xI :x2 )+,..+ ( X0-1 +x)]=-=.!!b 0-1L f )f (x-1...........l:+x 0 2 n j~O 2 0- 1 (x. +x. ) b or!'J.x L f -'__1+_1 where !'J.x = ;a. j~O 2 y I 1 I JL , ~ , ,1/ , , , , \ , , ,IL , , , :'\, , ,j , , , , , , , , , , , ,, , , , , x a l x l x2 l x, 1 X4 b I I I i. Riemann sums i. The height of each rectangle is f(c,), where Cj is any point in each subinterval, [Xj, Xj+l], for 0 :s i:S (n - I). o ii. Riemann sum is Lf(c,)(x, - XH) or o j~1 b !'J.xLf(c,), where /j.x=(Xj-Xi-\) = ;a. ;=1 2. Trapezoid method a. Divide the total interval into trapezoids with parallel sides of equal subinterval b-alengths, --,,-- . b. This results in n + 1 points on the x-axis. . b-ahc. T ese pomts are x. = a, XI =a+--,,-, x 2 =a+2(b-a) x =a+n(b-a)n , ... , n n ' d. The area of each trapezoid is the average of the vertical left and right (parallel) sides multiplied by the horizontal subinterval length (distance between them). e. The average of the two parallel sides is f(xj)+ f(xj+l ) forO :S i:S(n-l), 2 f. The sum of these trapezoid areas is b-;,a [f(xo); f(x\) + f(xl ); f(x2 ) + ... + f(xn-I~+ f(xo)] = b [0 0-1 ]2-na j~f(xj)+j~f(x,) . 3. Simpson's Parabolic Rule a. The interval [a, b] is divided into subintervals of length, b-a, where n is a positive even integer. n b. This results in n + 1 points on the x-axis. . b-aThc. ese pomts are x. = a. XI =a+--,,-. x 2 =a+2(b;a), .. . , xo=a+n(b;a) . d. Every three consecutive points, (Xj, f(xj)). on the curve are also points on a parabola when 0 < i<n. e. The su~ of these parabola areas is b3-na (f(xo) + 4f(x.) + 2f(X2) + 4f(x,) +.. ..+ 2f(x_2) + 4f(x_.) +f(x.)]. f. j) [NOTICE: Inside the brackets, the coefficients of f(xo) and f(x.) are both 1, but the coefficients of the f(xj)'s with odd subscripts are all 4, and the coefficients of thef(x,),s with even SUbscripts are all 2.] II. Definite Integral A. If the limit exists, then the function has an integral on the interval [a, b]. B. J is the integral sign. C.f(x) is the integrand. D.a is the lower limit of integration. E. b is the upper limit of integration. 1. Using Riemann sums [see bottom left ofpage]. f: f(x)dx= Iim!'J.xi:, f(cJ n--+.... ;=1 2. Using the trapezoid method, s: f(x)dx =!~'?o t; [j~ f(xj)+:f f(x j )1 3. Using the Simpson's Parabolic Rule, fb f(x)dx= lim !'J.x [((Xt) + 4f(x.) + 2f(X2) fI n --+co 3 + 4f(x,) +...+ 2f(x_2) + 4f(x_tl+ f(x.)]. 4. fb f(x)dx is the net signed area between the c~rve and the x-axis. a. When this area is above the x-axis, it is considered to be ~. b. When this area is below the x-axis, it is considered to be ~. c. @JForexample, when a is a positive number, the curve for y = .Ja2 - x 2 is a semicircle with a radius ofaunitsanda centerof(O, 0); therefore, in the interval [-iI. a], this semicircle and thex axis create a bounded region having an area of 2 1t~ square units, so the integral orone-halfof this region is f: .Ja2 _x2 dx= 1t: . Ill. Indefinite Integral A.F(x) is called an antiderivatlve off(x) if r(x) =f(x). B. There is a family of antiderivatives F(x) + C, where C is a constant, because all such functions have the same derivative. C. & Jf(x~ = F(x) + C. IF and onlv IF, r(x) =f(x). IV. Fundamental Theorem of Calculus A. If f(x) is an integrable continuous function on the interval [a, b], and if F(x) is a continuous function in [a, b] such that rex) = fix), then f: f(x)dx= F(x)l! =F(b)- F(a). 