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. $$f(x)=\tan ^{-1} x \approx x$$

Short 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.

Step-by-step solution

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.

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 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:

• \(\text{Error} = |\text{Approximation} - \text{Actual Value}|\)

In part (a), for \(x = 0.1\), we estimated the error to be 0.0003.
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 \(\frac{\pi}{2}\).

• For small values of \(x\), \(\tan^{-1}(x)\) can closely be approximated by \(x\).
• This is because the tangent function is linear around zero.

Understanding the behavior of the arctangent function around zero is key to grasping why \(\tan^{-1}(x)\approx x\) holds for small \(x\), providing a simpler way to calculate values.

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.

• 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.

Using this method provides a balance between precision and simplicity, helping to compute values quickly when working with small values of \(x\). Keep in mind, though, the trade-off between simplicity of the approximation and precision — the more terms you compute in a Taylor series, the more accurately it approximates the function for larger \(x\).

Hence, understanding these methods equips students with the tools needed for effective problem-solving in calculus and beyond.