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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 7.4, Problem 56E
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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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Finding and Evaluating Second Partial Derivatives In Exercises 53-56, find the four second partial derivatives and evaluate each at the given point.

f ( x , y ) = ln ( x − y ) ;   ( 0 , − 2 )

To determine

To calculate: The four second partial derivatives and four second partial derivatives at the point (0,−2) for the function f(x,y)=ln(x−y).

Explanation

Given information:

The provided function is f(x,y)=ln(x−y).

Formula used:

Consider the function z=f(x,y) then for the value of ∂z∂x consider y to be constant and differentiate with respect to x and the value of ∂z∂y consider x to be constant and differentiate with respect to y.

According to Higher-Order Partial Derivatives,

∂∂x(∂f∂x)=∂2f∂x2=fxx∂∂y(∂f∂y)=∂2f∂y2=fyy∂∂y(∂f∂x)=∂2f∂y∂x=fxy∂∂x(∂f∂y)=∂2f∂x∂y=fyx

Calculation:

Consider the provided function is,

f(x,y)=ln(x−y)

Partially derivative of the function f(x,y)=ln(x−y) with respect to x.

fx(x,y)=∂∂x(ln(x−y))=1(x−y)⋅∂∂x(x−y)=1(x−y)⋅(1)=1(x−y)

Partially derivative of the function f(x,y)=ln(x−y) with respect to y.

fy(x,y)=∂∂y(ln(x−y))=1(x−y)⋅∂∂y(x−y)=1(x−y)⋅(−1)=−1(x−y)

Again, partially derivative of the function fx(x,y)=1(x−y) with respect to x.

fxx(x,y)=∂∂x(1(x−y))=−1(x−y)2⋅∂∂x(x−y)=−1(x−y)2

Substitute (x,y)=(0,−2) for the derivative fxx(x,y)=−1(x−y)2.

fxx(0,−2)=−1(0−(−2))2=−1(2)2=−14

Again, partially derivative of the function fx(x,y)=1(x−y) with respect to y.

fxy(x,y)=∂∂y(1(x−y))=−1(x−y)2⋅∂∂y(x−y)=−1(x−y)2⋅(−1)=1(x−y)2

Substitute (x,y)=(0,−2) for the derivative fxy(x,y)=1(x−y)2

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Sect-7.1 P-1CPSect-7.1 P-2CPSect-7.1 P-3CPSect-7.1 P-4CPSect-7.1 P-5CPSect-7.1 P-6CPSect-7.1 P-7CPSect-7.1 P-1SWUSect-7.1 P-2SWUSect-7.1 P-3SWU
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(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with a...

Multivariable Calculus

Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = 2cos t + sin 2t, [0, /...

Calculus: Early Transcendentals

Evaluating Trigonometric FunctionsIn Exercises 17-20, sketch a right triangle corresponding to the trigonometri...

Calculus: Early Transcendental Functions

Draw a polygon for the distribution of scores shown in the following table. X f 6 2 5 5 4 3 3 2 2 1

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

SOC Twenty-five students completed a questionnaire that measured their attitudes toward interpersonal violence....

Essentials Of Statistics

Solve the initial-value problem. y2y3y=0,y(0)=2,y(0)=2

Calculus (MindTap Course List)

let f(x) = x3 + 5, g(x) = x2 2, and h(x)= 2x + 4. Find the rule for each function. 5. fg

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Sketch the graph of the function. 50. g(x) = ||x| 1|

Single Variable Calculus

Describe how the presence of possible outliers might be identified on (a) histograms. (b) dot plots. (c) stem-a...

Understanding Basic Statistics

Rewrite each expression in Exercises 116 as a single rational expression, simplified as much as possible. 1(x+y...

Finite Mathematics

Schmidt (1994) conducted a series of experiments examining the effects of humor on memory. In one study, partic...

Statistics for The Behavioral Sciences (MindTap Course List)

Counterexamples Find a counterexample to prove that the inductive argument in a. Exercise 31 is incorrect. b. E...

Mathematical Excursions (MindTap Course List)

For Problems 1-10, identify each statement as true or false. Objective 1 Every rational number is a real number...

Intermediate Algebra

Some parts of California are particularly earthquake-prone. Suppose that in one metropolitan area. 25% of all h...

Probability and Statistics for Engineering and the Sciences

In problems 23-58, perform the indicated operations and simplify. 50.

Mathematical Applications for the Management, Life, and Social Sciences

A canister is dropped from a helicopter 500 m above the ground. Its parachute does not open, but the canister h...

Single Variable Calculus: Early Transcendentals

For the following second mortgage applications, calculate the percentage of appraised value and the potential c...

Contemporary Mathematics for Business & Consumers

Sketch the graph of each equation from x=0 to x=4. y=3sinx+cos2x

Trigonometry (MindTap Course List)

True or False Label each of the following statements as either true or false. 8. Any group of order must be cy...

Elements Of Modern Algebra

In Review Exercises 1 to 6, use the figure shown. Construct an isosceles triangle with vertex angle B and an al...

Elementary Geometry For College Students, 7e

Solve each equation: 8x9=56

Elementary Technical Mathematics

Using a Function In Exercises 3742, consider the function f(x,y)=3x3y2. Find Duf(3, 2), where u=vv, using each ...

Calculus: Early Transcendental Functions (MindTap Course List)

Finding a Limit In Exercises 39-50, find the limit (if it exists). If it does not exist, explain why. limx2f(x)...

Calculus of a Single Variable

In Exercises 34-44, use your calculator value of unless otherwise stated. Round answers to two decimal places....

Elementary Geometry for College Students

In Exercise 1731, solve each linear programming problem by the simplex method. Maximize P=x+2yz. Subject to 2x+...

Finite Mathematics for the Managerial, Life, and Social Sciences

Finding and Analyzing Derivatives Using Technology In Exercises 49-54, (a) use a computer algebra system to dif...

Calculus

The third partial sum of is:

Study Guide for Stewart's Multivariable Calculus, 8th

Let limxaf(x)=L. For 0 pictured at the right, what is the best choice for in the limit definition? a) p b) q c...

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Limit and Continuity In Exercises 11-24, find the limit and discuss the continuity of the function. lim(x,y)(2,...

Multivariable Calculus

Evaluating a Definite Integral In Exercises 936, evaluate the definite integral. Use a graphing utility to veri...

Calculus (MindTap Course List)

Define counterbalancing and explain how it is used to minimize or eliminate threats to internal validity from t...

Research Methods for the Behavioral Sciences (MindTap Course List)

Compute 8516453.

Mathematics For Machine Technology

Critical Thinking Determine if the statement is true or false. If the statement is false, then correct it and m...

College Algebra (MindTap Course List)

A partial computer output from a regression analysis follows. a. Compute the missing entries in this output. b....

Statistics for Business & Economics, Revised (MindTap Course List)

Find your colleges IRB guidelines and procedures (you probably will need to search online through your colleges...

Research Methods for the Behavioral Sciences (MindTap Course List)

The report Findings From the 2014 College Senior Survey (Higher Education Research Institute, December 2014) su...

Introduction To Statistics And Data Analysis

On a syndicated television show the two hosts often create the impression that they strongly disagree about whi...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

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