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Financial Maths Sem2
Regression & Correlation, Time Series, Financial Maths, Business Maths
| Question | Answer |
|---|---|
| What is Interpolation? | Estimation carried out within a range of values given for the independent variable. |
| If we use 79 - 98 as range values for "X", to calculate the Y on the X line, give an example of an Interpolated estimate | Any number between 79- 98, e.g. "X "= 80. |
| Interpolated estimates are usually regarded with... | ...a degree of confidence. (Interpolated Estimates are usually accurate due to them being in the data set). |
| What is Extrapolation? | Estimation that is based on the values of the independent variable that has not been considered in the calculation of the of the appropriate regression line. (Value for "X" outside the range) |
| If we use 62 - 84 as range values for "X", to calculate the Y on the X line, give an example of an Extrapolated estimate | Any number outside the range, e.g. "X"= 50 or 93. |
| What is Extrapolation usually used for? | Forecasting. |
| What is Forecasting? | Values of a variable described over time. |
| What is Forecasting also known as? | Time Series. |
| Why is Extrapolation undertaken with care? | One can never be certain that the regression line calculated from the data given will still be appropriate in regions of values not used in the calculation of the line. |
| If the x-value on which it is based lies within the range of the given data x-values, what is Y1? | Y1 is an Interpolated Estimate. |
| If the x-value on which it is based lies outside the range of given data x-values, what is Y2? | Y2 is an Extrapolated Estimate. |
| What is a Scatter Plot? | A Scatter Plot uses dots to represent values for two different numeric variables (x and y). |
| What is Scatter Plots also known as? | Scatter Chart or Scatter Graph. |
| What does the position of the dots on the horizontal and vertical axis indicate? | Indicates values for an individual data point. |
| What are Scatter Plots used for? | Scatter plots are used to observe relationships between variables. |
| What are Scatter Plots' primary uses? | Scatter plots’ primary uses are to observe and show relationships between two numeric variables (X and Y). |
| What do the dots on a Scatter Plot report? | The values of individual data points, but also patterns when the data are taken as a whole. |
| On a Scatter Plot what do we call data plotted on the graph? | Cartesian (X,Y) Coordinates. |
| How do we mark a point using Cartesian Coordinates on a graph? | Mark by how far along the X-axis it is and how far up the Y-axis it is. |
| Why do we draw a picture to represent the data on a Scatter Plot? | It makes it easier to interpret. |
| What is another name for "Line of Best Fit"? | Trend Line. |
| What is a Trend Line used for? | Used to represent the behaviour of a set of data to determine if there is a certain pattern. |
| What is a Trend Line used with a Scatter Plot mostly used for? | To see if there is a relationship between two variables. |
| What is the Purpose of a Trend Line? | 1) Determining if a set of points exhibits a positive trend, a negative trend, or no trend at all. 2) Predicting unknown or future data points. |
| How do you calculate a line using Least Squares Regression? | 𝐿𝑖𝑛𝑒 𝑜𝑓 𝐵𝑒𝑠𝑡 𝐹𝑖𝑡 ∶ 𝑦 = 𝑎 + 𝑏𝑥 𝑆𝑙𝑜𝑝𝑒: 𝑏 = (𝑛 ∑ 𝑥𝑦 −∑𝑥 ∑𝑦) ÷ (𝑛 ∑ 𝑥² − (∑𝑥)²) 𝑎 =((∑ 𝑦) ÷ 𝑛)− (𝑏 (∑𝑥) ÷ 𝑛) |
| What does Y= a + bx mean in relation to Least Squares Regression? | It's the formula for "Line of Best Fit". Where y = how far up; x = how far along; b = Slope; a = the Y Intercept |
| How does Least Squares Regression work? | It works by making the total of the square of the errors as small as possible (that is why it is called "least squares"). |
| What are the steps to calculate Least Squares Regression? | Step 1: For each (x, y) point calculate x²and xy Step 2: Sum all x, y, x² and xy, which gives us Σx, Σy, Σx2 and Σxy (Σ means "sum up“) Step 3: Calculate Slope b Step 4: Calculate Intercept a Step 5: Assemble the equation of a line: y = a + bx |
| What is the Formula for Correlation? | 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡: 𝑟 = (𝑛 ∑𝑥𝑦 −∑𝑥∑𝑦)÷ √𝑛 ∑ 𝑥² − (∑𝑥)² 𝑛∑𝑦²− (∑𝑦)² |
| What is the Coefficient of Determination more commonly known as? | R-squared (or R²). |
| What does the Coefficient of determination assess? | Assesses how strong the linear relationship is between two variables, and is heavily relied on by researchers when conducting trend analysis. |
