{"id":183518,"date":"2025-01-16T14:07:55","date_gmt":"2025-01-16T14:07:55","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=183518"},"modified":"2025-01-16T14:07:57","modified_gmt":"2025-01-16T14:07:57","slug":"below-is-a-contour-map-for-a-function-fx-y-that-describes-the-topography-of-a-park","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/16\/below-is-a-contour-map-for-a-function-fx-y-that-describes-the-topography-of-a-park\/","title":{"rendered":"Below is a contour map for a function f(x, y) that describes the topography of a park"},"content":{"rendered":"\n<p>Below is a contour map for a function f(x, y) that describes the topography of a park. (a) Estimate the values of f. f and fy at the point (x, y) = (0.6.0.4). (b) What is the equation for the tangent plane to the graph of f at the point corresponding to (x,y) (0.6,0.4)? (c) Estimate far, fry, fyz and fyy at the point (x, y) = (0.6.0.4).<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/01\/image-274.png\" alt=\"\" class=\"wp-image-183519\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To address the problem involving the contour map of a function ( f(x, y) ) representing the topography of a park, we will estimate the values of ( f ), ( f_x ), and ( f_y ) at the point ( (x, y) = (0.6, 0.4) ), determine the equation of the tangent plane at this point, and estimate the second-order partial derivatives ( f_{xx} ), ( f_{xy} ), ( f_{yx} ), and ( f_{yy} ).<\/p>\n\n\n\n<p><strong>(a) Estimating ( f ), ( f_x ), and ( f_y ) at ( (0.6, 0.4) ):<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Estimating ( f(0.6, 0.4) ):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>From the contour map, locate the contour lines near ( (0.6, 0.4) ). The contour values at ( y = 0.4 ) are approximately 30 at ( x = 0.6 ), 15 at ( x = 0.4 ), and 59 at ( x = 0.8 ). Interpolating these values suggests that ( f(0.6, 0.4) \\approx 30 ).<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Estimating ( f_x(0.6, 0.4) ):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The partial derivative ( f_x ) represents the rate of change of ( f ) with respect to ( x ) at a fixed ( y ). To estimate ( f_x ) at ( (0.6, 0.4) ), consider the change in ( f ) between ( x = 0.4 ) and ( x = 0.8 ) at ( y = 0.4 ):<br>[<br>f_x(0.6, 0.4) \\approx \\frac{f(0.8, 0.4) &#8211; f(0.4, 0.4)}{0.8 &#8211; 0.4} = \\frac{59 &#8211; 15}{0.4} = \\frac{44}{0.4} = 110<br>]<br>Thus, ( f_x(0.6, 0.4) \\approx 110 ).<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Estimating ( f_y(0.6, 0.4) ):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The partial derivative ( f_y ) represents the rate of change of ( f ) with respect to ( y ) at a fixed ( x ). To estimate ( f_y ) at ( (0.6, 0.4) ), consider the change in ( f ) between ( y = 0.2 ) and ( y = 0.6 ) at ( x = 0.6 ):<br>[<br>f_y(0.6, 0.4) \\approx \\frac{f(0.6, 0.6) &#8211; f(0.6, 0.2)}{0.6 &#8211; 0.2} = \\frac{86 &#8211; (-13)}{0.4} = \\frac{99}{0.4} = 247.5<br>]<br>Thus, ( f_y(0.6, 0.4) \\approx 247.5 ).<\/li>\n<\/ul>\n\n\n\n<p><strong>(b) Equation of the Tangent Plane at ( (0.6, 0.4) ):<\/strong><\/p>\n\n\n\n<p>The equation of the tangent plane to the surface ( z = f(x, y) ) at the point ( (x_0, y_0, z_0) ) is given by:<br>[<br>z &#8211; z_0 = f_x(x_0, y_0)(x &#8211; x_0) + f_y(x_0, y_0)(y &#8211; y_0)<br>]<br>Substituting the estimated values:<br>[<br>z &#8211; 30 = 110(x &#8211; 0.6) + 247.5(y &#8211; 0.4)<br>]<br>Simplifying:<br>[<br>z = 30 + 110(x &#8211; 0.6) + 247.5(y &#8211; 0.4)<br>]<br>This is the equation of the tangent plane at ( (0.6, 0.4) ).<\/p>\n\n\n\n<p><strong>(c) Estimating ( f_{xx} ), ( f_{xy} ), ( f_{yx} ), and ( f_{yy} ) at ( (0.6, 0.4) ):<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Estimating ( f_{xx}(0.6, 0.4) ):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The second partial derivative ( f_{xx} ) represents the rate of change of ( f_x ) with respect to ( x ). To estimate ( f_{xx} ) at ( (0.6, 0.4) ), consider the change in ( f_x ) between ( x = 0.4 ) and ( x = 0.8 ) at ( y = 0.4 ):<br>[<br>f_{xx}(0.6, 0.4) \\approx \\frac{f_x(0.8, 0.4) &#8211; f_x(0.4, 0.4)}{0.8 &#8211; 0.4} = \\frac{110 &#8211; 110}{0.4} = 0<br>]<br>Thus, ( f_{xx}(0.6, 0.4) \\approx 0 ).<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Estimating ( f_{xy}(0.6, 0.4) ):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The mixed partial derivative ( f_{xy} ) represents the rate of change of ( f_x ) with respect to ( y ). To estimate ( f_{xy} ) at ( (0.6, 0.4) ), consider the change in ( f_x ) between ( y = 0.2 ) and ( y = 0.6 ) at ( x = 0.6 ):<br>[<br>f_{xy}(0.6, 0.4) \\approx \\frac{f_x(0.6, 0.6) &#8211; f_x(0.6, 0.2)}{0.6 &#8211; 0.2} = \\frac{110 &#8211; 110}{0.4} = 0<br>]<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Below is a contour map for a function f(x, y) that describes the topography of a park. (a) Estimate the values of f. f and fy at the point (x, y) = (0.6.0.4). (b) What is the equation for the tangent plane to the graph of f at the point corresponding to (x,y) (0.6,0.4)? (c) [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-183518","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183518","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=183518"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183518\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=183518"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=183518"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=183518"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}