{"id":253471,"date":"2025-07-12T05:23:29","date_gmt":"2025-07-12T05:23:29","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=253471"},"modified":"2025-07-12T05:23:31","modified_gmt":"2025-07-12T05:23:31","slug":"the-graph-of-the-curve-shown-is-the-graph-of-the-implicit-equation","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/12\/the-graph-of-the-curve-shown-is-the-graph-of-the-implicit-equation\/","title":{"rendered":"The graph of the curve shown is the graph of the implicit equation"},"content":{"rendered":"\n
The graph of the curve shown is the graph of the implicit equation<\/pre>\n\n\n\n
\"\"<\/figure>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
Here are the solutions to the problems shown in the image.<\/p>\n\n\n\n
Problem 1:<\/strong>
The question asks for the y-coordinate of the points on the curve x\u00b2 – 20x + 8y\u00b2 – 48y = 250 where the tangent line is vertical. A vertical tangent line occurs where the slope, dy\/dx, is undefined. This typically happens when the denominator of the derivative expression is zero.<\/p>\n\n\n\n
To find the derivative, we use implicit differentiation on the given equation with respect to x:
d\/dx (x\u00b2 – 20x + 8y\u00b2 – 48y) = d\/dx (250)<\/p>\n\n\n\n
Differentiating each term gives us:
2x – 20 + 16y(dy\/dx) – 48(dy\/dx) = 0<\/p>\n\n\n\n
Now, we solve for dy\/dx. First, we group the terms containing dy\/dx:
16y(dy\/dx) – 48(dy\/dx) = 20 – 2x<\/p>\n\n\n\n
Factor out dy\/dx:
(16y – 48) dy\/dx = 20 – 2x<\/p>\n\n\n\n
Isolate dy\/dx by dividing:
dy\/dx = (20 – 2x) \/ (16y – 48)<\/p>\n\n\n\n
The slope dy\/dx is undefined when the denominator is zero. We set the denominator equal to zero and solve for the y-coordinate:
16y – 48 = 0
16y = 48
y = 48 \/ 16
y = 3<\/p>\n\n\n\n
Answer:<\/strong> 3<\/p>\n\n\n\n
\n\n\n\n
Problem 2:<\/strong>
This question asks for the instantaneous rate of change of y = ln(4x\u00b2 + 3) when x = 3. The instantaneous rate of change is found by calculating the derivative of the function, dy\/dx, and then evaluating it at the specified x-value.<\/p>\n\n\n\n
We need to differentiate y = ln(4x\u00b2 + 3) using the chain rule. The derivative of a natural logarithm function, ln(u), is u’\/u, where u is the inner function.
Here, the inner function is u = 4x\u00b2 + 3.
The derivative of the inner function is u’ = d\/dx(4x\u00b2 + 3) = 8x.<\/p>\n\n\n\n
Applying the chain rule, the derivative of y is:
dy\/dx = u’\/u = 8x \/ (4x\u00b2 + 3)<\/p>\n\n\n\n
Next, we evaluate this derivative at x = 3:
dy\/dx |_(x=3) = (8 * 3) \/ (4 * (3)\u00b2 + 3)
= 24 \/ (4 * 9 + 3)
= 24 \/ (36 + 3)
= 24 \/ 39<\/p>\n\n\n\n
The question requires the answer to be a simplified fraction. Both the numerator and the denominator are divisible by 3:
24 \u00f7 3 = 8
39 \u00f7 3 = 13
The simplified fraction is 8\/13.<\/p>\n\n\n\n
Answer:<\/strong> 8\/13<\/p>\n\n\n\n
\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
The graph of the curve shown is the graph of the implicit equation The Correct Answer and Explanation is: Here are the solutions to the problems shown in the image. Problem 1:The question asks for the y-coordinate of the points on the curve x\u00b2 – 20x + 8y\u00b2 – 48y = 250 where the tangent […]<\/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-253471","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/253471","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=253471"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/253471\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=253471"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=253471"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=253471"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}