Midterm Study Guide
Midterm Study Guide M119
Popular in Brief Survey of Calculus
One Day of Notes
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Popular in Mathematics (M)
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This 6 page Study Guide was uploaded by Rachel McCord on Tuesday March 3, 2015. The Study Guide belongs to M119 at Indiana University taught by William Orrick in Spring2015. Since its upload, it has received 458 views. For similar materials see Brief Survey of Calculus in Mathematics (M) at Indiana University.
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Date Created: 03/03/15
M119 A Brief Survey of Calculus Midterm Study Guide 1 Uquot 10 11 12 13 14 15 16 17 18 19 20 21 22 Function Notation yxmxb a Y depends on X b Independent variable what can change within the problem in the parentheses c Dependent variable Changes depending on the independent variable Significant digits number of digits counting from the first nonzero digit Function Composition When you replace the independent variable of a function with a different equa on a Rt1t and ArTtr2 so a function composition would be Art T1t2 Domain of a function The set of values that the independent variable can take Range of a function Set of values that the dependent variable can take Interval Notation a ab the set of all values x in interval altxltb b ab the set of all values x in interval altxltb c There are also cases of ab Discrete quantity when the problem involves whole numbers such as population of a country Continuous quantity when the problem can have decimalsfractions such as the radius or area of a circle Vertical Line Test In order to determine if a graph is a function no vertical line should pass through more than one point of the function A function assigns a single value to every number in the domain Constant Rate of Change functions where the rate of change is always the same slopes and rates are related Slope riserun or the change in Y the change in X or y2y1x2x1 Arithmetic sequence A sequence of equally spaced values able to determine from a table if a function is potentially linear X and Y are arithmetic Constant Slope Determine slope of points to determine if a function is linear f slope stays constant then it is linear Speed distance travelled time elapsed Average rate of change the same equation as slope Secant line The straight line representing the distance for a trip made at constant speed Also known as the reference line joins curve at 2 points Increasingdecreasing functions A function is increasing if the graph is rising when looked at from left to right In the same way a function is decreasing if the graph is falling Many functions can be neither or both Concave down The rate of change decreases with time which causes the graph to bend down Slope is decreasing Concave up The rate of change increases with time which causes the graph to bend up Slope is increasing Relative change absolute change initial quantity p1 pO p0 is a fraction or percentage Linear depreciation the decrease in value is the same each year Fixed Cost Constant cost that may be included whether producing or not 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Variable cost vcq where cthe cost of production and qquantity independent variable based on quantity produced Total cost FV Fcq Revenue Function represents the income from the sale of q goods assuming all goods are sole It depends on price p of the goods Rqpq a Price refers to the selling price while cost refers to the cost to the producer for manufacturing the goods Profit Function the difference between the money taken in by a firm and the money going out RqCq Marginal Quantities a Marginal cost the additional cost entailed in producing one additional unit q1 units b Marginal revenue profit the additional revenue that would be obtained if q1 goods were produced c Marginal cost at production level q is 1q1cq d Marginal quantities are the slopes of the corresponding function Producers A higher selling price which means more producers means a greater supply of goods Consumers A higher selling price means less consumers spending money which results in lower demand for goods Supply Sp is an increasing function and Demand Dp is a decreasing function and represent quantities of goods a Price on vertical axis and quantity on horizontal axis Law of Supply and Demand states that in the longrun prices and quantities will go to the equilibrium Specific Tax a fixed dollar amount per product Sales Tax a fixed percentage of the purchase price Effective tax Even when the tax responsibility if on the consumerproducer both end up paying part of the tax Producers pay what they lose in revenue from having to lower prices and consumers pay the rest Monopoly If a company has no competition it is not subject to the market price so it will maximize its revenue Exponential functions PtPoat will later be written as POaquott the independent variable is in the exponent a PO initial quantity since the initial time t0 called the vertical intercept of Pt b A the base of the exponential function It is the factor of change of the function in each time step Relative Rate of Change r is the fractional or percentage change of the function in each time step For exponential functions r is constant Ra1 Differences between linear and exponential functions a Linear ymxb described by two parameters slopem and the vertical intercept b absolute rate is constant b Exponential PP0aquott described by two parameters base a and vertical intercept PO relative rate is constant 39 40 41 42 43 44 45 46 47 48 49 50 51 Geometric sequence A sequence of numbers in which the ratio between successive numbers stays the same Properties of Exponents a XquotaXquotbXquotab XquotaXquotbXquotab Xquota1Xquota assume x doesn t equal 0 XquotO1 XquotaquotbXquotab Xquot1nnsquarerootX XYquotaXquotaYquota xyquotaXquotayquota assume x doesn t equal 0 Interest and compounding various investments savingsmoney market accountsbonds etc are examples of exponential growth as long as the interest earned is kept in the account rather than withdrawn Nominal rate rate r Annual percentage rate APR annual rate Effective annual yield how much you actually earn In general with an annual rate r n times yearly compounding interest rn is paid n times Add interest rate to principle yielding factor of increase of 1rn Compounding base The base of exponential function describing compound interest a1rnquotn Compound interest formula If an initial investment of PO dollars is left to earn interest for t years compounded n times yearly so number of interest payments is nt we get PtP01rnquotnt Continuous Compounding Pt11nquotn a As frequency of compounding gets bigger and bigger the final balance approaches a definite value As n goes towards infinity the result is increased by a decreasing amount b Approaches a known mathematical value e Limiting factor of increase as n approaches infinity it is equotr d If an initial investment of P dollars earns interest at a nominal rate r compounded continuously the balance after t years is PtP0equotrt Summary of Notation n compound interest problems r