| What is the Equation of a Line, in relation to Regression and Correlation? | Equation of the line, y = mx + b |
| What is the Dependent Variable and the Independent Variable in relation to the equation of a line? | Dependent Variable = Y Independent Variable = MX |
| What is a Time Series? | A time series is a name given to numerical data that is described over a uniform set of points. |
| What are some examples of Time Series occurring naturally? | 1. Annual turnover for a firm for a number of years. 2. Numbers unemployed in each quarter for any given year. 3. Total monthly sales for a company for any given time. 4. Daily takings in a supermarket for a number of days in a row. |
| What is a Time Series Cycle? | Normally a time series exhibits a pattern which broadly repeats, called a cycle. |
| What is a Time Series Model? | A time series model is a set of data points ordered in time, and it’s used in forecasting the future. |
| What are some reasons why companies use Time Series Model? | • Business records and in particular certain time series of sales and purchases need to be kept by law. • They are also used to help control current and future business activities. • In order to explain the movements of time series data. |
| What is Time Series Analysis? | The evaluation and extraction of the components of a model that break down a particular series into understandable and explainable portions. |
| What does Time Series Analysis Identify? | • Trends • Factors that need to be eliminated • Forecasts that need to be made. |
| What is the general formula for the Time Series Additive Model? | y = t + s + r Where • y = given time series value • t = the trend component • s = the seasonal component • r = the residual component |
| What is the general formula for the Time Series Multiplicative Model? | y = t*s*r Where • y = given time series value • t = the trend component • s = the seasonal component • r = the residual component |
| What does Trend mean in relation to Time Series? | Trend - The underlying long-term tendency of the data. |
| What does Seasonal Variation mean in relation to Time Series? | Seasonal Variation - Short term cyclical fluctuations in the data about the trend which take their name from the standard business quarters of the year. Seasons may be days, months, quarters or any defined time period. |
| What does Residual Variation mean in relation to Time Series? | Residual variation - These include other factors not explained by trend or seasonal variation. |
| What are the 2 components Residual Variation normally consist of? | 1. Random Factors- Disturbances due to everyday unpredictable influences e.g., weather, illness, etc. 2. Long term cyclical factor -This is due to underlying economic causes outside the scope of the immediate environment. E.g. minor recessions. |
| How do you graph a Time Series? | • The standard graph is a line plot. • Time series points on the y axis and time on the x axis. • Single points are joined by straight line segments. • Trends can then be added to our plots. |
| What are the methods used to extract Trends? | • Semi averages -Simplest technique, calculate 2 averages, plot onto a graph and join to form a straight line. • Least squares regression -Results in a straight line • Moving averages -Most commonly used method, calculates a set of averages. |
| What is a Moving Average? | The basic method used in measuring seasonal fluctuation, described later. Each average is calculated by moving from one set of overlapping values to the next. The number of values in each set is the same and is known as the period of the moving average. |
| What is a Moving Average useful for? | A moving average is useful in smoothing a time series to see its trend. |
| Period of the moving average must coincide with... | ...the cycle of the series. |
| Each moving average trend value must coincide with... | ...an appropriate time point – the median of the time points being averaged. |
| What is an Even Period, in relation to Moving Averages? | There is no center time period -the moving totals are positioned between two time periods. E.g. Sales from 2010, 2011, 2012, 2013 ÷ 4 = Average over 4 Years Sales from 2011, 2012, 2013, 2014 ÷ 4 = Average Next 4 Years Avg 1 + Avg 2 ÷ 2=Even Period |
| What is Weighted Moving Averages? | This involves selecting a different weight for each data value and then computing a weighted average of the most recent n values as the smoothed value. The most recent observation receives the most weight and the weight decreases for older data values. |
