TOTAL DIFFERENTIALS. In Section B-4, we observed that.c is a pretty good approximation for? C, which is the increment of

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1 TOTAL DIFFERENTIALS The differential was introduced in Section -4. Recall that the differential of a function œ0ðñis w. œ 0 ÐÑ.Þ Here. is the differential with respect to the independent variable and is equal to?ß the increment of Þ In Section -4, we observed that. is a pretty good approximation for?, which is the increment of, and defined as? œ 0Ð? Ñ 0ÐÑÞ This differential. approximates? with respect to one variable. Now suppose D œ 0Ðß Ñ is a function of two variables. What is a good approximation for? D if changes from to? and changes from to?? TOTAL DIFFERENTIAL Definition: Let D œ 0Ðß Ñ be a function of and. Let. and. be real numbers. Then the total differential of D is.d œ 0ÐßÑ. 0ÐßÑ. EXAMPLE 1 À $ $ onsider Dœ0ÐßÑœ"! ) % Þ (a) Find the total differential.dþ (b) Evaluate.D when œ ß œ "ß. œ Þ!"ß and. œ Þ!Þ (a) First we find 0 Ðß Ñ and 0 Ðß ÑÞ 0 Ðß Ñ œ $! "' and 0 Ðß Ñ œ ) " So by definition,.d œ a$! "' b. a ) " b. (b) Putting in the values into the result of (a) we get.d œ Ð))ÑÐÞ!"Ñ Ð!ÑÐ Þ!Ñ œ"þ) The result here tells us that if you increase by Þ!" and decrease by.02, D œ 0Ðß Ñ will increase by approximately Just as in the one variable case, we can use.d as an approximation for? DÞ We define the increment of the dependent variable D as? Dœ0Ð? ß? Ñ 0ÐßÑœ0Ð.ß.Ñ 0ÐßÑ

2 MHIE À For relatively small values of? and?,.d? D To see this, consider the following: È EXAMPLE 2: Let Dœ0ÐßÑœ (a) Use the total differential.d to approximate? D when changes from $ to Þ*) and changes from % to %Þ!"Þ (b) alculate the actual change?d. (a) First we find.d À.D œ 0 Ðß Ñ. 0 Ðß Ñ. œ Š. Š. È Note that. œ? œ Þ*) $ œ Þ! and. œ? œ %Þ!" % œ Þ!"Þ So our approximation is Ð$Ñ È$ % Ð%Ñ È$ %.D œ Š a Þ! b Š aþ!" b œ!þ!!) (b) To calculate the actual change, we find? Dœ0Ð? ß? Ñ 0ÐßÑ: È?D œ 0ÐÞ*)ß %Þ!"Ñ 0Ð$ß %Ñ œ ÈÞ*) %Þ!" È$ % œ!þ!!(*!$"$ So our approximation for?d is pretty good. Now since 0Ð.ß.Ñ 0ÐßÑœ? D.D œ0ðßñ. 0ÐßÑ.Þ 0Ð.ß.Ñ 0ÐßÑ.D Then "Î$ EXAMPLE 3: Approximate a"þ* Þ" b First we note that Dœ0ÐßÑœÈ$ œa b "Î$ ÞWe start with œßœß. œ Þ!) and. œ!þ"þ Now we calculate.dþ.d œ a b. a b. Evaluating at.d at the above values yields Þ $ $.D œ $ a b ÐÑÐÑÐ!Þ!)Ñ $ a b ÐÑÐÑÐ!Þ"Ñ œ!þ!!'( So we approximate 0Ð"Þ*ß Þ"Ñ 0Ðß Ñ.D œ!þ!!'( œ Þ!!'( Definition À If Aœ0ÐßßDÑ is a function of three variables, the total differential.ais.aœ0 ÐßßDÑ. 0 ÐßßDÑ. 0 ÐßßDÑ.D D

