Problem 25 - Essential Calculus

Question

In Exercises 25 and \(26,\) use a computer algebra system to find the indicated Taylor polynomials for the function \(f\). Graph the function and the Taylor polynomials. \(f(x)=\tan x\) (a) \(n=3, \quad c=0\) (b) \(n=3, \quad c=\pi / 4\)

Step-by-Step Solution

Answer

The third degree Taylor polynomial for \(f(x) = \tan (x)\) at \(c = 0\) is \(P_3(x) = x + \frac{x^3}{3}\). At \(c = \pi / 4\), it is \(P_3(x) = 1 + 2 \cdot (x-\pi / 4) + 2 \cdot (x-\pi / 4)^2 + 3 \cdot (x-\pi / 4)^3\)

Step 1 for problem (a)

This problem has asked for a Taylor polynomial of order 3 around \(c = 0\). The Taylor polynomial \(P_n (x)\) of order \(n\) of a function \(f\) around the point \(c\) is given by: \[P_n(x) = f(c) + f'(c)(x - c) + f''(c) \frac{(x-c)^2}{2!} + f'''(c) \frac{(x-c)^3}{3!} + ......\]Since \(n = 3\) only terms up to \(n = 3\) will be considered and anything after that will be ignored. As, \(f(x)= \tan (x)\), \(f(c=0)= \tan(0)=0\), \(f'(c=0)= \sec^2(0)=1\), \(f''(c=0)= 2\tan(0)\sec^2(0)=0\) and \(f'''(c=0)= 2\sec^4(0)+2\tan(0)\sec^2(0)=2\). Hence the Taylor polynomial becomes:\[P_3(x) = 0 + 1 \cdot x + 0 \frac{x^2}{2!}+ 2 \frac{x^3}{3!}\] Simplifying, get the third degree Taylor polynomial for \(f(x)=\tan (x)\) at \(c=0\).

Step 2 for problem (b)

Here, \(c = \pi / 4\). Same Taylor polynomial formula from Step 1 will be applied, substituting \(c = \pi / 4\). As, \(f(x)= \tan (x)\), \(f(c=\pi / 4)= \tan(\pi / 4)=1\), \(f'(c=\pi / 4)= \sec^2(\pi / 4)=2\), \(f''(c=\pi / 4)= 2\tan(\pi / 4)\sec^2(\pi / 4)=4\) and \(f'''(c=\pi / 4)= 2\sec^4(\pi / 4)+2\tan(\pi / 4)\sec^2(\pi / 4)=12\). Hence the Taylor polynomial becomes:\[P_3(x) = 1 + 2 \cdot (x-\pi / 4) + 4 \frac{(x-\pi / 4)^2}{2!}+ 12 \frac{(x-\pi / 4)^3}{3!}\] Simplifying, get the third degree Taylor polynomial for \(f(x)=\tan (x)\) at \(c=\pi / 4\).

Step 3 Drawing the Graphs

To create the graphs for the original function and the Taylor polynomials, plot the function \(f(x) = \tan (x)\), the polynomial from Step 1 \(P_1(x) = x + \frac{x^3}{3}\), and the polynomial from Step 2 on the same graph. The graph show how closely the Taylor polynomial approximates the function near the point \(c\).

Key Concepts

Understanding Taylor Series Expansion

A Taylor series expansion allows us to approximate complex functions with polynomials, making them easier to analyze and compute.
This concept involves expanding a function into an infinite series, each term calculated from the derivatives of the function at a single point, usually denoted as \(c\).
Key aspects of the Taylor series:

• Central Point \(c\): The expansion is done around a central point \(c\) where the function's derivatives are evaluated.
• Order \(n\): The order determines how many terms of the derivative are used in the polynomial. For instance, when \(n=3\), only derivatives up to the third order are included.
• Formula: The Taylor polynomial \(P_n(x)\) is given by \[P_n(x) = f(c) + f'(c)(x - c) + f''(c) \frac{(x-c)^2}{2!} + \ldots + \frac{f^{(n)}(c)(x-c)^n}{n!}\]
• Approximation: The closer you are to the point \(c\), the better the polynomial approximates the function.

This approach is particularly useful when calculating values of trigonometric functions, transcending their typical periodic nature.

Exploring Trigonometric Functions

Trigonometric functions, like \(\tan(x)\), are fundamental in mathematics, representing the relationships between angles and sides of triangles.
These functions are periodic and rely on concepts like sine, cosine, and tangent.Key trigonometric functions:

• Sine \(\sin(x)\): This is the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• Cosine \(\cos(x)\): This represents the ratio of the adjacent side to the hypotenuse.
• Tangent \(\tan(x)\): Defined as \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), it's the ratio of the opposite side to the adjacent side.
• Secant \(\sec(x)\): This is the reciprocal of cosine, important for deriving derivatives like \(\sec^2(x)\).

Understanding these relationships helps in constructing Taylor polynomials for trigonometric functions, providing approximations valid in small intervals around \(c\).
This can simplify complex calculations in calculus and engineering.

A Glimpse into Calculus

Calculus is the mathematical study of continuous change, built on the concepts of derivatives and integrals.
It is essential in understanding the behavior of functions and their rates of change.Key aspects:

• Derivatives: These measure how a function changes as its input changes. For example, the derivative of \(\tan(x)\) is \(\sec^2(x)\), a vital element in Taylor expansions.
• Integrals: While derivatives handle rates of change, integrals deal with accumulation, such as finding areas under curves.
• Limits: Fundamental to defining both derivatives and integrals, limits help describe the behavior of functions as they approach specific points.

Using Taylor series in calculus allows us to approximate and analyze function behaviors near specific points. This technique simplifies complex function manipulations, making them accessible through polynomial forms.