Business Calculus Final Exam Review Unit 1 Average Velocity o Average Velocity Slope o The slope m of the line through the points x1 y1 and x 2 y 2 is given by the equation For example Find the average velocity of the line segment jointing the points 1 4 and 4 2 Label the points as x1 1 y1 4 x2 4 and y2 2 To find the slope m of the line segment joining the points use the slope formula Solution m 6 5 Direct Substitution Limit Problems o Plug in the limit values that it is approaching into x in the equation To solve for a limit we want to calculate the y value for the function at a particular x value Example In this case a limit is a y value and the y value is obtained by direct substitution Another Example o Visual Limit Problems One sided Limits Look at the value that the limit is approaching Find that x value on the graph and visualize a vertical line at x approached number Look at where the graph line approaches that vertical line at x approached number The y value at that location is your answer If asked to solve for a one sided limit the limit will always exist The limit will either be from the positive side or the negative side The positive side is on the right of the graph and the negative side is on the left of the graph o A limit from the positive side of number n is denoted by o A limit from the negative side of a number n is denoted by o o Example graph of o Another example as can be seen on the Given the following graph compute Solution The function is approaching a value of 2 as x moves in towards 4 from the right Limits by Factoring This method should be used when you find that when trying to solve the limit you are getting 0 0 When factoring for limits some of the factors will cancel and you will not be left with o Example However if you factor the numerator it would result in The term would cancel on the top and the bottom leaving you with which can be found by substitution as 4 o The solution will always be a number zero or infinity With that being said anything infinity infinity with the correct sign Locate the part of the polynomial that is raised to the highest exponent Take this part and plug in the number that the limit is approaching into this part of the equation Solve If you came up with a number that is the answer If you have a number infinity the answer is infinity depending on the signs that were multiplied Absolute Value limit problems If you encounter a limit involving absolute values the best approach is to start b getting rid of the absolute value signs You can do this by remembering that the definition of the absolute value function is Example or o o o o With absolute values it s important to be very careful about two sided limits since the function is x 1 from one side and x 1 from the other We should evaluate both one sided limits seperately to make certain that they re equal o Here since x is approaching 1 from the positive side we know that we ll always be in the domain where x 1 We can replace x 1 with x 1 o o o Since we re approaching 1 from the negative side we can replace x 1 with x 1 o The two one sided limits disagree so we conclude that The Find M Limit Problems Determine the values of constants a and b so that exists Begin by computing one sided limits at x 2 and setting each equal to 3 Thus and o Now solve the system of equations a 2b 3 and b 4a 3 Thus a 3 2b so that b 4 3 2b 3 iff b 12 8b 3 iff 9b 15 iff Then Derivatives o Marginal derivative o Note that correct solutions to an e grade problem will NEVER have the word marginal in it o The f x h f x and Definition of Derivative Problem Use the definition of derivative and the fact that f x h f x x 4h 6x 2h 2 2xh 6 9h 4 2h to find f x Eliminate any term with an h 2 or higher power After that divide all of what is left by h Derivative Rules Asymptotes o Finding the Vertical Asymptote Set the bottom of the equation to zero Do not cancel Solve Remember you cannot take the square root of a negative number so if that is the case the answer will be no vertical asymptote o Finding the Horizontal Asymptote Find the part of the polynomial that is raised to the highest exponent of the top of the fraction and write that over the part of the polynomial that is raised to the highest exponent of the bottom of the fraction Creating a whole new fraction Plug infinity into the new equation from the previous step solve If your answer from the last step is infinity the answer will be NO horizontal asymptote If your answer is a number divided by infinity the final answer will be zero Any number infinity infinity Y cannot divide by infinity If that is the case the answer will be no horizontal asymptote o If the top exponent from the new numerator in step 1 is bottom exponent from new denominator in step 1 answer is no horizontal asymptote o If top exponent bottom exponent answer is y 0 o If top exponent bottom exponent answer will be the coefficient of the top divided by the coefficient of the bottom Finding Inflection Points o Given f x f x and f x and asked to find the local extrema minimum maximum value and inflection points F x will give you the y coordinate f x will help you determine where the graph is increasing decreasing and extrema points F x will help you determine where the SLOPE is increasing decreasing and help you decide the concavity of the graph within a range and find possible inflection points Begin by finding out what you can from the first derivative Set the first derivative equal to zero and solve to find possible critical points Remember critical points are places that you can have a minimum maximum value If given a fraction set the top 0 and solve because the bottom of a fraction cannot be equal to zero Because the bottom of the fraction cannot equal 0 it COULD be a critical point but will NEVER be a local minimum or maximum value Make a number line and label with critical values Take note of the values for which x cannot equal zero Break up the chart there along with ranges Evaluate each factor to determine positive negative sign for each part of the line chart Put f x on the top and f …