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Chapter 4: Problem 13

Estimate \(f(2.8)\) given that \(f(3)=2\) and \(f^{\prime}(x)=\left(x^{2}+5\right)^{1.5}\)

Short Answer

Expert verified
After calculating the right side of the final equation in step 3, the approximate value of \(f(2.8)\) will be obtained.

Step by step solution

01

Identify Known Values

We know that \(f(3) = 2\), \(f'(x) = (x^2 + 5)^{1.5}\), and we're trying to find \(f(2.8)\). Here, \(a = 3\), \(f(a) = 2\) and \(a+h = 2.8\). So, \(h = 2.8 - 3 = -0.2\).
02

Find the Derivative at Point a

Calculate \(f'(3)\) using the given derivative function \(f'(x) = (x^2 + 5)^{1.5}\). Hence, \(f'(3) = (3^2 + 5)^{1.5} = (9 + 5)^{1.5} = (14)^{1.5}\).
03

Use the Linear Approximation Formula

Now, use all of these values in the linear approximation formula \(f(a+h) \approx f(a) + hf'(a)\). Here, it becomes \(f(2.8) \approx 2 - 0.2 * (14)^{1.5}\). Calculate the right side of the equation to obtain the approximation value.

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Key Concepts

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

Derivative Calculation
Understanding how derivatives work is key to solving many mathematical problems, especially those related to rates of change and approximation. In this context, the derivative function tells us how fast something is changing at any given point along a curve. This slope can be used to make predictions about the function around that point.
Here, we have a derivative function given by:
  • \( f'(x) = (x^2 + 5)^{1.5} \)
To find the derivative at a specific point, say \( x=3 \), we substitute 3 into this equation:
  • Find \( f'(3) = ((3)^2 + 5)^{1.5} \)
  • This simplifies to \( (9 + 5)^{1.5} = 14^{1.5} \)
Computing \( 14^{1.5} \) requires knowledge of exponents, and the result is crucial for determining the slope at the given point.
Linearization
Linearization is a method used to approximate the value of a function near a given point. It uses the idea that close to this point, the function can be approximated by a straight line 鈥� its tangent line. This involves using the derivative to find the slope of this line.
The formula for linearization, or local linear approximation, is given by:
  • \( f(a+h) \approx f(a) + h \, f'(a) \)
Where:
  • \( f(a) \) is the known function value at \( x=a \)
  • \( h \) is the small step from \( a \) to the desired point
  • \( f'(a) \) is the derivative at \( x=a \)
In our problem, we use \( a = 3 \) and \( h = -0.2 \). This results in:
  • \( f(2.8) \approx 2 + (-0.2) \, f'(3) \)
This linear model allows us to estimate \( f(2.8) \) despite not having the exact function form.
Approximation Methods
Approximation methods are essential in mathematics and science because they allow us to estimate unknown values or solve complex equations using simpler, solvable forms. Linear approximation is one such method, particularly useful when we need a quick and reasonably accurate estimate of a function's value.
To successfully use linear approximation:
  • Identify a point where you know both the function and its derivative.
  • Calculate the derivative at your chosen point.
  • Choose a step size \( h \) that is relatively small, so the linear approximation remains accurate. Smaller steps generally result in better accuracy.
  • Apply the linear approximation formula: \( f(a+h) \approx f(a) + h \, f'(a) \).
In this exercise, \( f(3) = 2 \) is known, and with \( h = -0.2 \), the approximation formula provides \( f(2.8) \) by considering the slope given by the function derivative. Approximations offer insight, bridging gaps where exact solutions may be inaccessible or too cumbersome to find.

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Most popular questions from this chapter

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