Question

a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\cos x, a=\pi / 6$$

Short answer

Answer: \(T_0(x) = \frac{\sqrt{3}}{2}\) 2. What is the 1st-order Taylor polynomial for the function \(f(x) = \cos x\) centered at the point \(a = \frac{\pi}{6}\)? Answer: \(T_1(x) = \frac{\sqrt{3}}{2} -\frac{1}{2}(x-\frac{\pi}{6})\) 3. What is the 2nd-order Taylor polynomial for the function \(f(x) = \cos x\) centered at the point \(a = \frac{\pi}{6}\)? Answer: \(T_2(x) =\frac{\sqrt{3}}{2} -\frac{1}{2}(x-\frac{\pi}{6}) - \frac{\sqrt{3}}{4}(x-\frac{\pi}{6})^2\) 4. How can you graph the function \(f(x) = \cos x\) and its Taylor polynomials \(T_0(x), T_1(x),\) and \(T_2(x)\) together? Answer: You can graph the function and its Taylor polynomials by using graphing tools like Desmos, GeoGebra, or even graph paper. Plot the function \(f(x) = \cos x\) and the Taylor polynomials \(T_0(x), T_1(x),\) and \(T_2(x)\) using the equations found in the solution.

Step-by-step solution

Review the Taylor polynomial formula

Recall that the nth-order Taylor polynomial centered at a point \(a\) is given by the formula: $$T_n(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^n$$ To find the Taylor polynomial centered at \(a=\frac{\pi}{6}\) for \(f(x)=\cos x\), we need to find the \(n\)th derivative \(f^{(n)}(x)\) and evaluate it at \(a=\frac{\pi}{6}\).

Find the derivatives

Calculate the first few derivatives of \(f(x)=\cos x\). 1. \(f'(x) = -\sin x\) 2. \(f''(x) = -\cos x\) 3. \(f^{(3)}(x) = \sin x\) The derivatives repeat this pattern in a cyclic manner. Now, let's plug in the given value, \(a=\frac{\pi}{6}\), into these derivatives.

Evaluate the derivatives at \(a = \frac{\pi}{6}\)

Now, plug in the value of \(a=\frac{\pi}{6}\) into the derivatives: 1. \(f(\frac{\pi}{6}) = \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\) 2. \(f'(\frac{\pi}{6}) = -\sin(\frac{\pi}{6}) = -\frac{1}{2}\) 3. \(f''(\frac{\pi}{6}) = -\cos(\frac{\pi}{6}) = -\frac{\sqrt{3}}{2}\) 4. \(f^{(3)}(\frac{\pi}{6}) = \sin(\frac{\pi}{6}) = \frac{1}{2}\) From these, we can now construct our Taylor polynomials.

Find the Taylor polynomials

Using the Taylor polynomial formula, substitute the derivatives and \(a=\frac{\pi}{6}\) into the formula for \(T_n(x)\) for \(n = 0, 1,\) and \(2\): For \(n=0\): $$T_0(x) = f(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$$ For \(n=1\): $$T_1(x) = f(\frac{\pi}{6}) + f'(\frac{\pi}{6})(x-\frac{\pi}{6}) = \frac{\sqrt{3}}{2} -\frac{1}{2}(x-\frac{\pi}{6})$$ For \(n=2\): $$T_2(x) = f(\frac{\pi}{6}) + f'(\frac{\pi}{6})(x-\frac{\pi}{6}) + \frac{f''(\frac{\pi}{6})}{2!}(x-\frac{\pi}{6})^2 =\frac{\sqrt{3}}{2} -\frac{1}{2}(x-\frac{\pi}{6}) - \frac{\sqrt{3}}{4}(x-\frac{\pi}{6})^2$$

Graphing the Taylor polynomials and the function

Now, you can graph the function \(f(x)=\cos x\) and its Taylor polynomials \(T_0(x), T_1(x),\) and \(T_2(x)\) together. To plot these graphs, you may use graphing tools like Desmos, GeoGebra, or even graph paper. Make sure to use the equations found in Step 4 for each Taylor polynomial. Along with the function \(f(x)=\cos x\), you will notice how the function is better approximated by each Taylor polynomial as the order increases.

Key concepts

Cosine Function

The cosine function, denoted as \(\cos x\), is a fundamental trigonometric function that is periodic in nature. It describes the x-coordinate of a point on the unit circle as the angle \(x\), measured in radians, varies. Cosine values range from -1 to 1 and are useful in various applications, from engineering to animation. The graph of \(\cos x\) is a wave that repeats every \(2\pi\) radians.Understanding how the cosine function behaves is crucial in fields like physics and engineering, where wave phenomena are prevalent. It's characterized by its even symmetry, meaning the cosine function is symmetrical about the y-axis: \(\cos(-x) = \cos(x)\). This property aids in simplifying many mathematical computations.Moreover, as part of Euler's formula, the cosine function is intricately linked with the exponential function, enriching its theoretical significance.

Nth-order Derivative

Derivatives of functions describe how the function changes as the input changes. For trigonometric functions like cosine, derivatives exhibit a cyclical pattern. For \(f(x) = \cos x\):

• The first derivative, \(f'(x) = -\sin x\), shows how \(\cos x\) slopes downwards at zero crossing points.
• The second derivative, \(f''(x) = -\cos x\), mirrors the original function but with an inverted sign.
• The third derivative, \(f^{(3)}(x) = \sin x\), reverts the sine function positively.
• The fourth derivative returns to the original function: \(f^{(4)}(x) = \cos x\).

This cyclic behavior repeats every four derivatives, which is particularly useful in creating Taylor polynomials, allowing predictions and approximations of function behavior around specific points. Understanding these cyclic patterns aids in simplifying complex calculus operations involving trigonometric functions.

Polynomial Approximation

Polynomial approximation involves using polynomials to estimate the values of a more complex function. This is especially useful when dealing with transcendental functions like the cosine function. The Taylor series is a powerful tool that helps in constructing these approximations.The nth-order Taylor polynomial approximates a function around a specified point \(a\), using its derivatives at this point to construct:\[ T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k \]This polynomial provides a simpler mathematical form that is much easier to work with than most functions, capturing the function's behavior, especially for values of \(x\) close to \(a\). As the order \(n\) increases, the approximation becomes more accurate over a larger interval. The Taylor polynomial aids in computations across various fields, from computer science to engineering, where precise calculations with functions like \(\cos x\) are necessary.

Graphical Representation

Graphing helps visualize the differences and similarities between the original function and its polynomial approximations. When plotted together, the function \(f(x)=\cos x\) and its Taylor polynomials \(T_0(x)\), \(T_1(x)\), and \(T_2(x)\) provide insight into how the approximation improves with increasing order.

• \(T_0(x)\) represents a horizontal line at \(f(\frac{\pi}{6})\), capturing only the function's value at the point.
• \(T_1(x)\) brings in the linear element, providing a better fit by considering the slope (first derivative) at \(\frac{\pi}{6}\).
• \(T_2(x)\) includes a quadratic term, making the curve fit closely to the \(\cos x\) around \(\frac{\pi}{6}\).

As you graph each higher-order polynomial, observe how the curve more closely follows the peaks and valleys of \(\cos x\). This demonstrates how Taylor polynomials can effectively approximate complex functions within a local region.