4 B. Iff(x) is a continuous function on the interval [a. b]. then the function F(x)= f: f(t)dt is an antiderivative off(x) on [a. b]. V. Mean Value Theorem for Integrals A.If f(x) is a continuous function on the interval [a, b], then there is a point m in the interval such that f(m)=If--fbf(x)dx or (b-a)f(m)= f:f(x)dx . -a ' VI. Basic Definite Integral Theorems A. f; f(x)dx=O B. f: f(x)dx=-f: f(x)dx when a < b. C. f!cf(x)dx=c f: f(x)dx when c is a constant. D. f:[f(xl±g(x)]dx=f: f(x)dx±f:g(x)dx E. f!cdx=db-a) F. f: f(x)dx <f: g(x)dx whenf(x) < g(x). G. f: f(x)dx=f: f(x)dx+ f: f(x)dx H.@JForexample: e.slnxdx =f~.sinxdx+ f;sinxdx= - f~.(-sinx)dx+f;sinxdx= -f;sinxdx+f;sinxdx=O because f~.sinxdx=- f~.(-sinx)dx; and, from a geometric point of view, y 2 / "\ \ y = x - slnx -It: !} / 71 2 y 2 y=slnx / "\ -If\. / -\ 71 x 2 f~. (-sinx)dx can be considered the region bounded by the x-axis and the curve of y = -sinx. which has the same area as the region bounded by the x-axis and the curve y = sin x; therefore, f~. (-sinx)dx= f;(sinx)dx, and hence, r.sinxdx = f~.sinxdx+ f;sinxdx= - f;sinxdx+ f;sinxdx= O. VII. Common Integration Formulas A,fdu =u + C "..+1B. f"..du = -- when,;t - 1. ,+1 C. Jeos udu = sin u D,fsin udu = -eos u E. Jtan udu = Inlsee ul F. Jeot udu = Inlsin ul G.Jsee udu = Inlsee u + tan ul H. Jese udu = Inlese u - eot ul I. Jsee2udu = tan u J. Jese2 udu = ~ot u K,fsec u tan udu =see u L. Jese u eot udu = ~se u M.Je"du= e" N·f!du=f d,: =Inlu! O fbudu=L . Inb P. Jlnluldu =u(lnlul-l) -0)' f has a at an on a is 1<' · a. here aylor a =O, or all : )xn )=1, Integration (con tinued) Q.f .Ja~~u2 =arCSin(*)=sin-. (*) R.f a2~u2 =~arctan(*)=~tan-I(*) S f~=sinh-I(.!!.)=ln(U+~) . .Ja2 +u2 a a T f~=COSh-I(.!!.)=ln(U+~) . .Ju2-a2 a a U f ~=.ltanh-l (.!!.)=-.Lln(a+u) . a2-u2 a a 2a a-u when u2<a2 V. J~=-.lcoth-I (.!!.)=_...Lln(a+u)u2_a2 a a 2a u-a when a2 < u2 W.f.Ju2±a2du= .!!..Ju2+a2±E..lnlu+.Ju2+a2 12 - 2 X. J) Note that the Fundamental Theorem tells how to evaluate the integral of a function J by using any antiderivative ofJ(x); and, the value of the integral can be obtained no matter which antiderivative ofJ(x) is used. I.~For example, the integral f: cosxdx can be evaluated using the antiderivative sin x or sin x + 7, since D..(sinx) = cosx and D..(sinx + 7) = cosx. 2. So, f: cosxdx=(sinx)l: = sinb-sina. 3. Also, f: cosxdx=(sinx+7)1: = (sinb+ 7)-(sin a+ 7)= sinb-sina. 4. Therefore, since the solutions are the same, it is wise to choose the simplest antiderivative when stating an integration formula and solution. VIII. Integration Techniques A. Substitution I. This is a method of using the Chain Rule to calculate integrals and find antiderivatives. 2. Use f.f(g(x»)g'(x)ttt = J.f(u)du, where u = g(x). 3. IfJi(x)ttt= F(x) + C, then useJ.f(g(x»g·(x)dx = F(g(x» + C. B. Integration by Parts I. Use the abbreviated notation, Ji(x)g'(x)dx = .f(x)g(x) - Jg(xV'(x)ttt, from the Fundamental Theorem of Calculus; set u = J(x) and v = g(x), sorex) = DxU and g'(x) = Dxv; then, use the resulting formula fudv = uv -Jvdu. 2. For definite integrals, use f: J(x)g'(x)dx= J(x)g(x)l: - f: g(x) j'(x)dx. C. Partial Fractions I. Decompose rational functions whenever the denominator is factorable. 2. Integrate each partial fraction that results. D. Improper Integrals I. Integrals over an infinite interval or having an infinite range. 2. Some converge, having a finite limit that exists. 3. Some diverge, having a finite limit that does not exist. 