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- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Calculus_with_Excel_(May_and_Bart)/02%3A_Business_Applications/2.03%3A_Modeling_Revenue_Costs_and_ProfitIf the slope of the demand curve is 0, the consumers have a fixed price they will pay for however much of the product is available. To find the break-even point when we are given data instead of an eq...If the slope of the demand curve is 0, the consumers have a fixed price they will pay for however much of the product is available. To find the break-even point when we are given data instead of an equation, we usually follow this procedure: Find the best fitting equations for price and cost. The fixed cost is $10 (the constant/fixed part of the cost function), and the variable cost is $3 per item.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Calculus_with_Excel_(May_and_Bart)/06%3A_Functions_of_Several_Variables/6.04%3A_Optimization_and_Best_Fitting_CurvesIf we are trying to fit the data with a different model we want to choose the equation from that model that minimizes the sum of the squares of the error. We add in the error, which is the difference ...If we are trying to fit the data with a different model we want to choose the equation from that model that minimizes the sum of the squares of the error. We add in the error, which is the difference between the predicted y and the actual y, and the square of the error. We add extra columns for the predicted population, the error between the prediction and the actual population, the square of the error.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Calculus_with_Excel_(May_and_Bart)/07%3A_Integration/7.03%3A_Basic_AntidifferentiationWe first replace \(n\) with \(n+1\) to get \(\frac{d}{dx} x^{n+1}=(n+1)x^n\text{.}\) We then divide both sides by \(n+1\) to obtain \(x^n=\frac{d}{dx} x^{n+1}/(n+1)\text{.}\) Finally, we note that add...We first replace \(n\) with \(n+1\) to get \(\frac{d}{dx} x^{n+1}=(n+1)x^n\text{.}\) We then divide both sides by \(n+1\) to obtain \(x^n=\frac{d}{dx} x^{n+1}/(n+1)\text{.}\) Finally, we note that adding a constant \(C\) does not change the derivative, so \(x^n=\frac{d}{dx} (x^{n+1}/(n+1)+C)\text{.}\) Since we have divided by \(n+1\text{,}\) we need to insist that \(n+1\ne 0\text{.}\) Using the notation of indefinite integrals we obtain our power rule formula:
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Calculus_with_Excel_(May_and_Bart)/04%3A_Symbolic_Differentiation/4.01%3A_Elementary_Derivatives\begin{gather*} f'(x)=lim_{h\to0}\frac{f(x+h)-f(x)}{h} =lim_{h\to0}\frac{(a (x+h)^2)-(a x^2)}{h}\\ =lim_{h\to0}\frac{(a x^2+2 a h x +a h^2)-(a x^2)}{h}\\ =lim_{h\to0}\frac{2 a h x +a h^2}{h} =lim_{h\t...\begin{gather*} f'(x)=lim_{h\to0}\frac{f(x+h)-f(x)}{h} =lim_{h\to0}\frac{(a (x+h)^2)-(a x^2)}{h}\\ =lim_{h\to0}\frac{(a x^2+2 a h x +a h^2)-(a x^2)}{h}\\ =lim_{h\to0}\frac{2 a h x +a h^2}{h} =lim_{h\to0}(2ax+h)=2ax. \end{gather*}
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Calculus_with_Excel_(May_and_Bart)/05%3A_Differentiation_Techniques_and_Applications/5.02%3A_Related_RatesAs we have seen, \(\frac{dy}{dx}\) is the instantaneous rate of change of \(y\) with respect to \(x\text{.}\) In chapter 4 we learned techniques for finding \(\frac{dy}{dx}\) when \(y\) is defined as ...As we have seen, \(\frac{dy}{dx}\) is the instantaneous rate of change of \(y\) with respect to \(x\text{.}\) In chapter 4 we learned techniques for finding \(\frac{dy}{dx}\) when \(y\) is defined as a function of \(x\text{.}\) In the last section we learned how to use implicit differentiation to find \(\frac{dy}{dx}\) when we were given an equation in \(x\) and \(y\text{.}\) In this section we want find \(\frac{dy}{dx}\) when \(x\) and \(y\) are both described in terms of another variable.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Calculus_with_Excel_(May_and_Bart)/06%3A_Functions_of_Several_Variables/6.01%3A_Evaluating_and_Graphing_Functions_of_Several_VariablesCreate a table that expresses the future value of a deposit as a function of the annual interest rate and the number of years the deposit is held, with the amount of the initial deposit and the number...Create a table that expresses the future value of a deposit as a function of the annual interest rate and the number of years the deposit is held, with the amount of the initial deposit and the number of times per year that the interest is compounded being treated as parameters, where the interest on a deposit of $10,000 is compounded quarterly, and the deposit is held for 20 to 40 years at interest rates ranging from 3% to 5%.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Calculus_with_Excel_(May_and_Bart)/02%3A_Business_Applications/2.01%3A_Market_Equilibrium_ProblemsIf the price goes up we can assume that all the old suppliers are still willing to sell at the higher price, but some more suppliers may enter the market. If the price goes down we can assume that all...If the price goes up we can assume that all the old suppliers are still willing to sell at the higher price, but some more suppliers may enter the market. If the price goes down we can assume that all the old consumers are still willing to buy at the lower price, but some more consumers may enter the market. When we look at a graph of the supply price graph and the demand price graph on the same graph, we know the supply curve goes up as we go left to right, while the demand curve goes down.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Calculus_with_Excel_(May_and_Bart)/01%3A_Functions_Graphs_and_Excel/1.01%3A_Linear_Functions_and_ModelsAvoiding the intercepts we can choose \(x\) to be any number other than \(x = 0\) and \(x=3/4\text{.}\) We may look for nice values of \(x\) that make the arithmetic come out nice: for example choose ...Avoiding the intercepts we can choose \(x\) to be any number other than \(x = 0\) and \(x=3/4\text{.}\) We may look for nice values of \(x\) that make the arithmetic come out nice: for example choose \(x=3\) and \(x=4\text{.}\) When \(x=3\text{,}\) \(y=9\text{,}\) and we get the point \((x,y)=(3,9)\text{.}\) When \(x=4\text{,}\) \(y=13\text{,}\) and we get the point \((x,y)=(4,13)\text{.}\)
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Calculus_with_Excel_(May_and_Bart)/06%3A_Functions_of_Several_VariablesFrom a simple point of view, the future amount is a function of 4 variables, the initial deposit, annual rate, periods per year, and number of years. To consider this as a function of a single variabl...From a simple point of view, the future amount is a function of 4 variables, the initial deposit, annual rate, periods per year, and number of years. To consider this as a function of a single variable, we fixed 3 of the 4 variables as constants for a particular problem. Before we look at functions of several variables, we want to create a list of tasks we have learned to accomplish with functions of one variable:
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Calculus_with_Excel_(May_and_Bart)/02%3A_Business_Applications
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Calculus_with_Excel_(May_and_Bart)/04%3A_Symbolic_DifferentiationFor most functions that gave an easy approximation without any rules other than the conceptual understanding that we obtained the derivative by zooming in far enough for the graph to look like a strai...For most functions that gave an easy approximation without any rules other than the conceptual understanding that we obtained the derivative by zooming in far enough for the graph to look like a straight line. When we looked at the derivative at many points we found that for polynomials of degree 2 or less, the derivative seems to be a polynomial of one degree lower.