represents the nominal rate APR Annual factor of increase a the base relates to r differently depending on the compounding schemes Annual Compounding a1r Compounding n timesyear a1rnquotn Continuous Compounding aequotr Effective annual yield a1 In all other contexts r is the relative rate of change and a is the base given by a1r Logarithms The logarithm of a number n taken base a is the exponent of a needed to get n Written as logan Base conversion base conversion factor is always equal to the log of the original base The best choice of a base is e which is known as the natural base gem 999g 099939 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 a Written as xlnequotx andor aequotlna b The second is called the base conversion formula We use aequotlna to convert from any base a to base e c K is known as the base conversion factor lnorigina base na Connection with continuous compounding k is the nominal rate that produces a factor of change of a under continuous compounding a KltO alt1 means decay b Kgt1 agt1 means growth c Do not mix up relative rate of change r and the continuous rate k i Ra1 ii Klna Properties of Logarithms a LnABnAlnB b LnABlnAlnB c LnAquotnnlnA Half Life and Doubling Time t will often be nfactor of changek Instantaneous Speed Exponential functions are characterized by relative rate of change Average speed distance travelledelapsed time Displacement location where you are on a number line Velocity measured by net displacementelapsed time Displacement is always zero when you return to initial starting point Displacement is also known as the derivative of the function To find derivative you can use smaller and smaller intervals There is no limit to the accuracy we can obtain All we need to do is make the time interval shorter and shorter Instantaneous velocity is an example of the more general concept of instantaneous rate of change denoted with called prime Derivative ft2 ft1 t2t1 Always think about what the unit of measure is Right left and twosided estimates are good for finding derivatives Two sided estimates are only accurate if intervals are equally spaced Otherwise use left and right and then find average of the two Geometric meaning of the average rate of change is the slope of the secant line connecting the two points on the graph of a function corresponding to the endpoints of the interval over which the average is taken As the two endpoints move closer to the point of interest the secant line moves slightly thus changing the slope Geometric meaning of derivative is the slope of the tangent line to the curve at the point of interest Derivative function is when y t is a function of t Derivative has many aspects 1 If x represents time and fx represents displacement of an object then the derivative describes the velocity of the object at time x 2Derivative function describes how the instantaneous rate of change of fx depends on the independent variable x velocity function When the f x graph is given 72 73 74 75 F x gt 0 then fx is increasing F x 0 then fx is contant F x lt 0 then fx is decreasing Look at turning points Look where the function is horizontal derivative has horizontal intercept there i Increasing slope is positive ii Decreasing slope is negative e You can find the shape of fx but not the exact graph from the derivative f Peaks and valleys correspond with changes in concavity Calculus has 2 main discoverers Isaac Newton and Gottfried Wilhelm Leibniz a The derivative is a limit of the ratio of deltaYdeltaX b Can also be written as ddx Local linear approximation find the point using slope form but it is only an approximation because in reality the slope changes a little between the two points We don t know the shape and steepness of the curve so it could be an under or over estimate a Fx change of x fx f x change of x Relative change in f between t2 and t1 is ft2ft1ft1 f t1 and t2 are spaced on unit apart t2t11 a Relevant change per unit time is ft2ft1ft1t2t1 b The limit that t2 approaches t this ratio becomes the relative rate of change f t1ft1 c Relative rate of change is equal to constant continuous rate k IF the function is exponential to kx fx equotkx Graphical meanings a F x is positive fx increasing F x negative fx decreasing F x is zero at a point and changes sign there then fx has a turning point at that point F x is zero but doesn t change sign fx has horizontal slope not a turning point Magnitude of f x tells us about the steepness Concave up if slope is increasing f x is increasing g Concave down is slope is decreasing fx is decreasing goo9 The 76 The derivative of the derivative is called the second derivative a Helps with determining concavity b If f x is positive then f x is increasing then fx is concave up c If f x is negative then f x is decreasing then fx is concave down d A point where concavity changes is called an inflection point i If f x changes sign at a particular value of x then fx has an inflection point there ii To find if we have an inflection point we can look at places where f x is zero e F negative 9 slope off is decreasing 9 f is undetermined 77 The derivative of a function doesn t change if the function is shifted 78 Realistic cost of revenue function cost function is nonlinear changing slope due to efficiency and revenue is proportionality profit is largest where the gap is largest and revenue is above cost a Marginal Revenue the selling price slope of revenue function Marginal cost the additional incremental cost for producing one extra unit Marginal revenue and all marginal quantities are derivatives of their related functions If the cost has a flatter slope than revenue increase production If the cost has a steeper slope than revenue decrease production f Where cost and revenue have equal slopes usually want to produce there 79 Formulas for derivative functions a Power function derivative ddx Xquotp pXquotp1 Constant Multiplicative factor ddx Cfx C ddxfx Cf x Sum or Difference ddxfx or gx f x or g x The derivative of equotx is equotx it is itself The derivative of a constant is always zero Given fx find equation of tangent line at Xa i Point of tangency a fa ii Slope is f a iii Use point slope to find equation 80 Derivatives of exponential functions a ddx equotx equotx b ddx aquotx lnaaquotx c ddx lnx 1x 81 Chain rule for the derivative of a composite function a Taking the derivative of a composite function b The chain rule states that the derivative of a composite function with inside function gx is the derivative of the outside function with gx treated as the variable multiplied by g lX c ddX equotgX equotgX g lX d Chain rule when the outside function is a power function ddx gx n ngxquotn1 g lX Chain rule when the outside function is a natural log ddx lngx 1gx g x f By the chain rule the derivative of lngx is simply the relative rate of change gx i Relative rate f f 82 Product rule is the product of elementary functions ux and vx ddx f x u xvx uxv x 83 Quotient Rule You can rewrite as a product and use product rule or use formula a F x vu uv vquot2 84 Derivative of fx axbcxd is adbccxdquot2 0903 799097
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