4. If J(x) is continuous on [a, 00) and integrable on [a, b) for all b > a, then J;" J(x)dx=limf!J(x)dx,if the limit exists. b .... 5. If .f(x) is continuous on (-<Xl, b] and integrable on (a, b1 for all a < b, then f~_J(x)dx= lim fJ(x)dx, if the limit exists. •....- 6. IfJ(x) is continuous on (-<Xl, 00) and integrable, then [j(x)dx=f~_ J(x)dx+f: J(x)dx, for any a, if the limit exists. IX. Applications of Integrals A.Areas I . The area, viewed as horizontal rectangles, in the interval [a, b] between two curves with J(x) > g(x), making .f(x) the top curve, is f:[J(x)- g(x)]dx. 2. The area, viewed as vertical rectangles, in the interval [V" Y2] where y, < Y2 between two curves.f(v) > g(y), makingJ(y) the right curve, is f:[J(y)- g(y)]dy. x B.Volumes 1. Solids oriented to an axis with cross-sections on planes that are perpendicular to that axis; the area of a cross-section is given by the function A(x). 2. The volume of a cross-section slice of thickness I:!.p is A(x)l:!.p. 3. The volume of the solid bounded by the planes at the ends of the interval [a, b] is V = f: A(p)dp. 4. Disk Method: The volume of a solid of revolution created by the curve .f(x) as it revolves around an axis in the interval [a, b] is V= 7t f:[J(xW dx. J) [NOTICE: [f(x)P is the radius squared.] y 5. Washer Method: The volume ofthe solid of revolution created between two curves with J(x) >g(x) as they revolve around an axis in the interval [a, b] is V= 7tf:[J(X)2 -g(x)2]dx. J) [NOTICE: [f(X)2 - g(X)2] is outer radius squared minus inner radius squared.] 6. Shell Method: This method is used when it is difficult to compute the inside or the outside radius of a cross-section. a. A radial coordinate r, with a < r < b, along an axis perpendicular to the axis of revolution, produces the heights her) of cylindrical sections or shells of the solid parallel to the axis of revolution. b. The area of this shell at r is A(r) = 27trh(r). c. The volume of this solid shell is V =f: A(r)dr=f:27trh(r)dr. C. Surface Area 1. Solids of revolution created by revolving the function y =J(x) around the x-axis in the interval (a, b). 2. Then, between x = a and x = b, the surface area is S = f:27tJ(x)~1 + [j'(X)]2 dx . 3. If the generated curve C is parametrized by (x(t), yet»~, with a :s t :s b, and revolves around the x-axis, then the surface area is S= f:27ty(t)~[X'(t)]2 +[y'(t)]2 dt. D.Arc Length I. Ifa graphy = J(x) has a continuous derivative in the interval (a, b), then, between x = a andx = b, the graph has length L= f: ~1 + [j'(x)]2 dx. 5 DIFFERENTIAL EQUATIONS I. Differential equations are equations that involve derivatives of unknown functions. II. General solutions represent a family of curves if it has unspecified constants. III. A basic differential equation involving the dependent variable isr(x) = kf(x) ory ' = ky or ddY = ky, giving dy= kdt where lYl = kt + e . I y IV.A differential equation that is linear in the dependent variable and involves only the first order derivative isr(x) + p(I)J(x) = q(t). V. Separation of variables can be written as ;: =J(x)g(y) orJ(x)dx= g(ylv· VI. Chain Rule Equation: DJ(u) = D'/(u)DxU. VII. IfF'(x) =J(x), then F(x) is an antiderivative of J(x). VIII.F'(x) =J(x) is solved when the Fundamental Theorem of Calculus is used to evaluate f:J(x)dx . IX. Always consider if the differential equation has a solution and if there is only one solution. A.