{"id":40234,"date":"2025-06-27T14:50:04","date_gmt":"2025-06-27T14:50:04","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=40234"},"modified":"2025-06-27T14:50:06","modified_gmt":"2025-06-27T14:50:06","slug":"the-limit-below-represents-a-derivative-f","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/the-limit-below-represents-a-derivative-f\/","title":{"rendered":"The limit below represents a derivative f’"},"content":{"rendered":"\n

The limit below represents a derivative f’ (a). Find f(x) and a (2 + h)4 16 lim h-0 h f(e<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

The given expression is:<\/p>\n\n\n\n

lim\u2061h\u21920(2+h)4\u221216h\\lim_{{h \\to 0}} \\frac{{(2 + h)^4 – 16}}{h}<\/p>\n\n\n\n

This matches the formal definition of the derivative of a function f(x)f(x) at a point aa:<\/p>\n\n\n\n

f\u2032(a)=lim\u2061h\u21920f(a+h)\u2212f(a)hf'(a) = \\lim_{{h \\to 0}} \\frac{{f(a + h) – f(a)}}{h}<\/p>\n\n\n\n

To identify f(x)f(x) and aa, observe the structure of the numerator:<\/p>\n\n\n\n

(2+h)4\u221216(2 + h)^4 – 16<\/p>\n\n\n\n

The expression (2+h)4(2 + h)^4 corresponds to f(a+h)f(a + h), and the constant term 16 represents f(a)f(a). Therefore, we deduce that the function evaluated at aa yields:<\/p>\n\n\n\n

f(a)=16f(a) = 16<\/p>\n\n\n\n

Looking at (2+h)4(2 + h)^4, it appears to be the function f(x)=x4f(x) = x^4, shifted to center at x=2x = 2. If we assume f(x)=x4f(x) = x^4, then:<\/p>\n\n\n\n

f(2)=24=16f(2) = 2^4 = 16<\/p>\n\n\n\n

This confirms our assumption. So, the function is f(x)=x4f(x) = x^4 and the point aa is 2.<\/p>\n\n\n\n

Final Answer:<\/strong><\/p>\n\n\n\n

f(x)=x4,a=2f(x) = x^4, \\quad a = 2<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

This problem exemplifies the definition of a derivative at a point. The general form measures the instantaneous rate of change of the function at a certain input. By comparing the structure of the given limit to the standard derivative form, we identify both the function rule and the specific input.<\/p>\n\n\n\n

In our case, f(a+h)=(2+h)4f(a + h) = (2 + h)^4 implies f(x)=x4f(x) = x^4, and since f(2)=16f(2) = 16, it verifies that a=2a = 2. The numerator represents the change in the function\u2019s output as the input varies slightly from 2 to 2+h2 + h, and dividing by hh yields the average rate of change, which becomes the derivative as hh approaches zero.<\/p>\n\n\n\n

This limit thus represents the derivative of f(x)=x4f(x) = x^4 evaluated at x=2x = 2, which is f\u2032(2)=4×3=32f'(2) = 4x^3 = 32, though the question only asks for f(x)f(x) and aa.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

The limit below represents a derivative f’ (a). Find f(x) and a (2 + h)4 16 lim h-0 h f(e The Correct Answer and Explanation is: The given expression is: lim\u2061h\u21920(2+h)4\u221216h\\lim_{{h \\to 0}} \\frac{{(2 + h)^4 – 16}}{h} This matches the formal definition of the derivative of a function f(x)f(x) at a point aa: f\u2032(a)=lim\u2061h\u21920f(a+h)\u2212f(a)hf'(a) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-40234","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40234","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=40234"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40234\/revisions"}],"predecessor-version":[{"id":40237,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40234\/revisions\/40237"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=40234"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=40234"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=40234"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}