| What do we use as a "Forecast for the future" in a majority of applications of Weighted Moving Averages? | In the majority of applications, we use the smoothed value as a forecast of the future. |
| In Weighted Moving Averages, what must the sum of weights be? | The sum of the weights must be equal to 1. |
| Summarize Moving Averages | To summarize the technique of using moving averages, its purpose is to help identify the long-term trend in a time series (because it will smooth out short-term fluctuations). • It is used to reveal any cyclical and seasonal fluctuations. |
| What is Seasonal Variation? | Gives an average effect on the trend which is solely attributable to the ‘season’ itself. Expressed in terms of deviations from (additive model) or percentages of (multiplicative model) the trend. |
| What is the technique to calculate Seasonal Variation - Additive Model? | •Given the original time series y, together with the trend t, •Calculate for each time point, the value of y-t. •For each ‘season’ in turn find the average of the y-t values •If total of averages differs from 0 adjust one or more so that total is 0. |
| What is the formula for Geometric Mean? | 𝐺𝑀 = √𝑥2 ∗ 𝑥1 ∗ ⋯ 𝑥n |
| What is Geometric Mean? | The geometric mean of a set of n positive numbers is defined as the nth root of the product of n values. |
| What is Geometric Mean used for? | The geometric mean is useful in finding the average change of percentages, ratios, indexes, or growth rates over time. |
| What is De-Seasonalising the Data used for? | •To adjust the data for seasonal effects. |
| How do you De-Seasonalising the Data using Additive Model? | • Additive Model: Adjustment is performed by subtracting the seasonal figure from the time series, y-s |
| How do you De-Seasonalising the Data using Multiplicative Model? | • Multiplicative model: Adjustment is performed by dividing the original time series values by the seasonal figure from, y/s |
| How to identify an Additive or Multiplicative Time Series from its variation? | If the magnitude of the seasonal component changes with time, then the series is multiplicative. Otherwise, the series is additive. |
| Explain how to do Forecasting | •Step 1-Estimate a trend value for the time point. •Step 2-Identify the seasonal variation value appropriate to the time point. •Step 3-Add (or multiply) these two values together to give the required forecast. |
| What is the formula for Time Series Forecasting using Additive method? | Y estimate = t estimate + s • Y estimate = estimated data value • t estimate = projected trend value • s = appropriate seasonal variation value |
| What is the formula for Time Series Forecasting using Multiplicative method? | Y estimate = t estimate * S • Y estimate = estimated data value • t estimate = projected trend value • s = appropriate seasonal variation value |
| Explain Principle Amount, P or PV | • The amount of money that is initially being considered . • It may be an amount about to be invested or loaned or may be the initial cost of plant or machinery. |
| Explain Accrued Amount, A or FVn | • A principle amount after some time has elapsed for which the interest has been calculated and added. • It may be an amount owed or invested. |
| Explain Rate of interest, i or r | • Interest is a proportionate amount of money which is added to some principle amount (invested or borrowed). • It is expressed as a percentage rate per annum |
| Explain Number of time periods, n | • The number of time periods over which the principle is being invested or borrowed. • It may be years, quarters, months or even days. |
| Explain what Simple and Compound Interest is used for | When an amount of money (Principle, P) is invested over a number of years the interest earned can be dealt with in two ways, |
| Explain Simple Interest | Any interest earned is NOT added back to the Principle invested. |
| Explain Compound Interest | Any interest earned IS added back to the Principle invested. |
| What is the Simple Interest General formula? | FVn = PV( 1 + r * n) Where: 𝐹𝑉n = the cumulative amount, the future value at time n𝑃𝑉 = the principle invested, the present value of the investment/loan 𝑟 = the interest rate 𝑛 = the number of time periods |
| What is the Simple Interest General formula as PV? | PV = FVn ÷ ( 1 + r * n) |
| What is the Simple Interest General formula as n? | n = ((FVn ÷ PV)-1) ÷ r |
