MACKENZIE HIMMELBAUER 1131 FINALS STUDYGUIDE 1 1131 Business Calculus Final Study Guide Notes Before you begin studying Mathematical definitions are provided by the Pearson Introductory Mathematical Analysis For Business Economics and the Life and Social Sciences thirteen Edition to coincide with your notes the sections are not in numerical order This is a GUIDE the only way to really know the material is to do practice This study guide is organized in the way that we learned it so as best problems but this can help you organize your thoughts 10 1 Limits o Definition of a limit The limit of f x as x approaches a is the number L as lim X values can be close to L if we take x as close to but different from a If there is no such number we say that the limit of f x as x approaches a DNE o Properties of Limits Factoring out multiplicative constant lim lim Limit of a sum or difference will work for more than just 2 functions lim lim lim Limits of products lim lim lim Division of limits lim lim lim provided that lim 0 Limit raised to a power lim lim Mackenzie Himmelbauer lim lim lim lim lim lim 1 10 2 One Sided Limits right handed limit formula lim left handed limit formula lim Other If both the left handed limit and the right handed limit equal the same value the limit does exist If both the left handed limit and the right handed limit equal different values the limit does not exist o Infinite Limits lim lim With infinite limits we can manipulate f x to be large and positive or large and negative to create values close to x a o Limits at Infinity for Rational Functions Degrees of the highest coefficient on top and bottom are the same 7 3 4 4 5 7 4 lim Degree of the highest coefficient is greater in the numerator makes limit either go to positive or negative infinity situation dependent 5 2 3 lim 7 5 EX EX Mackenzie Himmelbauer MACKENZIE HIMMELBAUER 1131 FINALS STUDYGUIDE 3 5 2 3 lim 7 5 EX Degree of the highest coefficient is greater in the denominator than the numerator makes the limit go to zero 4 2 3 lim 8 5 0 4 2 3 lim 8 5 0 number in the denominator which means that we will get incredibly close to zero positive vs neg infinity is irrelevant because we will have a higher o Tips for evaluation of limits Re order polynomial to find the leading coefficients Identify whether the highest degree of the coefficients are equal higher in the numerator or higher in the denominator If the degree of the coefficient is higher in the denominator the answer is automatically 0 regardless of If the degree of the coefficients in the numerator and denominator are equal take the numbers of the leading coefficients in fraction form If the degree of coefficient is higher in the numerator identify whether the coefficient is negative or positive and plug in the negative or positive infinity into the variable Identify if the answer is negative infinity or positive infinity based on the sign of the coefficients 10 3 Continuity o A function is continuous at these three conditions KNOW F a exists lim lim o the point of discontinuity is where f becomes discontinuous at a o rules A polynomial function is continuous at every point A rational function is discontinuous at points where the denominator is 0 and is continuous otherwise Mackenzie Himmelbauer 11 1 Derivatives o The Derivative A rational function is continuous on its domain The main point is a function that is continuous everywhere where it is defined and it is discontinuous wherever it is undefined o to find points of discontinuity there may not always be points of discontinuity set the denominator equal to 0 if the domain is all real numbers there is no point of discontinuity for piece wise defined functions you must utilize the rules of discontinuity to prove that it is continuous or discontinuous points of discount In piece wise defined functions can also be found where x is both less than a value or greater than a value o ex x 7 if x 4 13 4x if x 4 the point of discontinuity is at 4 The derivative of a function f is the function f f prime and defined by the following equation f x lim this is the definition of the derivative lim The derivative gives the slope of the tangent line f a is the slope of the line tangent to the graph of y f x at a f a If f is differentiable at a then continuous continuity does not imply differentiability o to use the definition of the derivative to find the derivative identify your y equation utilize the lim plug in x h wherever you see a x in the y formula formula you are finding the f x h portion of the limit take the results from the previous step and subtract the original formula you are formulating the top component of the f x h f x limit formula simplify divide by h plug in zero for any h value remaining and eliminate terms to find answer o to find the slope given y utilize your lim the definition of the derivative to solve resulting answer and insert given terms and steps to find derivative using Mackenzie Himmelbauer MACKENZIE HIMMELBAUER 1131 FINALS STUDYGUIDE 5 ex y 1 x2 at 1 0 derive 2x plug in 1 for x overall answer 2 o to find an equation of the tangent line to the curve at given point use definition of derivative to find derivative derivative with x variable plugged in slope m value in point slope formula insert given values into the point slope formula y y1 m x x1 x1 y1 are given to you final answer should be in y mx b format 11 2 Rules for differentiation simple way to differentiate o Rules 0 cf x cf x f x f x g x o to find the equation of the tangent line to the curve at the indicated point differentiate f x using given rules plug in given x value to the derivative you found gives your m value of the point slope formula plug your m into the original equation gives you the y value for your x y coordinates plug all found variables into point slope formula y y1 m x x1 11 3 The derivative as a Rate of Change Applications of Rate of Change to Economics o for mulas needed Total Cost Function c f q Marginal cost Average cost Relative rate of change of f x is Percent rate of change is 100 average cost per unit 11 4 The Product Quotient Rule Mackenzie Himmelbauer o If the two functions f x and g x are differentiable then the product is differentiable Product Rule f x g x f x g x f x g x derivative of first variable second variable first variable derivative of …