From Section 3.2- Comparing Functions and Numeric Derivatives

Skip to main content

\(\newcommand{\Volume}{\mathrm{Volume}} \newcommand{\xLow}{\mathrm{xLow}} \newcommand{\xHigh}{\mathrm{xHigh}} \newcommand{\Mcost}{\operatorname{Marginal Cost}} \newcommand{\cost}{\operatorname{Cost}} \newcommand{\Mrev}{\operatorname{Marginal Revenue}} \newcommand{\revenue}{\operatorname{Revenue}} \newcommand{\profit}{\operatorname{Profit}} \newcommand{\Mprofit}{\operatorname{Marginal Profit}} \newcommand{\Pprofit}{\operatorname{Predicted Profit}} \newcommand{\price}{\operatorname{Price}} \newcommand{\Dprice}{\operatorname{Demand Price}} \newcommand{\Pprice}{\operatorname{Purchase Price}} \newcommand{\Sprice}{\operatorname{Supply Price}} \newcommand{\marginal}{\operatorname{Marginal}\,} \newcommand{\linear}{\operatorname{Linear}\,} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\quantity}{\mathrm{Quantity}} \newcommand{\AnnualCost}{\operatorname{Annual Cost}} \newcommand{\AnnualExpense}{\operatorname{Annual Expense}} \newcommand{\RepairFactor}{\mathrm{Repair Factor}} \newcommand{\Sales}{\operatorname{Sales}} \newcommand{\prediction}{\operatorname{prediction}} \newcommand{\FutureValue}{\operatorname{Future Value}} \newcommand{\demand}{\operatorname{Demand}} \newcommand{\val}{\operatorname{Value}} \newcommand{\raw}{\mathrm{raw}} \newcommand{\gizmos}{\operatorname{gizmos}} \newcommand{\intermediate}{\mathrm{intermediate}} \newcommand{\tax}{\operatorname{tax}} \newcommand{\FaceValue}{\operatorname{Face Value}} \newcommand{\rate}{\mathrm{rate}} \newcommand{\years}{\mathrm{years}} \newcommand{\expense}{\operatorname{expense}} \newcommand{\FutureAmount}{\operatorname{FutureAmount}} \newcommand{\InitialDeposit}{\mathrm{InitialDeposit}} \newcommand{\ppy}{\mathrm{ppy}} \newcommand{\PresentAmount}{\operatorname{PresentAmount}} \newcommand{\PriceGizmo}{\mathrm{PriceGizmos}} \newcommand{\QuantityGizmo}{\mathrm{QuantityGizmos}} \newcommand{\QuantityGadget}{\mathrm{QuantityGadgets}} \newcommand{\QuantityWidget}{\mathrm{QuantityWidgets}} \newcommand{\PriceWidget}{\mathrm{PriceWidgets}} \newcommand{\PriceGadget}{\mathrm{PriceGadgets}} \newcommand{\QW}{\mathrm{QW}} \newcommand{\QG}{\mathrm{QG}} \newcommand{\PW}{\mathrm{PW}} \newcommand{\PG}{\mathrm{PG}} \newcommand{\Production}{\operatorname{Production}} \newcommand{\Labor}{\mathrm{Labor}} \newcommand{\Capital}{\mathrm{Capital}} \newcommand{\Linear}{\operatorname{Linear}\,} \newcommand{\PriceX}{\mathrm{PriceX}} \newcommand{\PriceY}{\mathrm{PriceY}} \newcommand{\predicted}{\operatorname{predicted}} \newcommand{\Time}{\mathrm{time}} \newcommand{\TotalAmount}{\operatorname{TotalAmount}} \newcommand{\TotalValue}{\operatorname{TotalValue}} \newcommand{\CashAmount}{\mathrm{CashAmount}} \newcommand{\DepositAmount}{\mathrm{DepositAmount}} \newcommand{\AvailableTrees}{\operatorname{AvailableTrees}} \newcommand{\Capacity}{\mathrm{Capacity}} \newcommand{\Amount}{\operatorname{Amount}} \newcommand{\InitialAmount}{\mathrm{InitialAmount}} \newcommand{\InvestmentAmount}{\mathrm{InvestmentAmount}} \newcommand{\Return}{\operatorname{Return}} \newcommand{\SD}{\mathrm{SD}} \newcommand{\Antideriv}{\operatorname{Antideriv}} \newcommand{\MC}{\operatorname{MC}} \newcommand{\MP}{\operatorname{MP}} \newcommand{\MR}{\operatorname{MR}} \newcommand{\CDFf}{\operatorname{CDFf}} \newcommand{\Principal}{\operatorname{Principal}} \newcommand{\ProducerSurplus}{\mathrm{ProducerSurplus}} \newcommand{\ConsumerSurplus}{\mathrm{ConsumerSurplus}} \newcommand{\TotalSocialGain}{\mathrm{TotalSocialGain}} \newcommand{\Elasticity}{\operatorname{Elasticity}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Section C.2 From Section 3.2- Comparing Functions and Numeric Derivatives

The spreadsheet skill in this section was to make a table and graph of a function and its derivative. We use a variety of approaches to find numeric derivatives.

Subsection C.2.1 Derivatives from the Intuitive Approach

We build a worksheet that plotsthe function and a secant curve with control over del x 3.2.5. We then reduce del x until the the graphs appear to be the same the graphs appear to be the same 3.2.6. Screencast of example using this approach. 3.2.3

Subsection C.2.2 Derivatives from Numerical Limits

Without using graphs, we can also look at the slope of the secant line as del x gets small. 3.2.7. from one row to the next. Screencast of example using this approach. 3.2.3

Subsection C.2.3 Graphing a Function with its Numeric Derivative

To build a chart of a function and its derivative and to grpah the functions together, we use a variant of the approach from the previous section. We set up successive colummes for x, x+del x, x-del x, f(x), f(x+del x) f(x-del x), and f'(x) 3.2.11. I then only have the enter the formula for the function one time, under f(x). Quick fill then provides the correct formula for f(x+del x) and f(x-del x). Screencast of example using this approach. 3.2.9

In practive, we usually set \(del x= 0.001\text{.}\)

Subsection C.2.4 Using Trendline to find Derivative Formulas

If the grpah of the numerical deerivative looks like a model we know, and one that trendline will produce, we can try to obtaina formula useing Trendline. Add a trendline and display the formula of the trendline and \(R^2\) If the model is correct, \(R^2=1\text{.}\) Screencast of example using this approach. 3.2.14