~For example, to solve the differential equation y' = J(x), use the Fundamental Theorem of Calculus to evaluate the integral f: J(x)dx as F(b) - F(a) (for this purpose, any solution will do). S.Such as, for f.23x2dx, F(x) can be x' or x1 +t or x'- 5, and so on, since all of them satisfy the differential equationy' = lr. C. & But, in most applications ofdifferential equations, it is not true that any solution will do; only one particular solution also satisfies some given initial conditions. D. Thus, in addition to the differential equation, specific numbers, a and e, might be given, such that the differential problemy' =J(x), with y = e when x = a is obtained. X.lf J is continuous in an interval containing point a, then y= f: J(t)dt+e satisfies some differential equation problems. XI. IfJand g are continuous andg(y) f. 0 fory in some interval containing e, then the differential problem , J(x) .. y = g (x) wherey = ewhenx = a III some Illterval containing a can be solved by solving the equation f: g(t)dt=f: J(t)dt fory. POLAR COORDINATES & GRAPHS I. Point P = (r, a) where: A. The pole, 0, is the center point (like the origin) of the polar coordinate system. B. r is the length of segment Op, C. 9 is the angle formed by segment OP (the terminal side) and the initial side (usually the positive x-axis). D. 9 is positive when segment OP moves counterclockwise; negative when it moves clockwise. II. Conversions A, J) From polar to Cartesian, use x = rcosa and y = rsina. S. J) From Cartesian to polar, use a =tan- ' 1'. and r =.Jx2+y2 . III. Graphs & Equations A.Circles 1. Cartesian equation (x - ecosa)2 + (y- esina)' = a2, has radius of a and polar coordinates center of (e, a). 2. Polar equation r - 2rccos(a - a) + c' = al, has radius ofa and polar coordinates center of(e, a). 3. Polar equation r = 2ecos(a - a); if the circle contains the origin, then c' = a2• x - - Polar Coordinates & Graphs (continued) 4. Polar equation r = 2ccos8 or r = 2csin8 if a = O. B. Roses I. Polar equations r = ccosn8 or r = csinn8. 2. n determines the number of petals on the rose. z C. Cardioids & Limacons I. Polar equations r = a ± bcos8 or r = a ± bsin8. w 2. When ~ > I, the Iimacon graph has an inner loop. 3.j) When ~=I, the limacon graph has NO~ inner loop, is heart-shaped, and is specifically O called a cardioid. .oiIIII a. fiP For example, r = 2(1 - cos8). 'I11III IV. Area A. The area bounded by the curve r = f(8) and enclosed by the rays 8 = a and 8 = ~ may be found using A=.!f~r2d8=.!f~ f(S)2dS . 2 (l 2 (l B. The area bounded by two polar graphs is A=.!f~(r.2 -r.2)d8.2 (l 2 I V. Arc Length A. The length of arc r = f(8) where 8 is in the interval [a, ~j is L = f~ r2 +(~~r dS. VI. Slope of Tangent A. The curve r = f(8), with coordinates x = f(S)cos8 and y = f(8)sin8 , has the slope of the tangent at dy sinS dr +rcos8 (x(S),y(8» of dy =AJi.= dS . dx dx cosS dr -rsinS dS d8 SEQUENCES & SERIES t.iIIII I. Sequences are functions that have domains that are 'I11III integers and ranges that are all real numbers. Z A. The integer in the nth position is called a term and denoted by the symbol a. rather than a(n}. W B. Consecutive arithmetic sequence terms have a ft common difference, d, with each term the result of ~ a.=a....l+d=al+d(n-I}. O C. Consecutive geometric sequence terms have a common ratio, r, with each term the result of a. = .oiIIII a...ir} = al(rrl . 