Problem 81 - Calculus Single Variable

Question

You are given a function \(f,\) a value \(c\) and \(a\) viewing rectangle \(R\) containing the point \(P=(c, f(c))\). In \(R,\) graph the four functions \(f, g, h,\) and \(k,\) where \(g(x)=\) \(f(c)+f^{\prime}(c)(x-c), h(x)=g(x)+1 / 2 f^{\prime \prime}(c)(x-c)^{2},\) and \(k(x)=\) \(h(x)+1 / 6 f^{\prime \prime \prime}(c) \quad(x-c)^{3} .\) The graphs of \(g, h,\) and \(k\) are, respectively, linear, parabolic, and cubic approximations of the graph of \(f\) near \(c\). The method of constructing these approximating functions, which are called Taylor polynomials of \(f\) with base point \(c,\) is studied in Chapter 8. \(f(x)=\sin ^{2}(x), c=\pi / 3, R=[0,2] \times[-0.2,1.2]\)

Step-by-Step Solution

Answer

Plot the functions \( f, g, h, \) and \( k \) within the specified viewing rectangle to see their approximations of \( f \) near \( c = \pi/3 \).

Step 1: Compute the derivatives of the function

The given function is \( f(x) = \sin^2(x) \). We need to find its first three derivatives evaluated at \( x = c = \pi/3 \). The first derivative is \( f'(x) = 2\sin(x)\cos(x) = \sin(2x) \). Evaluate at \( x = \pi/3 \): \( f'(\pi/3) = \sin(2(\pi/3)) = \sin(2\pi/3) = \sqrt{3}/2 \). The second derivative is \( f''(x) = 2\cos(2x) \). Evaluate at \( x = \pi/3 \): \( f''(\pi/3) = 2\cos(2\pi/3) = -2/2 = -1 \). The third derivative is \( f'''(x) = -4\sin(2x) \). Evaluate at \( x = \pi/3 \): \( f'''(\pi/3) = -4\sin(2\pi/3) = -2\sqrt{3} \).

Step 2: Construct the linear approximation g(x)

The linear approximation \( g(x) \) is given by the formula \[ g(x) = f(c) + f'(c)(x-c) \]. Calculate \( f(\pi/3) = \sin^2(\pi/3) = (\sqrt{3}/2)^2 = 3/4 \). Now apply the formula: \[ g(x) = \frac{3}{4} + \frac{\sqrt{3}}{2}(x - \pi/3). \]

Step 3: Construct the quadratic approximation h(x)

The quadratic approximation \( h(x) \) builds off \( g(x) \) and is given by \[ h(x) = g(x) + \frac{1}{2} f''(c)(x-c)^2 \]. Using \( f''(\pi/3) = -1 \), substitute into the formula: \[ h(x) = \left( \frac{3}{4} + \frac{\sqrt{3}}{2}(x - \pi/3) \right) - \frac{1}{2}(x - \pi/3)^2. \]

Step 4: Construct the cubic approximation k(x)

The cubic approximation \( k(x) \) builds off \( h(x) \) and is given by \[ k(x) = h(x) + \frac{1}{6} f'''(c)(x-c)^3 \]. Using \( f'''(\pi/3) = -2\sqrt{3} \), substitute into the formula: \[ k(x) = \left( \frac{3}{4} + \frac{\sqrt{3}}{2}(x - \pi/3) - \frac{1}{2}(x - \pi/3)^2 \right) - \frac{\sqrt{3}}{3}(x - \pi/3)^3. \]

Step 5: Graph the functions in the viewing rectangle

With the functions \( f(x) = \sin^2(x) \), \( g(x) \), \( h(x) \), and \( k(x) \) computed, plot them within the rectangle defined by \( x \) ranging from 0 to 2, and \( y \) ranging from -0.2 to 1.2. This plot will show the original function and its successive Taylor polynomial approximations around the point \( P = (\pi/3, f(\pi/3)) = (\pi/3, 3/4) \).

