/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 Find the linearization of $$ ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 69

Find the linearization of $$ f(x)=2-\int_{2}^{x+1} \frac{9}{1+t} d t $$ at \(x=1\)

Short Answer

Expert verified
The linearization of the function at \(x=1\) is \(L(x) = 5 - 3x\).

Step by step solution

01

Simplify the function

To find the linearization of the function, first, simplify it. The function is given as \[ f(x) = 2 - \int_{2}^{x+1} \frac{9}{1+t} dt. \] To differentiate the integral with respect to \(x\), it can be rewritten using the Fundamental Theorem of Calculus, which gives:\[ f(x) = 2 - \left[ G(x+1) - G(2) \right], \]where \( G(t) = \int \frac{9}{1+t} dt \). The derivative of \(f(x)\) then simplifies to:\[ f'(x) = -\frac{9}{1+x+1} = -\frac{9}{2+x}. \]
02

Evaluate the derivative at the point of interest

We need to evaluate \(f'(x)\) at \(x=1\) to find the slope of the tangent line, so:\[ f'(1) = -\frac{9}{2+1} = -\frac{9}{3} = -3. \]
03

Find the function value at the point of interest

To complete the linearization, determine \(f(x)\) at \(x=1\):\[ f(1) = 2 - \int_{2}^{2} \frac{9}{1+t} dt = 2 - 0 = 2. \]Since the integral over an interval with equal bounds is zero, \(f(1) = 2\).
04

Formulate the equation of the tangent line

The linearization \(L(x)\) of the function \(f(x)\) at \(x=1\) is given by the equation of the tangent line:\[ L(x) = f(1) + f'(1)\cdot(x - 1). \]Substitute \(f(1) = 2\) and \(f'(1) = -3\) into the equation:\[ L(x) = 2 - 3(x - 1) = 2 - 3x + 3 = 5 - 3x. \]
05

Conclusion

Thus, the linearization of the function \(f(x)\) at \(x=1\) is \(L(x) = 5 - 3x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a crucial concept in calculus. It connects differentiation and integration, two of the main branches of calculus.
  • This theorem states that, if you have a continuous function on an interval, the integral of its derivative over that interval gives back the original function minus a constant.
  • It effectively allows us to compute the derivative of an integral.
In the context of the exercise, the Fundamental Theorem of Calculus is used to simplify the integral part of the function. It helps transform the problem by making it easier to differentiate the function and continue with the linearization process.
Derivative
The derivative of a function is a measure of how that function changes as its input changes. It can be thought of as the slope of the function at a particular point.
  • Finding the derivative is critical for solving many calculus problems, including linearization.
  • This involves using rules of differentiation which might include product, quotient, and chain rules, depending on the function's form.
For the given problem, we find the derivative of the function after using the Fundamental Theorem of Calculus to handle the integral portion. Calculating the derivative gives us the slope of the tangent line at a specific point, in this case, x = 1. Once we have the derivative, we can proceed to find the linear approximation of the original function.
Tangent Line
The tangent line to a curve at a given point is the straight line that "just touches" the curve at that point. It represents the instantaneous rate of change of the function at that specific point.
  • The equation of a tangent line can be obtained using the point-slope form after determining the derivative at the point of interest.
  • This line provides a linear approximation for the function near that point, which is what we mean by its linearization.
For the exercise above, the linearization is essentially finding this tangent line. By evaluating the function and its derivative at the point where x = 1, we produce the linear approximation given by the tangent line formula: L(x) = f(a) + f'(a)(x - a). In this instance, this results in the equation 5 - 3x.
Integral Simplification
Integral simplification is a process to make an integral more manageable, mainly through the use of calculus laws and rules.
  • Simplifying the integral is often the first step in solving problems involving integration or differentiation of complex expressions.
  • Common techniques include variable substitution, breaking down the expression into simpler parts, and using the Fundamental Theorem of Calculus.
In the exercise, simplifying the integral \( \int_{2}^{x+1} \frac{9}{1+t} dt \) involved applying the Fundamental Theorem directly. This simplification allowed for easier differentiation and is essential for finding the linearization of the function at a specified point. Properly handling the integral result can significantly streamline solving the problem.

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