'I11III D. lf a sequence has a limit, then it converges; otherwise, it diverges. E. Ifa sequence converges, then it is bounded. F. If {a.} and {b.} are convergent sequences, then: I. ~r-(a. +b.)=~it,!!a. +~t,!!b. 2. Iim(a.b.) = lima. limb. .t- . f- . f a lima • .t_ . h I' b °3 . I· en 1mImb=~'w ~. "1_ ,, 1I11'~ II " j- " 4. limca =clima ,wherecmaybeanynumber. ,.roo " "roo II G. Increasing sequences have every term a", wi1h a. :5 a ... I' H. Decreasing sequences have every term a... wi1h "" ~ 11,,;-1' I. Together, decreasing sequences and increasing sequences form the group of monotone sequences. J. A sequence has an upper bound if every term of the sequence is less than some fixed number. K. A sequence has a lower bound if every term of the sequence is greater than some fixed number. L. A sequence that has both an upper bound and a lower bound is said to be bounded. ~ M. it. Monotone sequences converge. IF and only 'I11III IE. they are bounded. Z 1. The limit of an increasing sequence is its least upper bound. 2. The limit of a decreasing sequence is its greatestW lower bound. ft II.A series is a sequence obtained by adding the terms of ~ another sequence. o A. A sequence, {S.}. whose terms are defined by S = fa =0.. +a +a3 + ... +a , is a sequence of .... II k -=I' I 1 " 'I11III partial sums of each sequence {a.}. l.If the sequence, {S.}. converges, then the series converges; otherwise, the series diverges. B. Ifthe partial sums ot;.a sequence {a.} converge to the number, then S= L ak =al +a2+a3 + ... =limS.; k=1 nt this sum is an infinite series. C. There is the sequence {a.} of terms of the series. D. There is also the sequence {S.} of partial sums. E. it.The geometric series f ak (r)k-I is convergent, IF and only IF, I" < 1. k=1 I. In this case, the geometric series sum is S= fa (rt-I =---1L- . k=1 k I-r 2. The geometric series partial sum is • • I a(l-r·)S = I. a (r) - =---, when r1= I. • k=1 k l-r F. If !ak and fbk are convergent series,k=1 k=1 and if C and d are any real numbers, then f (cak+ db. )=c fa k+d f bk and it converges. k=1 k=1 k=1 G.lf f ak is a convergent series, then Iim(aJ=O; if k=1 . t - II. does not approach zero, then it diverges. H. it. An infinite series of non-negative terms converges, IF and only IF, its sequence of partial sums is bounded. l. Comparison test: If°:5 a.:5 b. and the series !bk _ hI converges, then the series Lak also converges; if _ _ k=1 _ Lak diverges, then Lbk diverges; the series Lbkk=1 .=1 k=1 dominates the series f IIk • k=1 J. it. Limit comparison test: Given the series of _ a~ positive terms, La. and Lbk, if lim bk =c when _ k=1 k=1 kt- k ~ c > 0, then ~ak converges, IF and only IF, Lbk k - I a - k=1 converges. If lim..,!- = 0 and if Lbk converges, _ .t_ Ok hI then I.ak converges. ~ K. If the series L~kl converges, then the series La. k=1 k=1 converges absolutely. L. If the series fak converges but f ~kl diverges, k=1 . =1 then the series f a. is conditionally convergent. k=1 M. it. Integral test: An infinite series, with terms that are the values at the positive integers of a decreasing function, f that does not take negative values in the interval [I, co) converges, IF and only lE, the improper integral r f(xflx converges. N.p-Series - 1 I. LkP converges whenp > I and diverges whenp :5 I. k=1 2. f -k1 is the harmonic series, from the p-series, k=1 where p = 1; always diverges. O. Ratio