Z y x 18 z y x 9 1 2 z z 3 z y x 9 1 xy 54 z y x f 29

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Chapter 6 / Exercise 2
Algebra and Trigonometry: Real Mathematics, Real People
Larson
Expert Verified
z y x 18 z y x 9 1 2 z z 3 z y x 9 1 xy 54 z , y , x f ( 29 ( 29 ( 29 3 2 2 2 2 2 2 2 2 2 2 113 4 2 2 2 162xy z 1 9x y z f x,y,z (1 9x y z 12x y z 1 9x y z + = - + - + ( 29 ( 29 3 2 2 2 2 113 3 2 2 2 162xy z 1 3x y z f 1 9x y z - = - + (b) ( 29 z , y , x f 123 ( 29 2 2 2 1 z y x 9 1 yz 3 z , y , x f + = ( 29 ( 29 ( 29 ( 29 ( 29 + - + = + = 2 2 2 2 2 2 2 2 2 2 2 2 2 12 z y x 9 1 yz x 18 y 1 z y x 9 1 z 3 z y x 9 1 yz 3 f z , y , x f ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 2 2 3 12 2 2 2 2 2 2 2 2 2 2 2 2 3z 1 9x y z 18x y z 3z 1 9x y z 3 z 9x y z f x,y,z 1 9x y z 1 9x y z 1 9x y z + - - - = = = + + + ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 123 4 2 2 2 3 1 9x y z 1 27x y z z 9x y z 2 1 9x y z 18x y z f x,y,z 1 9x y z + - - - + = + ( 29 ( 29 2 2 2 2 2 2 4 4 4 2 2 2 4 4 4 123 3 2 2 2 3 f x,y,z 1 9x y z 27x y z 243x y z 36x y z 324x y z 1 9x y z = + - - - + + 242
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Chapter 6 / Exercise 2
Algebra and Trigonometry: Real Mathematics, Real People
Larson
Expert Verified
( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 4 4 4 4 4 4 2 2 2 4 4 4 123 3 3 2 2 2 2 2 2 3 1 54x y z 243x y z 324x y z 3 1 54x y z 81x y z f x,y,z 1 9x y z 1 9x y z - - + - + = = + + 243
EXERCISES: A. In the following exercises, evaluate x u and y u , given: 1. xy y x u 2 2 - = 2. = - y x sinh u 1 3. y 2 cosh x 2 sinh u + = 4. 2 1 xy 4 cos e u - = 5. xy 1 x 4 log u 2 2 + = B. Find the indicated partial derivatives by two methods: (a) by Chain Rule; (b) by Substitution 6. y x 3 y 2 xy x 3 u 2 2 - + - + = ; s 3 r 2 x - = ; s r y + = ; find: r u and s u . 7. y x ln u 2 = ; r 4 se x = ; s 2 re y = ; find: r u and s u . 8. y x e u + = ; s cosh r x = ; r sinh s y 2 = ; find: r u and s u . 9. y cosh x sinh u + = ; r 2 ln s x = ; s 2 r ln y = ; find r u and s u . 10. y xe u - = ; ( 29 rst tan x 1 - = ; rst 2 ln y = ; find: r u , s u and t u . 11. = - x y tan u 1 ; r 5 x = ; r 25 y = ; find dr du 12. 2 2 2 z y x u + + = ; θ = tan x ; θ = cos y ; θ = sin z ; find θ d du . C. In each of the following, find: (a) ( 29 y , x f D 11 ; (b) ( 29 y , x f D 22 ; (c) show that ( 29 ( 29 y , x f D y , x f D 21 12 = 13. ( 29 2 2 x y y x y , x f - = 14. ( 29 2 1 x y 3 cos 2 y , x f - = 15. ( 29 x ye y cos x y , x f - = MODULE VI INDETERMINATE FORMS 244
CONTENTS: Lesson 27: Definitions, Kinds, Theorems L’Hopitals Rule, Indeterminate Forms 0 0 and Lesson 28: Other Indeterminate Forms 0 , - , 0 , 1 and 0 0 OVERVIEW OF THE MODULE: The lessons in this module cover the indeterminate forms, its definitions and kinds, as well as evaluation of its limits using L’Hopitals Rule. This module will test the students’ comprehension of the differentiation formulas as applied to the indeterminate forms and check whether the concepts of differentiation have been fully mastered. GENERAL OBJECTIVES: After completing the module, the students are expected to accomplish the following: define, determine, enumerate the different indeterminate forms of functions; evaluate the indeterminate forms 0 0 and using the theorem L’Hopital’s Rule; apply L’Hopital’s Rule on other indeterminate forms which at first may not be applicable; and master the different differentiation formulas which are needed in the application of the theorem L’Hopital’s Rule.