Question

Quadratic Approximations (a) Let \(Q(x)=b_{0}+b_{1}(x-a)+b_{2}(x-a)^{2}\) be a quadratic approximation to \(f(x)\) at \(x=a\) with the properties: \(\begin{aligned} \text { i. } Q(a) &=f(a) \\ \text { ii. } Q^{\prime}(a) &=f^{\prime}(a) \\ \text { ii. } & Q^{\prime \prime}(a)=f^{\prime \prime}(a) \end{aligned}\) Determine the coefficients \(b_{0}, b_{1},\) and \(b_{2}\) (b) Find the quadratic approximation to \(f(x)=1 /(1-x)\) at \(x=0 .\) (c) Graph \(f(x)=1 /(1-x)\) and its quadratic approximation at \(x=0 .\) Then zoom in on the two graphs at the point \((0,1) .\) Comment on what you see. (d) Find the quadratic approximation to \(g(x)=1 / x\) at \(x=1\) Graph \(g\) and its quadratic approximation together. Comment on what you see. (e) Find the quadratic approximation to \(h(x)=\sqrt{1+x}\) at \(x=0 .\) Graph \(h\) and its quadratic approximation together. Comment on what you see. (f) What are the linearizations of \(f, g,\) and \(h\) at the respective points in parts \((b),(d),\) and \((e) ?\)

Short answer

The quadratic approximations for the functions at their respective points are \(Q_f(x) = 1 + x + x^2\) for \(f(x)=1/(1-x)\) at \(x=0\), \(Q_g(x) = 1 - x + x^2\) for \(g(x) = 1 / x\) at \(x=1\) and \(Q_h(x) = 1 + x / 2 - x^2 / 4\) for \(h(x) = \sqrt{1+x}\) at \(x=0\). The linearizations are \(L_f(x) = 1 + x\) for \(f(x)\), \(L_g(x) = 2 - x\) for \(g(x)\), and \(L_h(x) = 1 + x/2\) for \(h(x)\). The quadratic approximations closely follow the actual functions close to the points they are approximating.

Step-by-step solution

Analysis of a Quadratic Approximation

The quadratic approximation to a function \(f(x)\) at a point \(x = a\) is given by \(Q(x) = b_0 + b_1(x-a) + b_2(x-a)^2\), where the coefficients \(b_0\), \(b_1\) and \(b_2\) satisfy: \(Q(a) = f(a)\), \(Q'(a) = f'(a)\), \(Q''(a) = f''(a)\). By matching these conditions to the quadratic, we can deduce \(b_0 = f(a)\), \(b_1 = f'(a)\), and \(b_2 = f''(a) / 2\).

Applying to \(f(x) = 1 / (1-x)\)

Differentiating \(f(x) = 1/(1-x)\) successively gives \(f'(x) = 1/(1-x)^2\) and \(f''(x) = 2/(1-x)^3\). Applying these at \(x = 0\) gives \(b_0 = 1\), \(b_1 = 1\) and \(b_2 = 2/2 = 1\), so the quadratic approximation at \(x=0\) is \(Q(x) = 1 + x + x^2\).

Graphing \(f(x) = 1 / (1-x)\) and its quadratic approximation

Plot both \(f(x) = 1 / (1-x)\) and its quadratic approximation \(Q(x) = 1 + x + x^2\). Upon zooming in at the point \((0,1)\), one will observe that the quadratic approximation closely mimics the actual function near this point.

Finding the quadratic approximation to \(g(x) = 1 / x\) at \(x=1\)

Following the same steps for \(g(x) = 1/x\) gives \(g'(x) = -1/x^2\) and \(g''(x) = 2/x^3\). Therefore \(b_0 = 1\), \(b_1 = -1\) and \(b_2 = 2/2 = 1\) so the quadratic approximation is \(Q(x) = 1 - (x-1) + (x-1)^2 = 1 - x + x^2\).

Graphing \(g(x) = 1 / x\) and its quadratic approximation

Now graph both g(x) and \(Q(x) = 1 - x + x^2\). The quadratic approximation will closely follow the actual function near the point \(x = 1\).

Finding the quadratic approximation to \(h(x) = sqrt{1+x}\) at \(x=0\)

The derivative of \(h(x) = sqrt{1+x}\) is \(h'(x) = 1/2sqrt{1+x}\) and the second derivative is \(h''(x) = -1/4(1+x)^{-3/2}\). Hence, \(b_0 = 1\), \(b_1 = 1/2\) and \(b_2 = -1/4\). So, the quadratic approximation is \(Q(x) = 1 + x/2 - x^2/4\).

Graphing \(h(x) = sqrt{1+x}\) and its quadratic approximation

Plot both \(h(x) = sqrt{1+x}\) and \(Q(x) = 1 + x/2 - x^2/4\). The quadratic approximation will trace the function closely around \(x=0\).

Finding the linearization

The linearization of a function at \(x=a\) is essentially the same as its quadratic approximation, except that only the first derivative is used. Hence the linearizations are \(L_f(x) = 1 + x\) for \(f(x)\), \(L_g(x) = 2 - x\) for \(g(x)\), and \(L_h(x) = 1 + x/2\) for \(h(x)\)

Key concepts

Taylor Polynomial

Understanding the Taylor Polynomial is crucial for approximating functions and analyzing their behavior near a given point. Essentially, a Taylor Polynomial is a sum of terms derived from the function's value and its derivatives at a single point. This approximation becomes more accurate as you include more terms from the function's Taylor series. For a quadratic approximation, the Taylor Polynomial includes terms up to the second derivative.

In the step-by-step solution, we used this concept to approximate the function at a point, which is a powerful tool in calculus for understanding complex functions using simpler polynomials. It's particularly useful because the coefficients of the polynomial directly correspond to the values and derivatives of the function at the point of approximation.

Function Differentiation

Function Differentiation is the process of finding the derivative, which represents the rate at which a function is changing at any given point. The derivative is a fundamental concept in calculus, providing critical information about the behavior of functions, such as rates of change and the slope of their graphs.

In our problem, differentiation helps us find the coefficients of the Taylor Polynomial by differentiating the function repeatedly. The first derivative gives us the coefficient for the linear term, and the second derivative provides the coefficient for the quadratic term. More specifically, by differentiating the functions given in the exercise, we align the polynomial's derivatives with those of the original function, ensuring that both the function and its approximation have the same slope and curvature at the point of interest.

Graphical Analysis of Functions

Graphical Analysis of Functions involves examining the visual representation of functions to understand their properties, including continuity, limits, and behavior near specific points. It's a fundamental aspect of calculus that gives an intuitive understanding of how functions behave.

The solution provided includes graphing both the original functions and their quadratic approximations to show how closely the approximation follows the actual function around the given point. When we 'zoom in' on the graphs at particular points, we can visually inspect how well the approximation mimics the true function. The closer the quadratic approximation is to the function on the graph, the more accurate our approximation is within a certain range of interest around the point.

Linearization of Functions

Linearization of Functions is the method of approximating a function near a given point using a line, which is the function's tangent at that point. It simplifies analysis and calculations by replacing a complex function with a simple straight line that closely matches the function's behavior around a specific value. The linearization is the first-degree Taylor Polynomial.

In our exercise, we calculated linearizations for each given function at the specified points. These linearizations serve as the simplest (first order) Taylor approximation. It's essentially the starting point of a Taylor series and is particularly helpful for making quick, rough approximations of function values near a point of interest, which can be crucial in both theoretical calculus and practical applications such as engineering and physics.