test: If lim ~~+III = L, then: kt- I"k ~ I. When L < I, the series La. converges absolutely. k=1 2. When L > I, theseries fak diverges. k=1 3. When L = I, more information is needed to determine whether or not the series fa. converges. k=1 - II P. Root test:lftheupperlimitlim~kl/k:lim~=L, then: kt_ k .... ~ I. When L < I, the series fak converges absolutely. k=1 2. When L > 1 or when L is the symbol co, the series fak diverges. k=1 3. WhenL = I, more information is needed to detennine whether or not the series !ak converges. Q. Power Series ~ k=! I. Form in (x-a) is Lck(x-a) where the terms ofthe k=O sequence {e.} are called the coefticients ofthe series. 2. Converges only for the choice x = a. 3. Converges absolutely for every number x. 4. There is a positive number r such that the series converges absolutely for every number x in the open interval (a - r, a + r) and diverges for every numberx outside the closed interval [a - r, a + rj; r is the radius of convergence. 5. fCk(x-4+ fdk(x-a)'= f(ck+dk)(x-atk=O k=O k=O 6. Ctck(x-4l~0dk(x-4)= k~OC~.c,dk _ , )(x-4 - k7. lf f(x)=k'~{k(x-a) has a radius of ~ convergence, r, then it is differentiable on (a - r, a + r) and /'(x) = fkck(x-at l. _ k-I k=1 8. /'(x) = L kCk(x-a) is a series that has a k=1 radius of convergence, r, but may ~ at an endpoint where (a - r, a + r) converged. 9.!'(x)=I,kck(x- at-1 is integrable on (a - r, k=1 a + r); its integral vanishing at a is f:f(t)dt=k~O kC:I (x-atl+C when ~-al <r. 10. Taylor Series - .r<k}(a) k a. L -,-(x-a) =f(a)+/'(a)(x-a)+ k=O k. f"(a) 2 --z!(x-a) +... b. It is centered at a. c. It converges at a or some interval around a. d. If r > ° and f(x)= kf =Ock(x-4 where ~--al < r, then the coefficients are the Taylor .r<kl(a) coefficients ck =~. e. MacLaurin series is the Taylor series with a = 0, .r<k}(O) k centered at 0; k~O~(x) . i. Basic MacLaurin series (a}-1=1+x+x2+ ... = f x· when Ixi < 1.1 -x 11=0 2 3 (b)ln(l+x)=x-X; +~ - ... = ( 1)·+1 •f - x when-I < x:5l. 11=1 n (C)ln!~~=2(X+~ +x; + ... ) - 211+1 = 2 L _x__ when Ix! < 1. .=o2n+1 xl ~ (d) arctanx=x-T+ ~ - ... ( I)· X 2.+1 = f - when Ix! < 1. . =0 2n+1 _ x 2 xl - x" (e) r -I +x+2f+3f+'''= .~o -;;r for all real x. ( ). _.1 x2 X4 - -1 x-. (f) cosx=l-x+Tr. +4'. - ... =~ ( ), for all real x. . - 0 2n. • Xl x~ - (-I)" x 2n" (g) slDx=x-3T+ST-"'= L ( ) for all real x. . . .=0 2n+ 1 ! f . . Binomi~ series p(p-I) 2 (p)x- I. (l+x) =1+px+-2-,-x + ... =! forp 1= °and Ix! < 1. . . 0 n ii. The binomial coem:ients are (:) = I , ( p)= , (p)= p(p-) and"p choose k" I P 2 2' is (p)= P(P-I)(p-2) ... (p-k+J) . iii.lfp i!a positive integer ~~ ( : ) = °for k >p. .... 11 ri lthl.ll rtuned. No pan ofthiJ pubt;u. lion may be reprodueed or [ r :uaJIII 'I~ In Iny form, <If b} an) rn-. elrctrQllic or mech"niClI, incloom& p/lvl.JoOOfI>, m:onh.. or In)' infor:nltion .storqt and Ktricnl 'YJlcm . ..... ithoul ·.orI( \CO pcnni~ion (rom the: puhliJher. C2eM fbtClluts.l .e. 0.409 NOTE 10 STttOEl\.: Thi,pnOc IS inlmdtd for,~-'1JIW11O'G1MIy OuclOilI6 condcmt;.Jfurm;l1. this autikroNlOtL'O'W1,.1,. ltU6,' fr~~~d'r~2ko~Pnt~s at s::!':r;:,:;~::,,"m-' qUICKStUay.com ~~.c;;::':;,,"':;~,:;,~,:; ~ U.S.$S.95 Author: Dr. S. B. KLzlik ~~ :/:Ie r:ro:..: Customer Hotline # 1.800.230.9522 cr~lttlincd'nthi'illidc. ISBN-13: 978-142320856-3 ISBN-10: 142320856-0 9 ~11ll~~IIII~~lllJlllJ IlfIIIlllIlil~I 6
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