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 Identify all relevant variables, concepts and constants Describe the problem situations using appropriate mathematical relationships, functions, formulas, equations or graphs Collect all essential information and data S ~ extra and unnecessarY information and data Derive a mathematical expression or statement for the problem, making sure all measurements are in' the correct unit Complete the appropriate mathematical manipulations and solution techniques 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! Lh FUNCTIONS I Definitions A A relation is a set of order 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 of y = x I Dependent variable is the output variable in an equation and depends on or is determined by the input variable 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 -x, x, -x2 ~ run 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 If a> 0, the graph opens up If a < 0, the graph opens down 4.±a are the slopes of the two sides of the graph C Square root: f(xl=a.Jx-h +k I (h, k) is the endpoint If a> 0, the graph goes to the right If a < 0, the graph goes to the left D Polynomial: f(x) = a"x' + a~,x~' + + a,x + ao ao is the y-intercept There are, at most, n zeros or x-intercepts where f2 p f(x) = o There are, at most, n - points of change or turns in the graph 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 (h, k) is the vertex If a> 0, the graph opens up If a < 0, the graph opens down S Use quadratic formula to solve for the zeros or x-intercepts: f f(x) = log, x = In x; this is the natural log r -~ g.log i.log.y=log.x-log.y j log.xY = ylog.x I Trigonometric Basics a j) Angles can be measured in degrees and radians 2a x-axis wherey = O If degree p(x) = degree q(x), the asymptote is y = (lead coefficient ofp(x»/(lead coefficient of q(x» 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.] If a > 1, the function is increasing If a < 1, the function is decreasing Rules for exponents: a x'" • x" = x"'" h XIII =x",-n ii I degree = (] degrees ;0) radians tanS=~ ~ ~ ~ III Z ~ (d) cscS=.! y (e) secS=.! x (f) cotS=,,! y vi j) use ful values r =x",n = degrees; t = radians', {) = unde fined d - ", 1_ - = (] !O) (c) xn X i I radian iii.proportion conversion of angle measurements: angle i n degrees angle in radians 180 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 P(I,O) (f) j ) Angles rotating one or more full times require adding ±21t for each rotation v function definitions (a)sinS=y (b) cos S =x qCx) l.p(x) and q(x) are polynomials, and q(x) f o If degree p(x) < degree q(x), the asymptote is the c (x'" log x logx Inx 10gb a = loga =III/l; this is the change-of-base rule h log.xy = log.x + log.y X= -b+~ () F Rational: f(x) = p x X= x'" 30 4S 60 90 180 t n .n .n .n It g'(;T=~: sin l J2 J3 2 h.x~ =~ =(!!/Xr cos I J3 J2 l 2 -] j ~=~ tan J3 J3 {) k.~ ="'!!/X 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 e _1_= x"' x - ", f (xy)"' =x"'y"' i ~=!!/X.~ I Ifx' =x', then a = b m Ifa x = b x , then a = b if a f O H Logarithmic:f(x) = log.x I.x>O a> and a f Lhf(x) = log.x, IF and only IF a ftx )= x a is the base S Logarithms are exponents Rules for logarithms: a log.I = b.log.a = I c.log.ax=x d a'og.x = x e Iflog.x = log.y, then x =y full cycle of the curve, or 2: c Horizontal shift or phase shift is c d Vertical shift or average value is d e Sine:f(x) = aslnb(x- c) + d f Cosine:f(x) = acosb(x - c) + d g Tangent:f(x) = atanb(x - c) + d [CAUTION! Tangent has no amplitude so a affects the vertical stretch and shrink only.] h Cosine is even; sine & tangent are odd Important identities & formulas Lh ~ " W m Z ~ Functions {Sontinued) a Pythagorean identities i sin' u + cos' U = 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 'f tan u tanv c Half-angle formulas sini=±p-~osu ii cosi=±~I+~OSU iii tan.!! = I-cosu =~ sinu l+cosu d Double-angle formulas i sin2u = 2sinu cosu ii cos2u = cos' u - sin' u = 1- 2sin' u = 2cos' u - iii.tan2u= 2tanu I-tan' u e Power-reducing formulas sin' u l-cos2u ii cos' U= iii tan' U= H 0/' ~ Forexample, when finding lim x -28 such thaI x-+l x­ K.Cosine:f(x) = cosx y I I I r =+ - - I I " -I If j I I j x x"# x - becomes (x - 2)(x + 2x + 4) and then , x-2 x-2' 2" (x' + 2x + 4); consequently, when x is close to 2, "'~ -2 I l I I M.General Transformations 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 vertical stretch if a> 1; vertical shrink if 0< a < 1; x-axis reflection if a is negative j(ax) has horizontal shrink if a> 1; horizontal stretch ifO < a < 1; y-axis reflection if a is negative LIMITS & CONTINUITY n > m, the lim j(x) = lim P «x» X ,)ooo X ) ~ ° B Linear identity:j(x) =x G.Rational: j(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 j ( ) p(x) x = Q(x) and Q(x) "# 0, then when: x- (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 > l-cos2u +cos2u Ill Basic Common Function Graphs A.Cons* 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 ° D.Square root: j(xb-.JX I Logarithmic: j(x)=lnx = 12 B lim c = c, when c is a constant ~A l+cos2u xl-: (xl + 2x + 4) is close to 12; therefore, lim III Rules x )2 x- y 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 X_CHI lim[j(x) g(x)]=(limj(x»)(limg(x») X )(I X-HI E Quadratic function: f(x) =x' j(x) lim j(x) D lim =~ provided g(x "# 0) and X > g(x) limg(x) ' X > limg(x) "" J Sine: j(x) = sinx y f 2+ +-+-f-IH 1'\ ­ I t-: r "+-t -+:v.:,,,x 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 -\ C The first derivative function notations include dy, DxY, and !L j(x); second dx fix • d1 y derivative notations mclude j (x), y, dX' f'(x), y', and D;y D The derivative atx = a is usually written as: /'(11), D(f)(II), or 1;L II RuleslFormulas A Assume fix) and g(x) are differentiable functions, :x D,f(x) = lex) = f'(x), and D.g(x) = ! g(x) = g'(x) for the following statements: 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) x +« X +d' t- 0, then lim lex) = lim f'(x) x~ x~ g(x) g'(x) 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) 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 Mean Value Theorem: If/is continuous in the closed interval [a, b], then there is a point m in the open mterval (a, b) such that /(b)- lea) () b-a f' m Lmg(x)=mg'(x) for all real numbers m dx L{J(xhg(x») = f'(xhg'(x) dx 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 Lc=O, when c is a constant dx 10.Lx=1 dx II L(l)=_ L dx 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 dx x - is defined 14 L dx £ = I, 2vx 15 L JI (x) I ; derivative 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=l dx t- 0, x 19.A log x=_I_ dx • xlna 20 L(sinx) = cosx dx 21 L(cosx)=-sinx dx 22 L (tanx)=sec l x dx 23 L (cotx)=-csc l x dx 24 L(secx) = secx' tanx dx 25 L(cscx)= -cscX'cotx dx , !-., 27.L(arccosx)=- , !-., dx vI-xl 26 L(arcsinx) = 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) dx x.Jxl-l of an III Applications A.lmplicit Differentiation Used when it is difficult or undesirable to solve an equation for y, such as x' + y' = Differentiate both sides of the equation with respect to x Apply the Chain Rule Substitute y' for : and for ::; Solve for y ' ~ For example, when finding the derivative of y= g(x) Jx~\ with .Jx+l' f(;)[;(X] dx g(x) 1) and becomes L[ (x-I)]= Jx+IDx (x-l)-(x-I) Dx Jx+I ; dx Jx+I [.Jx+lf then, using the powerformula to find D x Jx + I , the statement becomes f A critical point, (x,f(x», is a point ofthe graph of/ that satisfies one of these conditions: i f'(x) = ii.f'(x) does not exist; OR iii.(x,f{x» is an endpoint of the graph iv.~Y For example, on the graph of y = 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 s y Jx+l (1)- (x-I) 2.Jx+l L x+1 finally, using algebra to simplify the expression, the derivative becomes B Graphs p ( x+3 or (x+3),Jx+t 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 c If/ is continuous and/'(x) > at every point of an open interval (a, b), then/ is increasing in this interval d If/ is continuous andf' (x) < 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 Concavity a A curve or part of a curve is concave up if the curve lies above the lines that are 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 If j" (x) > at every point in an interval, then the graph off{x) is concave up in this interval d.lfj"(x) < at every point in an interval, then the graph of/ex) is concave down in this interval 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 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 I - / " R \ ./ Q f" / - I- -­ \ \ T N x g I 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 [CAUTION! This test does not apply if f"(a) = 0.] Helpful Hints for Sketching a Curve C I Determine the domain for the function,f{x) Analyze all points where lex) is not continuous Sketch all vertical, horizontal and oblique asymptotes, if there are any Evaluater(x) andj"(x) Find and plot all critical points, a, where f'(a) does not exist or wherer(a) = O Find and plot all relative maximum and all relative minimum points Find and plot all possible inflection points, b, wherej"(b) does not exist or wherej"(b) = O Find and plot the x-intercepts and the y-intercepts, if there are any Complete the sketch of the curve D Rate of Change I Average rate of change of/ over the interval [a,x]: P a I s /(x)- lea) 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 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, " = and I is time f(1), 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 a equal width of 