3 APPLIATION EXAMPLE 4: Approximate the change in the volume of a beverage can in the shape of a right circular cylinder as the radius changes from 3 to 2.5 and the height changes from 14 to We note the volume of the can is Z œ Z Ð<ß 2Ñ œ 1 < 2Þ We now calculate.z À 1 < 2Þ.Z œ Z< Ð<ß 2Ñ.< Z2Ð<ß 2Ñ.2 œ 1<2.< 1 <.2 So our function is in two variables, i.e. Given that < œ $ß 2 œ "%ß.< œ!þ&ß and.2 œ!þß we get.z œ 1Ð$ÑÐ"%ÑÐ!Þ&Ñ 1 Ð$ ÑÐ!ÞÑ "'Þ*! Thus by decreasing the radius by 0.5 and increasing the height by 0.2, the volume of the can decreases by approximately "'Þ*! cubic units. PRATIE If D œ 0ÐßÑ or A œ 0ÐßßDÑ, 0 ind the total differential.d or.aof the following functions: 1. 0Ðß Ñ œ $ & ' Þ 0Ðß Ñ œ È $ 3. 0ÐßÑ œ lna b $ ( 4. 0ÐßÑ œ / $ 5. 0ÐßÑ œ / ln $ 6. 0ÐßßDÑ œ D 7. 0ÐßßDÑ œ D D 8. 0Ðß ß DÑ œ / D Find the approximate change in D when the point Ðß Ñ changes from Ð! ß! Ñ to Ðß ÑÞ 9. 0Ðß Ñ œ & à from Ðß $Ñ to ÐÞ!"ß $Þ!Ñ "Î Î$ 10. 0Ðß Ñ œ 2 à from Ð%ß )Ñ to Ð$Þ*)ß )Þ!$Ñ

4 11. 0Ðß Ñ œ È $ à from Ð$ß $Ñ to Ð$Þ!"ß Þ)*Ñ "Þ 0Ðß Ñ œ à from Ð ß "Ñ to Ð Þ!&ß "Þ!&Ñ 13. 0Ðß Ñ œ $/ à from Ð!ß %Ñ to Ð!Þ!$ß %Þ!"Ñ 14. 0Ðß Ñ œ / à from Ð"ß "Ñ to Ð!Þ*)ß!Þ!)Ñ 15. 0Ðß Ñ œ ln ln à from Ð$ß &Ñ to ÐÞ*&ß &Þ!%Ñ 16. Use the total differential to approximate È 'Þ!( (Þ*& Þ 17. The monthly profit (in dollars) at a fancy department store depends on the level of inventory (in thousands of dollars) and the floor space (in thousands of square feet) available for display of the merchandise, as given by the equation T Ðß Ñ œ!þ! "& $* &!ß!!! urrently, the level of inventory is $4,000,000 ( œ %!!!Ñ, and the floor space is 150,000 square feet ( œ "&!Ñ. Find the expected change in monthly profit if management increases the level of inventory by $500,000 and decreases floor space for display of merchandise by 10,000 square feet. 18. The :<9.?->398 function for one country is D œ ß where stands for units of labor and for units of capital. At present, 50 units of labor and 29 units of capital are available. Use differentials to estimate the change in production as the number of units is increased to 52 and capital is decreased to 27 units.!þ'&!þ$&

5 1..D œ Ð' & 'Ñ. Ð &Ñ. Answers 2..D œ Œ $.. È$ Œ È$ 3..D œ Œ.. Œ $ ' $ 4..D œ Š / ". $/. 5..D œ Š / 68. Š /. $ $ 6..A œ D. $ D..D D D " 7..A œ...d Ð Ñ Ð Ñ Ð Ñ D D D 8..A œ Š / D. Š D/ D. Š /.D 9.!Þ"$ 10.! 11.!Þ%& 12.!Þ!!&' 13.!Þ$$ 14. Þ"()%& 15.!Þ!"(!) 16. "!Þ!! 17. an increase of $19,250 per month units

MATH 1301 (College Algebra) - Final Exam Review

MATH 1301 (College Algebra) - Final Exam Review This review is comprehensive but should not be the only material used to study for the final exam. It should not be considered a preview of the final exam.