| What is the Simple Interest General formula as r? | r = ((FVn ÷ PV)-1) ÷ n |
| What is the Calculating APR formula? | 𝐴𝑃𝑅 = [1 +(𝑖 ÷ n)]nth power - 1 -Nth power is like Squared or Cubed but with n |
| Explain Nominal Interest Rates | Expressing interest rates per annum even though the interest may be compounded over time periods that are less than a year (semi-annually, monthly, weekly, daily etc.). • This is what we call nominal rates of interest. |
| Explain Effective Interest Rates | The actual rate of interest is called the effective rate or the actual percentage rate (APR). • The APR will always be greater that the nominal rate. |
| What is Effective Interest Rates also known as? | Actual percentage rate (APR). |
| What is the Compound Interest General Formula? | 𝐹𝑉 = 𝑃𝑉(1 + (𝑟 ÷ t) to the power of nt 𝐹𝑉n= the cumulative amount = 300,000 𝑃𝑉 = the principle invested 𝑟= the interest rate = 0.1 𝑛= the number of time periods = 2(in 2 years time) 𝑡= the number of compounding periods = 2(semi-annual) |
| What is Depreciation? | An allowance made in estimates and on balance sheets (normally for wear and tear). It's normal accounting practice to depreciate the values of certain assets Depreciation is like the opposite of interest rates –values get smaller rather than bigger. |
| Explain 2 Techniques used for Depreciation | 1. Straight line depreciation 2. Reducing balance depreciation |
| Identify the differences between Straight Line Depreciation and Reducing Balance Depreciation | Straight line depreciation is like simple interest and reducing balance depreciation is more like compound interest. |
| What is the Straight Line Depreciation General Formula? | • To increase PV by a percentage we calculate 𝑃𝑉(1 + 𝑖) • If instead we wanted to decrease a value, PV we would calculate 𝑃𝑉(1 − 𝑖) Thus the general formula for straight line depreciation is: 𝐹𝑉n = PV(1 − 𝑟 ∗ 𝑛) |
| What is the formula we use to increase PV by a percentage, in relation to Reducing Balance depreciation? | • To increase PV by a percentage we calculate 𝑃𝑉(1 + 𝑖) • If we do this for n time periods it grows to, 𝑃𝑉(1 + 𝑖)to the power of n |
| What is the formula we use to decrease a value, in relation to Reducing Balance depreciation? | To decrease a value, B we would calculate 𝐵(1 − 𝑖) • If we do this for n time periods it shrinks to, 𝐵(1 − 𝑖)to the power of n |
| What is the Reducing Balance Depreciation General Formula? | D= 𝐵(1 − 𝑖)to the power of n 𝐷= 𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑒𝑛𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑛th time period B= Original Book Value 𝑖= 𝑑𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒 𝑛= 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 (e.g. 𝑦𝑒𝑎𝑟𝑠) |
| Define Present Values | An important relationship in financial management is that which exists between the value of money and time. |
| The value of any given amount of money will _____ over time | Depreciate |
| Explain how to calculate the present value of a cash flow that’s due to be paid out or received at some point in time in the future. | This is done by discounting – By applying a discount factor to the future value of the cash flow in question. |
| What is the Present Value formula? | PV= FVn ÷ ((1 + r) to the power of n) 𝐹𝑉n= Future value 𝑃𝑉 = Present value 𝑟 = annual discount rate 𝑛 = number of years |
| What does the Discount Technique allow Investors to do? | It allows the investor to compare the returns on any given investment with that which may be available from the alternatives. |
| What is the Discount Technique referred to, in relation to investing? | In these scenarios the discount rate is interpreted as the rate of return on the next best alternative investment and hence the minimum required rate of return on the investment being reviewed. |
| How do we compare returns in the current investment period rather than comparing them at some fixed point in the future? | Instead of comparing the returns on the same initial outlay after a given time period for the two investments we could compare the initial outlays required to give the same final return from the two investments after the given time period. |
| What does Negative NPV for an Investment indicate? | As a rule a negative NPV for an Investment always indicates that that Investment is inferior to the alternative to which it is being compared. |
| What does Positive NPV for an Investment indicate? | As a rule a positive NPV for an Investment always indicates that that Investment is superior to the alternative to which it is being compared. |