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), f The sum of these trapezoid areas is b-;,a [f(xo); f(x\) + f(x l ) ; f(x ) + + b;; f(xn-I~+ f(xo)] = b This results in n + points on the x-axis b-a c These pomts are Xo = a, XI = a + ;;- , x =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_.)] = n-I b ;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 :s i:S n g The sum of these rectangle areas is b-a n L f(xj ) o [f(x.) + f(X2) +, + f(x.)] = b~a 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 + ) for O:S i:S (n -I) iii The sum of these rectangle areas is [f( b:a f ( X 0-1 Xo ;XI )+ f( xI :x )+, + 0-1 +x)] b +x ) =-=.!! L f (x -1 l: n j~O +x.1+_1) where !'J.x = b ;a or!'J.x0L- 1f (x - ' j~O I JL , , 1/ , y IL j, , , , ,,, , , , I ~ , , ,, ,, , , , ,, ,, , , a l x l x2 l x, I I \ :'\ , , ,,, X4 [0 x =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 < i I, the Iimacon graph has an inner loop 3.j) ~=I, the limacon graph has When NO inner loop, is heart-shaped, and is specifically called a cardioid a For example, r = 2(1 - cos8) IV Area A The area bounded by the curve r = f(8) and enclosed by the rays = a and = ~ may be found using A=.!f~r2d8=.!f~ f(S)2 dS (l (l B The area bounded by two polar graphs is fiP A=.!f~(r.2 -r.I2)d8 2 (l V Arc Length A The length of arc r = f(8) where is in the interval ~j is L = f~ [a, +(~~r dS r2 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 (x(S),y(8» dy sinS dr +rcos8 of dy =AJi.= dS dx dx cosS dr -rsinS dS d8 SEQUENCES & SERIES t.iIIII 'I11III Z W ft ~ O oiIIII 'I11III I Sequences are functions that have domains that are integers and ranges that are all real numbers A The integer in the nth position is called a term and denoted by the symbol a rather than a(n} B Consecutive arithmetic sequence terms have a common difference, d, with each term the result of a.=a l+d=al+d(n-I} C Consecutive geometric sequence terms have a common ratio, r, with each term the result of a = a ir} = al(rr l 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: ~r-(a +b.)=~it,!!a +~t,!!b Iim(a.b.) = lima limb B Ifthe partial sums ot;.a sequence {a.} converge to the number, then f- f­ a lima • t_ hen I'1m b · IImb=~'w "1_ ,, 1I11'~ limca =clima ,.roo " "roo it ~ Z W ft ~ o ° ~ ,wherecmaybeanynumber II 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 Monotone sequences converge IF and only IE they are bounded The limit of an increasing sequence is its least upper bound The limit of a decreasing sequence is its greatest lower bound II.A series is a sequence obtained by adding the terms of another sequence A A sequence, {S.} whose terms are defined by it S 'I11III II = fa =0 k -=I ' I +a +a3 + +a , k=O C lf f(x)=k'~{k(x-a) (a - r, a + r) and /'(x) = fkck(x-at l _ k-I k=1 /'(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 t- 9.!'(x)=I,kck(x- a is integrable on k=1 (a - r, a + r); its integral vanishing at a is f:f(t)dt=k~O kC:I (x-atl+C when ~-al ~ al ~ K If the series L~kl converges, then the series k=1 converges absolutely f it centered at 0; La is a sequence of " partial sums of each sequence {a.} l.If the sequence, {S.} converges, then the series converges; otherwise, the series diverges =o2n+1 xl ~ (d) arctanx=x-T+ ~ - X + f (- I)· 2n+1 = =0 r • Xl I - (-I)" x 2n " ( ) 2n+ ! (p)x- p(p-I) (l+x) =1+px+ ,-x + =! forp 1= and Ix! < n ° ii The binomial coem:ients are (:) = I , , (p)= p(p-) ( p)= I P 2' is and"p choose k" (p)= P(P-I)(p-2) (p-k+J) ~~ ( : ) = °for k > p iii.lfp i! a positive integer converges absolutely k=1 When L > or when L is the symbol co, the series diverges WhenL = I, more information is needed to detennine !a x~ f Binomi~ series II k=! Ix! < (g) slDx=x-3T+ST-"'= L for all real x .=0 P Root test:lftheupperlimitlim~kl/ k:lim~=L, then: kt_ k ~ whether or not the series k converges ~ Q Power Series I Form in (x-a) is Lck(x-a) where the terms ofthe k=O sequence {e.} are called the coefticients ofthe series Converges only for the choice x = a Converges absolutely for every number x 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 number x outside the closed interval [a - r, a + rj; r is the radius of convergence when ­ _ x xl - x" -I +x+2f+3f+'''= ~o -;;r for all real x ( ) _.1 x2 X4 -1 x- (f) cosx=l-x+Tr +4' - =~ ( ), - 2n for all real x (e) determine whether or not the series fa converges k=1 k=1 x· when Ixi < 11=0 (C)ln!~~=2(X+~ +x; + ) 211+1 = L _x when Ix! < O Ratio test: If lim ~~+III = L, then: kt- I"k ~ I When L < I, the series La converges absolutely k=1 When L > I, the series fak diverges k=1 When L = I, more information is needed to fak f - f fak r I and diverges whenp :5 I LkP - f c (x-4 k=Ok i Basic MacLaurin series =1+x+x2+ = f (a}-1 r I f(x)= e MacLaurin series is the Taylor series with a = 0, f a and r