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 of order pairs; written (x,y) or (x, 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. 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 = = rise x 2 -x, x, -x 2 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) = - 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.±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. 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 + 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 of change 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= . () 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 of q(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"' =(!!/Xr 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 2. a> 0 and a f. 1. 3. Lhf(x) = log.x, IF and only IF. a ftx )= 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= 10g b a = loga =III/l; 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 I II (a)sinS=y Z (b) cos S =x (c) (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 -] .J3 tan 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 Z Lh [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
%&4For every student who has ever found the answer to a particular calculus equation elusive or a certain theorem impossible to remember, QuickStudy comes to the rescue! This 3-panel (6-page) comprehensive guide offers clear and concise examples, detailed explanations and colorful graphs―all guaranteed to make calculus a breeze! Easy-to-use icons help students go right to the equations and problems they need to learn, and call out helpful tips to use and common pitfalls to avoid.
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EssentialToolsfor UnderstandingCalculus- Rules,
Concepts,Variables, Equations,Examples, j)Helpful
Hints&LhCommon PitfallsSTRATEGY FORSOLVING PROBLEMS EFFECTIVELY
I.Understand the principle (business or scientific)required.II. j)
Developa mathematicalstrategy.A.There are eight useful steps that
will help youdevelopthecorrectstrategy.I.Sketch, diagram orchartthe
relationships andinformationthatissubjectof
theproblem.2.Identifyall relevant variables, concepts
andconstants.3.Describethe problem situations
usingappropriatemathematicalrelationships,functions,formulas,equationsorgraphs.4.Collectall
essentialinformationanddata.S.Lh
extraandunnecessarYinformationanddata.6.Derive a mathematical
expression or statementfor the problem, making sure all
measurementsarein'thecorrectunit.7.Complete the appropriate
mathematicalmanipulationsandsolutiontechniques.8.Check the final
answer by using the originalproblemand
informationtomakecertainthattheanswers, units, signs, magnitudes,
etc., all makesenseandarecorrect!FUNCTIONS I.
DefinitionsA.Arelationisasetoforderpairs;written(x,y)or(x,
B.Afunctionisarelationthathasx-valuesthatarealldifferent
fordifferenty-values.Averticallinetestcan be used todeterminea
function; everyverticallineintersectsthegraph,atmost,once.C.A
one-to-one function is a function that has y-values that are all
different for differentx-values.A horizontal linetestcan be used
todetermine
aone-to-onefunction;everyhorizontallineintersectsthegraph,atmost,once.D.Domainisthesetof
allx-valuesof arelation.E. Rangeisthesetof ally-valuesof a
relation.F. Afunction is anevenfunction iff(- x)=fix).G.Afunction
is anoddfunctioniff(- x)=
-f(x).H.Theone-to-onefunctionsf(x)andg(x)areinversefunctionsiff(g(x
=g(f(x =x;f'(x)andg-'(x)indicate the inverse functions offix) and
g(x),respectively. Inverse functions are reflections
overthelinegraphofy = x .I.Dependent variable is the output
variable in anequation and depends on or is determined by
theinputvariable.1.Independent variable is the input variable in
anequation.II. CommonFunctionSummaryA.Linear:f(x)= mx+bI.m
istheslope;m=Y2 - y, = y,- Y2 = = risex2-x, x,-x2 run
2.bisthey-intercept.3.lt is a constant function when m = 0; it is
ahorizontalline.B.Absolutevalue:f(x)= - hi+kI.(h,
k)isthevertex.2.If a>0, thegraphopensup.3.If
a0,thegraphgoestotheright.3.p If a. g(x) limg(x)'X-->.
limg(x)""0.E. ;:{J(x)"=(limj(x)"),providedn isapositiveX----)Q
X-)Qinteger.F. Iimj(x)=Ais equivalentto lim[j(x)-A]=O.x----).
X----)QG.lfj(x) < g(x) < hex) for every x in a
puncturedneighborhood of a (that is, x near a),
andlimj(x)=IimhCxl=A,then limg(x)=A.X-HI x--+a
X.....,IIf2whenfindinglimx3-28suchthaI0/'x-+l x-x3, x-2 x-2'x"#2 -
8 becomes (x - 2)(x2+2x +4) andthen(x'+ 2x+ 4); consequently, when
x is close to 2,(xl+2x+4)iscloseto12;therefore, limxl-: =12.III.
Rulesx----)2 x-A.For polynomialp(x) to the n'h power withthe lead
term ofax" and polynomial Q(x) tothe m'h power with the lead term
ofbX"', ifp(x)x =Q(x) and Q(x)"# 0, then when:j()l.n=m,the
limj(xl=
