Week 1 - Oats and Sugar - Law, Economics and Development



Formula Sheet for Mid-semester ExamWeek 1Chapter 1.2 – Graphs and LinesGeneral form of a line:y=mx+b “m” is the gradient of the line (m=riserun)Where “b” is the y intercept-bm is x interceptPoint Gradient Formulay-y1=m(x-x1)(x1,y1) is a point on the linem is the gradient of the line (m=riserun)Chapter 2.1 – FunctionsVertical transformationWhere fx=fx-kThe function is shifted downwards by “k” unitsHorizontal transformationWhere fx=f(x-k)The function is shifted to the right by “k” unitsStretching/squeezing the functionWhere fx=2f(x)The function rises/falls double as quicklyAs such, where fx=f(x)2, the function rises/falls half as quicklyReflectionWhere , the graph is reflected about the x axisWhere x and y are inverted (i.e. ) the graph is perpendicular to how the graph will have beenShapes of graphs:fx=mx+b line parabola hyperbola “v” shaped graphChapter 2.3 – Quadratic FunctionsThe Quadratic Function General form Intercept form vertex formTo find the x-intercepts of a quadratic function:Either: factorize to intercept form, then equate each bracket to zeroOr: use the quadratic formula:To find the vertex form of a quadratic function:Either: use the quadratic formula to find the x-intercepts, and use the midpoint between the two intercepts:Or: find where Or: complete the squaree.g. take “", and add/subtract it to the formulaFind numbers that multiplies to give you “” and adds to give you “b”: these are the roots of the equation.Multiplies for 16, sums to -8 = -4Analyze the transformations:Here, we have a parabola, inverted, shifted upwards by 7 units and across 4 units to the right.Week 2Chapter 2.4 – Polynomials and rational functionDegree of a polynomialIs the highest “power” in the polynomial chain:e.g. , degree of the polynomial is “20”Y-interceptsWhere x=0X-InterceptsWhere y=0Note, where the polynomial has been factorized, equating each bracket to zero will give the x-intercepts (/roots)e.g.Roots are: (,0)(6,0)(1,0)Finding asymptotesWhere Vertical Asymptotes:After cancelling common factors, where , there is a vertical asymptoteHorizontal AsymptotesIf the degree of the degree of , is the horizontal asymptoteIf the degree of the degree of , is the horizontal asymptote: is the leading coefficient of is the leading coefficient of If degree of degree of , there is no horizontal asymptoteChapter 2.5 – Exponential functionsNote: graphs shift as per regular functions, see REF _Ref165371323 \h Chapter 2.1 – Functions.A graph has an exponential shape where . Properties of All graphs go through for any base bThe graph is continuousTheaxis is a horizontal asymptoteWhere , increases as increasesWhere decreases as increasesExponent LawsWhere a and b are positive, , x & y are real.Further:iff where iffCommon bases10 & Note, growth/decay formulae are often in the form , c & k are constants, t is time.Chapter 2.6 – Logarithmic Functionsiff that is: Log propertiesWhere b, M and N are positive, and p & x are real numbersiff Changing base of log etc.Note, calculator has andChapter 3.1 – simple interest, where I is interest, P is principle, r is the annual simple interest rate, and t is the time in years.Note: do not forget to add the principle again when working out future value, since this formula only works out interestChapter 3.1 – compound interestThe compound interest formula: where A is the future value at the end of n periods, P is the principle, r is the annual the annual nominal rate of interest, m is the amount of compounding periods per year, i is the interest rate per compounding period, n is the total number of compounding periods.Note: make sure “i” and “n” are in the same units of time.Continuous compound interest, r is the annual compounding rate, t is time in puting growth timeSince ,.Annual percentage yield, or, if compounded continuously, Chapter 3.4 – AnnuitiesStrategy (make into flowchart for final notes?):Make a timeline of paymentsIf single payment: either simple or compound interestIf multiple:Payments into an account increasing in value (FV)Payments being made out of an account decreasing in value (PV)All amortization is PV.Future value of an ordinary annuity, where FV is the future value, PMT is the periodic payment, i is the rate per period, n is the number of periods/payments.Present value of an ordinary annuityWeek 3Chapter 10.4 – the derivativeSlope of a secant between two pointsAverage rate of change (slope of a secant between x and x+h)The derivative from first principlesnote: it is most probable that the h on the denominator will goThis will fail if the line is non-differentiable at a point, e.gg where the graph:is not continuoushas a sharp cornerhas a vertical tangentChapter 10.5 – basic differentiation propertiesConstantJust an xA power of xA constant*a functionSum/differenceChapter 11.2 – derivatives of logarithmic and exponential functionsBase e exponential Base e exponential with constant in powerOther exponentialNatural logOther logChapter 11.3 – product/quotient ruleThe product ruleThe quotient ruleChapter 11.4 – the chain rule easy to do via substitutionthe above formula means: derivative