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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Economics: Marginal Benefits and Costs The table gives the marginal benefit and marginal cost of producing x units of a product for a given company. (a) Plot the points in each column and use the regression feature of a graphing utility to find a linear model for marginal benefit and a quadratic model for marginal cost as functions of the units produced x. (b) Use integration to find the benefit B and cost C equations. Assume B(0) = 0 and C(0) = 425. (c) Find the intervals in which the benefit exceeds the cost of producing x units. Make a recommendation for how many units the company should produce based on your findings. (Source: Adapted from Taylor, Economics, Fifth Edition)

Number of units 1 2 3 4 5
Marginal benefit 330 320 290 270 250
Marginal cost 150 120 100 110 120
Number of units 6 7 8 9 10
Marginal benefit 230 210 190 170 160
Marginal cost 140 160 190 250 320

Spreadsheet at LarsonAppliedCalculus.com

(a)

To determine

To graph: The marginal benefit, marginal cost and number of units produce by the company and find a linear model for marginal benefit and a quadratic model for marginal cost as a function of the units produce x. The tabulated data for number of units, marginal benefit and marginal cost is given as,

Number of units 1 2 3 4 5 6 7 8 9 10
Marginal benefit 330 320 290 270 250 230 210 190 170 160
Marginal cost 150 120 100 110 120 140 160 190 250 320
Explanation

Given Information:

The tabulated data for number of units, marginal benefit and marginal cost is given as,

Number of units 1 2 3 4 5 6 7 8 9 10
Marginal benefit 330 320 290 270 250 230 210 190 170 160
Marginal cost 150 120 100 110 120 140 160 190 250 320

Formula used:

The sum rule of basic integration [f(x)+g(x)]dx=f(x)dx+g(x)dx.

The simple power rule of integration xndx=xn+1n+1+C.

Use ti-83, graphing calculator, to find the function that best fits the data.

Step-1: Open ti-83 graphing calculator and press ON button.

Step-2: Press “2nd” and then click “0” key, search and click “Diagnostic On” in “CATALOG” and press “Enter” twice.

Step-3: Press “STAT” key and press on “1” key to enter the table of values for number of units in “L1” column and for marginal benefit in “L2” column.

Step-4: Press “STAT” key and go to “CALC”.

Step-5: Choose “8:LinReg(a+bx)” (Linear Regression) and press “Enter” twice.

Step-6: Note down the values of Linear Regression model.

Here for this given data Linear Regression model (a+bx) has following values:

a=351b=19.9

So, the equation becomes y=351.333319.87878787x.

Step-1: Open ti-83 graphing calculator and press ON button.

Step-2: Press “2nd” and then click “0” key, search and click “DiagnosticOn” in “CATALOG” and press “Enter” twice.

Step-3: Press “STAT” key and press on “1” key to enter the table of values for number of units in “L1” column and for marginal benefit in “L2” column.

Step-4: Press “STAT” key and go to “CALC”.

Step-5: Choose “8:LinReg(a+bx)” (Linear Regression) and enter L1 and L2

Step-6: Note down the values of Linear Regression model

(b)

To determine

To calculate: The equation of the benefit B and cost C to produce x units of a product when B(0)=0 and C(0)=425. The data that shows the marginal benefit and marginal cost for the number of unit’s produce is tabulated below.

Number of units 1 2 3 4 5 6 7 8 9 10
Marginal benefits 330 320 290 270 250 230 210 190 170 160
Marginal cost 150 120 100 110 120 140 160 190 250 320

(c)

To determine

To calculate: The intervals for which the benefit produces by the company exceed the cost to produce x units and also estimate the number of units the company should produce. The data that shows the marginal benefit and marginal cost for the number of unit’s produce is tabulated below.

Number of units 1 2 3 4 5 6 7 8 9 10
Marginal benefits 330 320 290 270 250 230 210 190 170 160
Marginal cost 150 120 100 110 120 140 160 190 250 320

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