Problem 12 Find a linear approximation for ... [FREE SOLUTION]

Chapter 3: Problem 12

Find a linear approximation for $f(b)$ if the independent variable changes from $a$ to $b$. $$ f(x)=-3 x^{3}+8 x-7 ; \quad a=4, \quad b=3.96 $$

Short Answer

Expert verified

The linear approximation for $f(3.96)$ is approximately -161.56.

Step by step solution

Identify the Function and Points

We need to approximate the value of the function $f(x) = -3x^3 + 8x - 7$ at $b = 3.96$ by using a linear approximation around $a = 4$.

Calculate the Derivative

To find the linear approximation, we need the derivative of $f(x)$, which gives us the slope of the tangent line. Calculating the derivative: \[f'(x) = \frac{d}{dx}(-3x^3 + 8x - 7) = -9x^2 + 8.\]

Evaluate the Function and Derivative at $a$

Calculate $f(a)$ and $f'(a)$: - $f(4) = -3(4)^3 + 8(4) - 7 = -192 + 32 - 7 = -167$.- $f'(4) = -9(4)^2 + 8 = -144 + 8 = -136$.

Use Linear Approximation Formula

The linear approximation formula is $L(x) = f(a) + f'(a)(x-a)$. Substitute the values: \[L(x) = -167 + (-136)(x - 4).\]

Calculate $L(b)$

Now calculate $L(3.96)$ using the linear approximation: \[L(3.96) = -167 + (-136)(3.96 - 4) = -167 + (-136)(-0.04).\]\[L(3.96) = -167 + 5.44 = -161.56.\]

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Key Concepts

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

Derivative

The derivative of a function is a critical concept in calculus. It is essentially the rate at which the function's value changes as its input changes. For the function of interest, which is $ f(x) = -3x^3 + 8x - 7 $, the derivative is calculated as $ f'(x) = \frac{d}{dx}(-3x^3 + 8x - 7) = -9x^2 + 8 $. Here鈥檚 a breakdown of why this is important:

The derivative $ f'(x) $ tells us the slope of the tangent line to the curve at any point $ x $.
This slope represents how fast and in what direction the function value is changing at that particular point.
Having $ f'(x) $ allows us to predict changes in the function鈥檚 output around different values of $ x $.

Understanding the derivative helps us find the tangent line, which is central to creating a linear approximation.

Tangent Line

The tangent line at a particular point on a curve gives us a linear snapshot that approximates the curve near that point. In practice, with the function $ f(x) = -3x^3 + 8x - 7 $, the tangent line at $ x = 4 $ can be determined using the derivative $ f'(4) = -136 $ and the point $ f(4) = -167 $.

The equation of the tangent line is derived from the point-slope form: $ y = f(a) + f'(a)(x - a) $.
This line is essentially the function's linear approximation around $ a $.
For our function, the tangent line at $ x = 4 $ can be expressed as $ y = -167 - 136(x - 4) $.

The tangent line simplifies complex, non-linear behaviors within a small interval and is crucial in linear approximation techniques.

Function Evaluation

Evaluating a function involves substituting a particular value of $ x $ into $ f(x) $. This evaluation helps us understand the specific behavior of the function at a given point. In the context of linear approximation, we evaluate both the function and its derivative at $ x = a $.

First, we find $ f(a) $, the function value at $ a $, providing a "baseline" from which we approximate future values.
Similarly, $ f'(a) $ gives us the slope of the tangent line which helps to estimate $ f(b) $.

For $ f(x) = -3x^3 + 8x - 7 $, evaluating at $ a = 4 $, we get $ f(4) = -167 $ and $ f'(4) = -136 $. These evaluations form the foundation for constructing the approximation formula.

Approximation Formula

The approximation formula allows us to estimate values of a function near a known point using its tangent line. For small changes in $ x $ close to $ a $, this formula provides an accurate estimate of the function's value. The linear approximation formula is given by:\[ L(x) = f(a) + f'(a)(x - a) \]

$ L(x) $ represents the estimated value of the function at $ x = b $.
It utilizes both $ f(a) $ and the tangent line slope $ f'(a) $ to give the best linear estimate.
In our example, substituting $ a = 4 $, we find $ L(3.96) = -167 + 5.44 = -161.56 $.

The formula is simple yet powerful, enabling predictions and simplifications in various scientific and engineering applications.

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91影视

Find a linear approximation for \(f(b)\) if the independent variable changes from \(a\) to \(b\). $$ f(x)=-3 x^{3}+8 x-7 ; \quad a=4, \quad b=3.96 $$

Short Answer

Step by step solution

Identify the Function and Points

Calculate the Derivative

Evaluate the Function and Derivative at \(a\)

Use Linear Approximation Formula

Calculate \(L(b)\)

Key Concepts

Derivative

Tangent Line

Function Evaluation

Approximation Formula

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