Question

Linear and Quadratic Approximations In Exercises 33 and 34, use a computer algebra system to find the linear approximation $$P_{1}(x)=f(a)+f^{\prime}(a)(x-a)$$ and the quadratic approximation $$P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}$$ to the function \(f\) at \(x=a\). Sketch the graph of the function and its linear and quadratic approximations. $$ f(x)=\arctan x, \quad a=0 $$

Short answer

The linear approximation \(P_1(x)\) is \(x\) and the quadratic approximation \(P_2(x)\) is also \(x\). The graph of both the linear and quadratic approximations is a straight line emerging from the origin, while that of the actual function is an S-shaped curve.

Step-by-step solution

Compute the first and second derivatives of \(f\)

The derivative of \(f(x) = \arctan(x)\), using the chain rule, is \(f'(x) = \frac{1}{1+x^2}\). The second derivative, computed by applying the chain and quotient rules to \(f'(x)\), is \(f''(x) = \frac{-2x}{(1+x^2)^2}\).

Evaluate \(f(a)\), \(f'(a)\), and \(f''(a)\)

Substitute \(a=0\) into \(f(x)\), \(f'(x)\), and \(f''(x)\) to get \(f(0) = \arctan(0) = 0\), \(f'(0) = \frac{1}{1+0^2} = 1\), and \(f''(0) = \frac{-2*0}{(1+0^2)^2} = 0\).

Compute the linear and quadratic approximations

The linear approximation \(P_1(x)\) is given by: \(P_1(x) = f(0) + f'(0) * (x-0) = 0 + 1 * x = x\). The quadratic approximation \(P_2(x)\) is given by: \(P_2(x) = f(0) + f'(0) * (x-0) + 0.5 * f''(0) * (x-0)^2 = x + 0 = x\).

Sketch the graph

The function \(f(x) = \arctan(x)\) has a characteristic S-shaped curve that passes through the origin. Its linear and quadratic approximations around \(x=0\) are both given by \(x\), a straight line emerging from the origin.

Key concepts

Taylor Polynomials

Taylor polynomials serve as a bridge between polynomial functions and more intricate functions in calculus. They are derived from the Taylor series, which is a way of expressing a function as an infinite sum of terms calculated from the derivatives of the function at a single point.

The key to understanding Taylor polynomials lies in grasping the process of approximation. For a smooth function, we can approximate it near a point using a polynomial. The first-degree Taylor polynomial, or linear approximation, is the simplest case where we use the function's value and the slope at a particular point. The second-degree, or quadratic approximation, adds curvature by incorporating the second derivative.

In the exercise example, we approximated the arctan function using both linear (\(P_1(x)\)) and quadratic (\(P_2(x)\)) Taylor polynomials evaluated at the point where x equals a.

First Derivative

The first derivative of a function represents the slope of the tangent line to the function at any given point. It is a crucial concept in both differential calculus and graph sketching because it reveals information about the rate of change of the function.

In the context of Taylor polynomials, the first derivative at a specific point is used to determine the linear approximation of the function at that point. As seen in the given exercise, we calculated the first derivative of the arctan function, which was used to form the linear part of the Taylor polynomial for the function's approximation.

Second Derivative

The second derivative provides insight into the acceleration or concavity of a function. While the first derivative tells us about the function's slope, the second derivative informs us if that slope is increasing, decreasing, or staying constant. This is vital in graph sketching and understanding the function's behavior.

For the quadratic approximation in Taylor polynomials, the second derivative is important as it reflects the curvature of the function at a particular point. In the exercise, the second derivative of the arctan function came out to be zero when evaluated at zero, meaning that the quadratic term in the approximation did not contribute any additional information beyond the linear term.

Calculus

Calculus is an extensive branch of mathematics that deals with changes. It has two primary categories: differential calculus that focuses on derivatives and rates of change, and integral calculus, which deals with accumulation of quantities and the areas under and between curves.

The exercise provided utilizes key concepts from differential calculus, namely the first and second derivatives, to construct linear and quadratic approximations, essentially showcasing calculus' power in understanding and approximating the behavior of functions.

Graph Sketching

Graph sketching combines an understanding of derivatives and function behavior to visualize a function's curve. It entails identifying key characteristics such as intercepts, slopes, maxima, minima, and points of inflection.

In the exercise, we needed to sketch the graph of the arctan function along with its linear and quadratic approximations. Although the linear and quadratic approximations are identical in this particular case, sketching both the original function and its approximations can be vital for visualizing how close these polynomials come to the actual function over a small interval around the point of approximation.