| What is Discounting Factor also known as? | Present Value Factor. |
| What is a Capital Investment, in relation to Investment Appraisal? | • A capital investment is a project which consists of 1. An initial outlay of capital 2. A set of estimated cash inflows and outflows over the lifetime of the project. 3. Optionally, a resettlement value which may be an inflow or an outflow. |
| Name two ways to compare Capital Investments | 1. Discounted Cash Flow and Net Present Value 2. Internal Rate of Return |
| What is the Discounted Cash Flows Technique? | • The discounted cash flows technique of investment appraisal involves calculating the sum of the present values of all cash flows associated with a project. • This sum is called the NPV. |
| What does NPV > 0 mean? | Project is in profit (worthwhile) - earns more than discount rate. |
| What does NPV = 0 mean? | Project breaks even (no better than a safe investment) -earns exactly the discount rate. |
| What does NPV < 0 mean? | Project makes a loss (not worthwhile) - earns less than the discount rate. |
| What is the formula for estimating IRR? | IRR= ((N1 * I1) - (N2 * I2)) ÷ (N1 - N2) • where discount rate 𝐼1 gives NPV 𝑁1 • and discount rate 𝐼2 gives NPV 𝑁2 |
| What are some factors to be considered in Comparison of Projects? | • If the capital needs to be borrowed • Ability to pay interest •General liquidity •Uncertainty of estimated flows |
| What are the Techniques used for Comparison of Projects? | NPV or IRR |
| What is the NPV Technique for Comparison of Projects? | • Normal to choose the project which has the largest NPV as the most profitable. • Most suited to projects with similar patterns of cash flows over the same length of time |
| What is the IRR Technique for Comparison of Projects? | • Normal to choose the project which has the largest IRR. |
| Advantages of the NPV Technique for Comparison of Projects | • Practical and Relevant • Gives the results in real money terms. |
| Disadvantages of the NPV Technique for Comparison of Projects | • Relies on the choice of the discount factor |
| Advantages of the IRR Technique for Comparison of Projects | • Not dependent on external rates of interest. |
| Disadvantages of the IRR Technique for Comparison of Projects | • Returns a relative percentage value - does not differentiate between the scales of the projects. |
| What is an Annuity? | A sequence of fixed equal payments (or receipts) made over uniform time intervals. |
| When may Annuities be paid? | 1. at the end of payment interval (ordinary annuity) 2. at the beginning of payment interval (due annuity) |
| The Term of an Annuity may... | 1. begin and end on fixed dates (a certain annuity) 2. depend on some event that cannot be fixed (a contingent annuity) |
| What is a Perpetual Annuity? | A perpetual annuity is one that carries on indefinitely |
| Annuity Compound Interest Formula | 𝐴= 𝑃(1 + 𝑖)to the power of n |
| Annuity Sum of a Geometric Progression formula | Sn = (a (r (to the power of n) - 1) ÷ (r-1) |
| How to tackle problems with Annuities | 1. Using a year by year schedule - calculate the value of the fund on a yearly basis. Only practical when there are a relatively small number of years. 2. Using a geometric progression - normally used when there are a large number of years. |
| NPV of an Annuity | PVannuity = X[(1 ÷ r) - ((1) ÷ (r(1 + r) to the power of n))] = (x ÷ r) * [1 - (1 ÷ ((1 + r) to the power of n))] |
| What does Amortisation of an Annuity mean? | Repaying a debt. |
| What does an Amortisation of an Annuity consist of? | Regular repayments where the repayment covers both part of the principle and interest. |
| What is an Amortisation Schedule? | This is a specification -year by year- of the state of the debt. |
| What does an Amortisation Schedule show? | • Amount of debt outstanding at beginning of the year • Interest paid • Annual repayment • Amount of principal repaid (optional) |
| An Amortisation Schedule should only be produced for... | ...Relatively short periods e.g. up to about 6 |
| What is a Sinking Fund? | • A sinking fund is an annuity invested in order to meet a known commitment at some future date. |
| What are Sinking Funds commonly used for? | 1. Repayment of Debts 2. Providing funds to purchase a new asset when an existing one is fully depreciated |
