Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …
Question
**Problem Statement:** Find \( f \) if \( f'(x) = 12x^3 + 14x + 2 \) and \( f(1) = 2 \). \[ f(x) = \] **Explanation:** This problem asks for the original function \( f(x) \), given its derivative \( f'(x) = 12x^3 + 14x + 2 \) and the condition that \( f(1) = 2 \). The task involves integrating \( f'(x) \) to determine \( f(x) \) and using the initial condition \( f(1) = 2 \) to find the constant of integration.
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Transcribed Image Text:**Problem Statement:** Find \( f \) if \( f'(x) = 12x^3 + 14x + 2 \) and \( f(1) = 2 \). \[ f(x) = \] **Explanation:** This problem asks for the original function \( f(x) \), given its derivative \( f'(x) = 12x^3 + 14x + 2 \) and the condition that \( f(1) = 2 \). The task involves integrating \( f'(x) \) to determine \( f(x) \) and using the initial condition \( f(1) = 2 \) to find the constant of integration.
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