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Chapter 4: Problem 51

Linear approximation a. Write an equation of the line that represents the linear approximation to the following functions at a. b. Graph the function and the linear approximation at a. c. Use the linear approximation to estimate the given quantity. d. Compute the percent error in your approximation. $$f(x)=1 /(x+1) ; a=0 ; 1 / 1.1$$

Short Answer

Expert verified
Based on the given function, \(f(x) = \frac{1}{x + 1}\), we found out its linear approximation at a=0 to be \(L(x) = -x + 1\). Using the linear approximation, we estimated the value of \(\frac{1}{1.1}\) to be approximately 0.9. Comparing this to the exact value of 0.909090909..., we calculated the percent error to be approximately -1.00%. So, the linear approximation has an error of approximately -1.00%.

Step by step solution

01

Find the equation of the line for the linear approximation

To find the linear approximation, we need to find the slope and a point that the line must pass through. The point will be \((a, f(a))\) and the slope will be the derivative of the function evaluated at point 'a'. First, find the derivative of the function: $$f'(x) = -\frac{1}{(x+1)^2}$$ Now, evaluate the derivative at a=0 to find the slope: $$f'(0) = -\frac{1}{(0+1)^2} = -1$$ The point the line must pass through is \((a, f(a)) = (0, f(0)) = (0, 1/(0+1)) = (0, 1)\). Using the slope-point form of a linear equation, our linear approximation is: $$L(x) = f'(0)(x-0) + f(0) = -x + 1$$
02

Graph the function and the linear approximation

Unfortunately, as a text-based AI, I cannot graph the function and its linear approximation directly. But you can use a graphing calculator or any graphing software online (such as https://www.desmos.com/calculator) to graph the functions. Plot the given function, $$f(x) = \frac{1}{x+1}$$, and its linear approximation, $$L(x) = -x + 1$$. You will see that the line \(L(x)\) approximates the function \(f(x)\) near the point \((0, 1)\).
03

Use the linear approximation to estimate the given quantity

We are asked to approximate the value of $$\frac{1}{1.1}$$ using our linear approximation. To do this, we find the value of \(L(x)\) at \(x=0.1\): $$L(0.1) = -0.1 + 1 = 0.9$$ So, our linear approximation estimates that $$\frac{1}{1.1} \approx 0.9$$.
04

Compute the percent error in the approximation

To find the percent error, we compare our linear approximation to the exact value of the given quantity. First, calculate the exact value of $$\frac{1}{1.1}$$: $$\frac{1}{1.1} = 0.909090909...$$ Now, compute the percent error: $$\text{Percent Error} = \frac{\text{Approximate Value} - \text{Exact Value}}{\text{Exact Value}} \times 100\%$$ $$\text{Percent Error} = \frac{0.9 - 0.909090909...}{0.909090909...} \times 100\%$$ $$\text{Percent Error} \approx -1.00\%$$ Therefore, our linear approximation has an error of approximately -1.00%.

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Key Concepts

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

Derivative Calculation
Calculating the derivative is an essential step in finding a linear approximation. A derivative represents the slope of a function at any given point, showing us how the function is changing at that point.
To find the linear approximation, we first determine the derivative of the original function. In our example, given the function \( f(x) = \frac{1}{x+1} \), we need its derivative to know how it behaves at specific points.
  • The formula for the derivative is calculated using differentiation rules. Here, it results in: \( f'(x) = -\frac{1}{(x+1)^2} \).
  • With this derivative, the slope at any point 'a' can be determined by evaluating \( f'(a) \).
  • For this exercise, to find the slope at \( a = 0 \), evaluate \( f'(0) = -1 \).
This gives us a slope of -1 at \( x = 0 \), making it possible to form the linear equation with the function's value at that point.
Percent Error
Percent error is a helpful measure to understand how accurate an estimation or approximation is compared to the exact value. It allows us to quantify the difference between an approximate and an exact value in percentage terms.
Here鈥檚 how you can compute the percent error in our linear approximation:
  • The exact value of \( \frac{1}{1.1} \) is approximately 0.9091 when calculated more accurately using decimal division.
  • Our linear approximation gives us the value 0.9 for the same expression.
  • The formula to compute percent error is: distinct_percent_error_formula\[ \text{Percent Error} = \frac{\text{Approximate Value} - \text{Exact Value}}{\text{Exact Value}} \times 100\% \].
Using it, substitute the values:
  • Approximate Value = 0.9
  • Exact Value = 0.9091
  • This yields a percent error of approximately -1.00\%.
This tells us our approximation was slightly lower than the exact value, and thus we had a small negative error.
Function Graphing
Graphing functions and their linear approximations provides a visual representation of how close the approximation is to the actual function. It's an excellent way to see how well our linear model fits, especially around the point of tangency.
Let's break down the graphing process:
  • Start with the original function, \( f(x) = \frac{1}{x+1} \). This is a hyperbolic curve.
  • The linear approximation \( L(x) = -x + 1 \) is a straight line which approximates \( f(x) \) near the point \((0, 1)\).
  • The point of intersection or tangency is where both the function and the linear approximation are exactly equal at both the value and slope, that is at \( x = 0 \).
When you graph these two, you should see that the line hugs the curve closely near \( x = 0 \). With graphing tools like Desmos or a graphing calculator, it's clear to see where our linear approximation aligns well and where it might diverge鈥攖ypically further away from the point of tangency, the approximation becomes less accurate.

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