| Explain - Repayment of Debts, in Relation to Sinking Funds | A debt is incurred over a fixed period of time, subject to a given interest rate. The sinking fund is set up to mature to the outstanding amount of the debt. |
| What is the Sinking Fund Schedule for Repayment of Debts? | 1. the outstanding amount of the debt 2. any interest paid on the debt 3. the regular payment into the sinking fund 4. the interest earned 5. the total amount in the fund. |
| Explain Providing funds to purchase a new asset when an existing one is fully depreciated, in relation to Sinking Funds | Available for investment into a fund (a depreciation fund), which will mature to some predetermined value. |
| How do you determine the book value of the asset at the end of the year , in relation to Sinking Funds? | The book value of the asset at the end of the year can be determined by subtracting the current amount in the fund from the original book value of the asset. |
| What is the Sinking Fund Schedule for Value of Assets? | 1. the payment into the fund (depreciation charge) 2. the interest earned 3. the amount in the fund 4. the current book value of the asset |
| Explain Perpetuity | A perpetuity is an investment offering a fixed sum each year to infinity. |
| Perpetuity Formula | PV = (X ÷ ((1 + r)¹)) + (X ÷ ((1 + r)²)) + (X ÷ ((1 + r)³)) + ... |
| Perpetuity Formula over an Infinite Time Period | PV = X ÷ R |
| When would you use the Perpetuity Formula over an Infinite Time Period? | In the special case of a constant sum of money over an infinite time period, the formula can be shortened |
| What is Growing Perpetuities? | A growing perpetuity is the same as before except that now the sum increases by g% each year. |
| Growing Perpetuities Formula | PV =(X ÷(1+r)¹) + ((X + 1g)¹) ÷(1+r)²) + ((X + 1g)²) ÷(1+r)³) + ... |
| Growing Perpetuities Formula Shortened | PVgrowing perpetuity = (x÷ (r-g)) |
| What is a Function used for? | A function is a way of describing certain types of quantitative relationships mathematically. |
| What is a Function? | A function is an expression involving one or more variables |
| What is an Equation? | An equation specifies an exact relationship between two functions |
| What is a Linear Equation? | A linear equation involves no powers of the variable involved greater than the first and takes the general form𝑦 = 𝑎𝑥 + b |
| What is the Linear Equation formula? | 𝑦 = 𝑎𝑥 + 𝑏 • x is the variable • a and b are numeric coefficients and can take any value |
| What is a Quadratic Equation? | • A quadratic equation involves no powers of the variable involved greater than the second and takes the general form 𝑦 = 𝑎𝑥² + 𝑏𝑥 + c |
| What is The Quadratic Equation formula? | 𝑦 = 𝑎𝑥² + 𝑏𝑥 + c • x is the variable • a, b and c are numeric coefficients and can take any value |
| What is a Parabola? | A quadratic function when plotted on a graph is called a parabola and always takes the same distinct shape . |
| How to plot a Quadratic | Choose at least 4 or 5 relevant x-values. Substitute these into the quadratic function to find the corresponding y values. Plot these points on the graph. Join the points with a smooth curve. |
| General Formula Linear Equation with one Variable | 𝑎𝑥 + 𝑏 = 0 |
| General Formula Linear Equation with two Variables | 𝑎𝑥 + 𝑏𝑦 = 𝑐 𝑑𝑥 + 𝑒𝑦 = f |
| General Formula Linear Equation with three Variables | 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 𝑒𝑥 + 𝑓𝑦 + 𝑔𝑧 = ℎ 𝑗𝑥 + 𝑘𝑦 + 𝑙𝑧 = m |
| What are the Techniques to solve Simultaneous Equations? | 1. Analytical (algebraic) technique involves manipulating the equation always gives the exact answer 2. Graphical technique involves drawing the graph and finding point of intersection approximate solutions |
| Explain Manipulating Equations | Any linear equation can be multiplied or divided throughout by any number without altering the truth of the equation. Any two linear equations can be added or subtracted (one from the other) algebraically to give a third equally valid equation. |
| Procedure for solving 2x2 Simultaneous Equations | Step 1: Eliminate one of the variables Step 2: Solve the resulting equation Step 3: Substitute this value back into one of the original equations Step 4: Check the solutions in the other equation not used in |
| Explain the Graphical Method | Step 1: Graph the two lines represented by the two given equations. Step 2: Identify the x and y values at the intersection point of the two lines. |
| Quadratic Equations General Formula | 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 • Where a, b and c are any numeric values (i.e. positive or negative, whole number or fractions). |