of whole thing times derivative of bracketThe general derivative rulesWeek 4Note, in this chapter, sign charts make stuff easierChapter 12.1 – first derivatives and graphsThe first derivative gives the slope of a graph at a point: a positive first derivative will give an upward slope, and vice versa. A “0” or undefined first derivative gives a partition value. It is also a critical value when it appears in both the domain of f’(x) and f(x), e.g. an asymptote is a partition value but not a critical value.Local extremaWhere the first derivative is 0, and the sign of the first derivative changes around it, it is a local extrema:– 0 + minimum+ 0 - maximum– 0 – or + 0 + not a local extremaNote, where , finding can also identify whether it is a local extrema: where , it is a local minimum; where , it is a local maximum. This test is invalid where .Chapter 12.2 – second derivatives and graphsThe second derivative describes the concavity of a graph (where , the concavity is positive , and (/the slope) is increasing; where , the concavity is negative and (/the slope) is decreasing.Point of inflexionA point of inflexion is where the concavity of the graph changes (and, as such, the sign of the derivative, too, will change. This occurs where (or, if it is a vertical point of inflexion, undefined) the line is continuous and the sign of the second derivative changes about that point.Graph sketchingAnalyze , find domain and interceptsAnalyze , find partition numbers and critical values and construct a sign chart (to find increasing/decreasing segments and local extrema)Analyze , find partition numbers and construct a sign chart (to find concave up and down segments and to find inflexion points)Sketch : locate intercepts, maxima and minima and inflexion points: if still in doubt, sub points into Chapter 12.6 – optimization Introduce variables, look for relationships among variables, and construct a mathematical model of the form Maximize/minimize on the interval I.Find critical values of .Find absolute maxima/minima: this will occur at a critical value or at an endpoint of an intervalCheck that the function is continuous over an intervalEvaluate at the endpoints of the intervalFind the critical values of The absolute maximum is the largest value found in step “b” or “c”.Chapter 4.1 – Systems of linear equations in two variablesSimultaneous equations of two lines: isolate a variable and substitute.Week 5 Chapter 13.1 – antiderivatives and indefinite integralsAntiderivative is symbolized by , and may be accompanied by any constant.Indefinite integral is a family of antiderivativesIndefinite integrals of basic functionsx to the power of ne to the power of xx as a denominatorIndefinite integrals of a constant multiplied by a function, or, two functionsChapter 13.2 – integration by substitutionBased on the chain rule: (derivative of outside function multiplied by the derivative of the inside function)Thus General indefinite integral formulaeIntegration by substitutionSometimes it is hard to recognize the form of the function to be integrated (that is: to see which of the above formulae apply to it). So, we substitute the messy part for “u” and integrate with respect to “u”, rather than x.Where General indefinite integral formulaeMethod of integration by substitutionSelect a substitution to simplify the integrand: one such that u and du (the derivative of u) are presentExpress the integrand in terms of u and du, completely eliminating x and dxEvaluate the new integralRe-substitute from u to x.Note, if this is incomplete (i.e. du is not present) you may multiply by the constant factor and divide, outside of the integral, by its inverse: e.g. Where integral of u is 1, du=dx. (and find x as “u-k”)Chapter 13.4 – the definite integralis the definite integral of from x=a to x=b. Worked out by subtracting where x=a from where x=b. Note, these are not absolute values: above x axis is positive, below is negative: opposite if other direction.Properties of a definite integral, where k is a constantError BoundsWhere f(x) is above the x-axis: The fundamental theorem of calculusAverage value of a continuous function over a periodWeek 6Chapter 15.1 – functions of several variablesSubstitute (x,y,z,…,et.) into the equation given.Find the shape of the graph by looking at cross sections (e.g. y=0, y=1, x=0, x=1). Chapter 15.2 – partial derivativesDerivatives with respect to a certain symbol: watch for signs, all other symbols count as constants derived with respect to x ||| derived first with respect to x, then yChapter 15.3 – maxima and minimaExpress the function as Find , and simultaneously equate them to find critical valuesFind (A, B, and C, respectively)Find A, and .IF AC-B*B>0 & A<0, f(a,b,) is local maximumIF AC-B*B>0 & A>0, f(a,b) is local minimumIF AC-B*B<0, f(a,b) is a saddle pointIF AC-B*B=0, test failsChapter 15.4 – maxima and minima using Lagrange multipliersWrite problem in formForm the function Derive with respect to x, y and lambdaSimultaneously equate answersIf more than 1 answer, find z values and deduce which is max/min. ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download