a. Explanation Given: The function f ( x , y ) = x 2 sin ( x + y ) , a point P 0 ( π 2 , π 2 , 0 ) and the direction parallel to the x z - plane . Concept Used: For a function that defines a surface z = f ( x , y ) differentiated at a point ( x 0 , y 0 ) . The equation of the tangent plane to the surface z = f ( x , y ) at a point ( x 0 , y 0 ) is given by z = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x − x 0 ) + f y ( x 0 , y 0 ) ( y − y 0 ) Calculation: For the givensurface f ( x , y ) = x 2 sin ( x + y ) and a point P 0 ( π 2 , π 2 , 0 ) Consider f ( x , y ) = x 2 sin ( x + y ) Calculating the partial derivatives f x ( x , y ) = x 2 cos ( x + y ) ( 1 ) + 2 x sin ( x + y ) = x 2 cos ( x + y ) + 2 x sin ( x + y ) and f y ( x , y ) = x 2 cos ( x + y ) ( 1 ) = x 2 cos ( x + y ) At point P 0 ( π 2 , π 2 , 0 ) f ( π 2 , π 2 ) = ( π 2 ) 2 sin ( π 2 + π 2 ) = π 2 4 sin ( π ) = π 2 4 ( 0 ) = 0 f x ( π 2 , π 2 ) = ( π 2 ) 2 cos ( π 2 + π 2 ) + 2 ( π 2 ) sin ( π 2 + π 2 ) = π 2 4 cos ( π ) + π sin ( π ) = b. To determine To find: The slope of the tangent line to the graph of f ( x , y ) = x 2 sin ( x + y ) at point P 0 ( π 2 , π 2 , 0 ) in the direction parallel to the y z - plane .

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

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The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 11.3, Problem 31PS

To determine

To find:The slope of the tangent line to the graph of f(x,y)=x2sin(x+y) at point P0(π2,π2,0) in the direction parallel to the xz-plane .

To determine

To find:The slope of the tangent line to the graph of f(x,y)=x2sin(x+y) at point P0(π2,π2,0) in the direction parallel to the yz-plane .

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Chapter 11.3, Problem 30PS

Chapter 11.3, Problem 32PS

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The slope of the tangent line to the graph of f ( x , y ) = x 2 sin ( x + y ) at point P 0 ( π 2 , π 2 , 0 ) in the direction parallel to the x z - plane .

Solution Summary: The author calculates the slope of the tangent line to the graph of f(x,y)=x2mathrm