Problem 80
Question
Consider the following common approximations when \(x\) is near zero. a. Estimate \(f(0.1)\) and give the maximum error in the approximation. b. Estimate \(f(0.2)\) and give the maximum error in the approximation. $$f(x)=\tan ^{-1} x \approx x$$
Step-by-Step Solution
Verified Answer
Question: Estimate the values of the inverse tangent function, \(f(x) = \tan^{-1}(x)\), at \(x=0.1\) and \(x=0.2\) using the approximation \(f(x) \approx x\), and provide the maximum errors in each case.
Answer: The estimated value of \(f(0.1)\) is 0.1 with a maximum error of 0.0003, and the estimated value of \(f(0.2)\) is 0.2 with a maximum error of 0.0026.
1Step 1: Part a: Estimating \(f(0.1)\) and the Maximum Error
Using the approximation provided, when \(x=0.1\), we estimate the value of \(f(x)\) as follows:
$$f(x) \approx x$$ so, $$f(0.1) \approx 0.1$$
To find the maximum error, we must first compute the actual value of \(f(0.1)\) using the \(\tan^{-1}\) function and then compare it with our approximation.
$$f(0.1) = \tan^{-1}(0.1)$$
Using a calculator, we find:
$$\tan^{-1}(0.1) \approx 0.0997$$
Now, let's calculate the maximum error:
$$\text{Error} = |\text{Approximation} - \text{Actual Value}| = |0.1 - 0.0997| = 0.0003$$
Hence, the estimated value of \(f(0.1)\) is 0.1, and the maximum error in the approximation is 0.0003.
2Step 2: Part b: Estimating \(f(0.2)\) and the Maximum Error
Using the approximation provided, when \(x=0.2\), we estimate the value of \(f(x)\) as follows:
$$f(x) \approx x$$ so, $$f(0.2) \approx 0.2$$
To find the maximum error, we must first compute the actual value of \(f(0.2)\) using the \(\tan^{-1}\) function and then compare it with our approximation.
$$f(0.2) = \tan^{-1}(0.2)$$
Using a calculator, we find:
$$\tan^{-1}(0.2) \approx 0.1974$$
Now, let's calculate the maximum error:
$$\text{Error} = |\text{Approximation} - \text{Actual Value}| = |0.2 - 0.1974| = 0.0026$$
Hence, the estimated value of \(f(0.2)\) is 0.2, and the maximum error in the approximation is 0.0026.
Key Concepts
Error EstimationInverse Trigonometric FunctionsApproximation Methods
Error Estimation
Whenever we approximate functions, it is crucial to understand how accurate this approximation is. This is where error estimation comes into play.
In the given exercise, we approximate the function \(f(x) = \tan^{-1}(x)\) using \(f(x) \approx x\) for small values of \(x\). The error in this approximation measures how close our estimated value is to the actual value, calculated using the difference:
While in part (b), for \(x = 0.2\), it increased to 0.0026, demonstrating that as \(x\) becomes larger, the error may also increase.
Therefore, understanding and calculating this error helps us gauge the reliability of our approximation, ensuring that when it matters, we use precise values.
In the given exercise, we approximate the function \(f(x) = \tan^{-1}(x)\) using \(f(x) \approx x\) for small values of \(x\). The error in this approximation measures how close our estimated value is to the actual value, calculated using the difference:
- \(\text{Error} = |\text{Approximation} - \text{Actual Value}|\)
While in part (b), for \(x = 0.2\), it increased to 0.0026, demonstrating that as \(x\) becomes larger, the error may also increase.
Therefore, understanding and calculating this error helps us gauge the reliability of our approximation, ensuring that when it matters, we use precise values.
Inverse Trigonometric Functions
Inverse trigonometric functions are the inverse functions of the trigonometric functions such as sine, cosine, and tangent. These functions are crucial in calculus and trigonometry because they allow us to find angles when the values of trigonometric ratios are known.
The function we are dealing with in this exercise is \(\tan^{-1}(x)\), also known as the arctangent function. It is used to find the angle whose tangent is \(x\). This function produces outputs ranging from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
The function we are dealing with in this exercise is \(\tan^{-1}(x)\), also known as the arctangent function. It is used to find the angle whose tangent is \(x\). This function produces outputs ranging from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
- For small values of \(x\), \(\tan^{-1}(x)\) can closely be approximated by \(x\).
- This is because the tangent function is linear around zero.
Approximation Methods
Approximation methods play an essential role in mathematics as they allow us to simplify complex functions to make them easier to work with, especially when exact solutions are challenging to obtain. In calculus, one common technique is using Taylor Series.
For the function \(f(x) = \tan^{-1}(x)\), one might use a Taylor series expansion, which expresses the function as an infinite sum of terms calculated from the values of its derivatives at a single point.
Hence, understanding these methods equips students with the tools needed for effective problem-solving in calculus and beyond.
For the function \(f(x) = \tan^{-1}(x)\), one might use a Taylor series expansion, which expresses the function as an infinite sum of terms calculated from the values of its derivatives at a single point.
- The simplest form we used is the linear approximation, \(f(x)\approx x\), suitable when \(x\) is near zero.
- A more accurate approximation could involve more terms of the series.
Hence, understanding these methods equips students with the tools needed for effective problem-solving in calculus and beyond.
Other exercises in this chapter
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