Question

Use a graphing utility to graph the function. Then graph the linear and quadratic approximations \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) in the same viewing window. Compare the values of \(f, P_{1},\) and \(P_{2}\) and their first derivatives at \(x=a .\) How do the approximations change as you move farther away from \(x=a\) ? \(\begin{array}{ll}\text { Function } & \frac{\text { Value of } a}{a} \\\ f(x)=\arctan x & a=-1\end{array}\)

Short answer

The first and second derivatives of \(f(x) = \arctan x\) are \(f'(x) = \frac{1}{1+x^2}\) and \(f''(x) = \frac{-2x}{(1+x^2)^2}\) respectively. At \(x = a = -1\), these values are \(f'(-1) = \frac{1}{2}\) and \(f''(-1) = 1\). Thus, the linear Taylor series approximation at \(x = a = -1\) is \(P_1(x) = f(-1) + f'(-1)(x - (-1))\), and the quadratic approximation is \(P_2(x) = f(-1) + f'(-1)(x - (-1)) + \frac{1}{2}f''(-1)(x - (-1))^2\). The approximations become less accurate as \(x\) moves away from -1.

Step-by-step solution

Compute derivatives

First, compute the first and second derivative of the function \(f(x) = \arctan x\). The first derivative \(f'(x) = \frac{1}{1+x^2}\) and the second derivative \(f''(x) = \frac{-2x}{(1+x^2)^2}\). Compute these derivatives at \(x = a = -1\), to get \(f'(-1) = \frac{1}{2}\) and \(f''(-1) = 1\).

Compute the Taylor series approximations

Compute the linear approximation \(P_1(x) = f(a) + f'(a)(x - a)\) and the quadratic approximation \(P_2(x) = f(a) + f'(a)(x - a) + \frac{1}{2}f''(a)(x - a)^2\). Substituting \(a = -1\), \(f(a) = -\frac{\pi}{4}\), \(f'(-1) = \frac{1}{2}\), and \(f''(-1) = 1\), compute \(P_1(x)\) and \(P_2(x)\)

Graph the function and approximations

Use a graphing calculator or software to graph \(f(x) = \arctan x\) along with the linear and quadratic approximations \(P_1(x)\) and \(P_2(x)\) computed in Step 2. You will clearly see the function and its approximations on the same graph.

Analyze the graphs

Analyze the graphs to see how the approximations \(P_1(x)\) and \(P_2(x)\) behave in relation to \(f(x)\) as you move further away from \(x = a\). You'll notice that the approximations are more accurate close to \(a = -1\) and they deviate more as \(x\) moves away from -1.

Key concepts

Graphing Utility

Graphing utilities such as calculators or specialized software are invaluable tools forvisualizing functions and their approximations. They allow users to input equations anddisplay their corresponding graphs, often in a matter of seconds. Besides providing avisual representation, many graphing tools also offer additional analytical features, suchas computing derivatives or evaluating functions at specific points.

When graphing the function \(f(x) = \arctan x\) together with its linear and quadratic approximations, a graphing utility can help students see exactly where these approximations align with the actual function, and where they diverge. This visual comparison makes it easier to understand the reliability of the approximations within the vicinity of the chosen value of \(a\).

By seeing the graphs of \(P_{1}(x)\) and \(P_{2}(x)\) together with \(f(x)\), students can directly observe the effects of the first and second derivatives on the shape and curvature of the approximations, enhancing their comprehension of calculus concepts.

Linear Approximation

Linear approximation is a method used to estimate the value of a function \(f(x)\) at a particular point \(x\) by using the value of the function and its first derivative at a close point \(a\). The formula \(P_1(x) = f(a) + f'(a)(x - a)\) serves as the linear approximation, which is essentially the equation of the tangent line at the point \(a\) on the function's curve.

The process of linear approximation simplifies the function to a straight line, allowing for easier calculations. However, it is most accurate near the point \(a\) and can become less reliable as \(x\) moves further from \(a\). The first derivative in the approximation provides information on the slope of the tangent line—representing the instantaneous rate of change of the function at \(a\).

Quadratic Approximation

Quadratic approximation builds upon the concept of linear approximation by including the second derivative, which accounts for the curvature of the function's graph. The formula \(P_2(x) = f(a) + f'(a)(x - a) + \frac{1}{2}f''(a)(x - a)^2\) gives a parabola that better fits the original function around the point \(a\), especially when the function itself has a curved trajectory.

By incorporating the second derivative, quadratic approximation allows for a more accurate estimate of \(f(x)\) over a wider range of \(x\) values around \(a\). The second derivative describes the concavity of the function—if it is pointing upwards or downwards—which helps in predicting how the function behaves beyond the immediate vicinity of \(a\). This greater accuracy near the point of approximation usually leads to better overall estimates of the original function.

First Derivative

The first derivative of a function, represented as \(f'(x)\), is crucial when discussing approximations and the behavior of functions. It describes the rate at which the function's value is changing at any given point, and it's the slope of the tangent line to the function's curve at that point.

In our exercise, the first derivative of \(\arctan x\) is \(f'(x) = \frac{1}{1+x^2}\). At the point \(a = -1\), the first derivative evaluates to \(\frac{1}{2}\). This means that the slope of the tangent line to \(f(x)\) at \(x = -1\) is \(\frac{1}{2}\). This information alone gives us the linear approximation—how the function changes linearly at that specific point. Understanding the first derivative is essential for not only creating the linear approximation but also for interpreting the immediate rate of change of the function.

Second Derivative

The second derivative of a function, expressed as \(f''(x)\), tells us about the acceleration of the function's rate of change; that is, how the slope itself is changing. It is a measure of the curvature of the function's graph. For the function \(f(x) = \arctan x\), the second derivative is \(f''(x) = \frac{-2x}{(1+x^2)^2}\), and at \(x = a = -1\), it is equal to 1. This indicates that at \(x = -1\), the slope of \(f(x)\) is becoming steeper at a rate proportional to the value of the second derivative.

When creating the quadratic approximation \(P_2(x)\), the second derivative gives us a more refined view of how the function behaves around \(a\), not just in terms of direction but also how it bends. For students, understanding the second derivative is key to grasping why a quadratic approximation provides a better estimate than the linear one, particularly for functions with noticeable curvature.