Key Concepts

Derivative Calculations

To construct Taylor Polynomials, we start by calculating the derivatives of the function. These derivatives help us understand how the function behaves near a point. For the exercise given, the function is \( f(x) = \sin^2(x) \). Derivatives are essentially the function's rates of change.
Here's the breakdown of derivative calculations:

• First Derivative \( f'(x) \): This represents the slope of the tangent line to the function at any point \( x \). For our function, the first derivative is \( f'(x) = 2 \sin(x) \cos(x) \), which simplifies to \( \sin(2x) \). This first derivative is evaluated at \( x = \pi/3 \), giving \( f'(\pi/3) = \sqrt{3}/2 \).


• Second Derivative \( f''(x) \): This gives us information about the curvature or concavity of the function. The second derivative for our function is \( f''(x) = 2 \cos(2x) \). At \( x = \pi/3 \), this evaluates to \( f''(\pi/3) = -1 \).


• Third Derivative \( f'''(x) \): This derivative helps to understand the rate at which the second derivative changes, relating to the function's **'jerk'** in motion terms. It is calculated as \( f'''(x) = -4 \sin(2x) \). Evaluated at \( x = \pi/3 \), it results in \( f'''(\pi/3) = -2 \sqrt{3} \).

Understanding these derivatives is essential for building Taylor polynomials that offer approximation methods around a particular point.

Approximation Methods

Approximation methods use simpler polynomial functions to closely mimic a more complex function near a specific point. Taylor Polynomials are fantastic tools for doing such approximations. In our exercise, we apply this method to the function \( f(x) = \sin^2(x) \) around \( x = \pi/3 \).
The essential approximations are:

• Linear Approximation \( g(x) \): This is the simplest type of approximation, akin to creating a tangent line. The formula uses the function's value and its first derivative at the point \( x = c \). It's given by \( g(x) = f(c) + f'(c)(x-c) \). For our problem, it becomes \( g(x) = 3/4 + (\sqrt{3}/2)(x - \pi/3) \).


• Quadratic Approximation \( h(x) \): Building on the linear approximation, this adds the second derivative, offering a parabolic shape to better reach the actual curve of \( f(x) \). The formula is \( h(x) = g(x) + 1/2 f''(c)(x-c)^2 \), transforming into \( h(x) = (3/4 + (\sqrt{3}/2)(x - \pi/3)) - 1/2(x - \pi/3)^2 \).


• Cubic Approximation \( k(x) \): Here, we integrate the third derivative, capturing more of the function's complexity. The cubic approximation is \( k(x) = h(x) + 1/6 f'''(c)(x-c)^3 \), leading to \( k(x) = (3/4 + (\sqrt{3}/2)(x-\pi/3) - 1/2(x-\pi/3)^2) - \sqrt{3}/3 (x-\pi/3)^3 \).

These approximations provide increasingly better estimates of the function \( f(x) \) around the point \( c \), offering a useful method to assess more substantial functions.

Graphing Functions

Graphing these functions allows us to visualize how well Taylor polynomial approximations mimic the original function \( f(x) = \sin^2(x) \). The specified viewing rectangle \( R = [0, 2] \times [-0.2, 1.2] \) effectively frames the region around the point \( P = (\pi/3, 3/4) \).
Creating each graph, and analyzing them visually:

• For the function \( f(x) = \sin^2(x) \), we see a smooth wave based on the sine function. This is our baseline for further approximations.


• With the linear approximation \( g(x) \), the graph appears as a straight line tangent, touching \( f(x) \) at \( x = \pi/3 \). It gives an immediate estimation only at \( c \).


• The quadratic approximation \( h(x) \) introduces a slight curve. Its parabolic form extends the approximation accuracy beyond just a single point.


• The cubic approximation \( k(x) \) more closely follows the curve, showing even better fidelity around the point \( c \) by incorporating more function behavior.

Plotting these polynomials makes it clear how each approximation builds complexity for a better fit, showcasing the power of Taylor polynomials in mimicking function behavior in calculus.