| Methods for Solving Quadratic Equations | 1. Algebraically i. Using a formula ii. Factorising 2. Graphically Remember! There are at most 2 solutions to every quadratic equations |
| Solving Quadratic Equations using a formula | [x=dfrac {-b pm sqrt {b^ {2}-4 a c}} {2 a} nonumber ] • Note: • If 𝑏² < 4𝑎𝑐 there is no solution • If 𝑏² = 4𝑎𝑐 there is one solution • If 𝑏² > 4𝑎𝑐 there are two distinct solutions |
| Solving Quadratic Equations by Factorisation | If a quadratic equation can be factorised i.e. written in the form (𝑥 − 𝑎)(𝑥 − 𝑏) = 0 then it can be solved. |
| Solving quadratic equations graphically | The procedure for solving a quadratic of the form 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 Step 1: Plot the graph 𝑦 = 𝑎𝑥² + 𝑏𝑥 + 𝑐 Step 2: Determine the x values of the points where the graph meets the x-axis |
| What can Differentiation be used for? | Can be used to find the minimum or maximum points of the curves of certain business functions e.g. cost and revenue functions. Can also be used to measure rate of change which is applied to cost and revenue functions to obtain marginal cost or revenue |
| Explain what Differentiation is | • Differentiation can be thought of as a process, which transforms one function into a different one. |
| What is a Derivative? | When one function transforms into another one. |
| What applications does Differentiation have when used for Total Cost, Total Revenue, Profit? | • Total Cost: We want to know where the minimum point on the TC is. • Total Revenue: we want to know where the maximum point on the TR curve is (P x X). • Profit: We want to maximise this (TR – TC). |
| Simple Functions | A simple function of x is defined to have the form 𝑎𝑥(to the power of b) where a & b can take on any numerical value. |
| Interpreting Differentiation | The larger the value of the gradient of a straight line (y=a+bx), the greater the rate of change of y with respect to x. We measure the rate of change by the gradient or slope of the line. Y=10+4x Gradient of line dy/dx=4 i.e. rate of change |
| What does the derivative of any function measure? | The rate of change. It is a function of x and can vary from one point to the next |
| What does Differentiation determine the position of? | • Differentiation determines the position of any turning points of the curve defined by a given function. • While a curve can be drawn on a graph given a particular function, differentiation will find them precisely. |
| What is Integration? | Integration can be regarded as the opposite process to differentiation, and can be used to identify specific revenue and cost functions, given marginal revenue and cost functions. E.g. differentiating 2𝑥² gives 6𝑥² integrating 6𝑥² gives 2𝑥³ |
| What is the rule for Integrating a Simple Function? | a) Increase the power of x by 1 b) Divide the whole function by b + 1 c) Add an arbitrary constant C : this reflects the fact that differentiating a constant gives zero. |
| Definition of Profit | Profit (P) = Revenue(R) - Cost(C) Mathematically: P(x) = R(x) - C(x) where x is the quantity of items demanded (supplied or produced) P(x) = Profit in terms of x R(x) = Revenue in terms of x C(x) = Cost in terms of x |
| Explain Revenue | • Revenue is equal to the number of units sold times the price per unit. |
| Revenue Function | R(x) = x * p(x) To obtain the revenue function, multiply the output level by the price function. (sometimes called the demand function) |
| Explain Fixed Costs | Associated with the purchase, rent or lease of equipment and fixed overheads. |
| Explain Variable Costs | Associated with the supply of raw materials and overheads necessary to manufacture each product. |
| Explain Special Costs | Might cover costs such as storage and maintenance. |
| What is the Form of a Cost Function? | C(x) = a + bx + cx² • x is the quantity if items demanded • a is the fixed cost associated with the product • b is the variable cost associated with the product • c is the (optional) special cost associated with the product |
| What is the Maximum Profit Function? | dP ÷ dx = 0 |
| Explain the Power Rule of Differentiation | • The rule is that the power in the variable of the original function becomes a multiple of the variable in the derivative and the power of the variable in the original function is reduced by 1. |
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DamienHart
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