Problem 1 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Find a bound on the error of the Taylor's polynomial approximation anchored at \(a=0\) and of indicated degree to a. \(f(x)=\cos x,\) of degree \(2,\) on \([0, \pi / 4]\). b. \(f(x)=e^{x},\) of degree 4 on [0,1] . c. \(f(x)=\cos x,\) of degree 4 on \([0, \pi / 2]\). d. \(f(x)=\sin x,\) of degree \(5,\) on \([0, \pi / 2]\). e. \(f(x)=\ln (1+x)\) of degree 5 on \([0,1 / 2]\). f. \(f(x)=\sin x\) of degree 4 on \([0, \pi / 4\).

Step-by-Step Solution

Answer

a. \(\frac{\sqrt{2}\cdot\pi^3}{384}\), b. \(\frac{e}{120}\), c. \(\frac{\pi^5}{3840}\), d. \(\frac{\pi^6}{46080}\), e. \(\frac{15}{64}\), f. \(\frac{\pi^5}{122880}\).

Step 1: Introduction to Error Bound Formula

The error bound for the Taylor polynomial of degree \( n \) is given by \( R_n(x) = \frac{M|x - a|^{n+1}}{(n+1)!} \), where \( M \) is an upper bound for the \( (n+1) \)-th derivative of \( f \) on the interval.

Step 2: Part A: Determine Error Bound for \( f(x) = \cos x \) of Degree 2 on [0, π/4]

1. The third derivative of \( f(x) = \cos x \) is \( f^{(3)}(x) = \, \sin x \), which is maximized at \( \pi/4 \) with absolute value \( \sqrt{2}/2 \). 2. Using \( M = \sqrt{2}/2 \), \( R_2(x) = \frac{\sqrt{2}/2 \cdot (\pi/4)^{3}}{3!} = \frac{\sqrt{2} \cdot \pi^3}{384} \).

Step 3: Part B: Determine Error Bound for \( f(x) = e^x \) of Degree 4 on [0,1]

1. The fifth derivative of \( f(x) = e^x \) is \( e^x \). An upper bound for \( e^x \) on \([0, 1]\) is \( e \). 2. Using \( M = e \), \( R_4(x) = \frac{e \cdot 1^{5}}{5!} = \frac{e}{120} \).

Step 4: Part C: Determine Error Bound for \( f(x) = \cos x \) of Degree 4 on [0, π/2]

1. The fifth derivative of \( \cos x \) is \( \sin x \), maximized at \( \pi/2 \) with absolute value \( 1 \). 2. Using \( M = 1 \), \( R_4(x) = \frac{1 \cdot (\pi/2)^5}{5!} = \frac{\pi^5}{3840} \).

Step 5: Part D: Determine Error Bound for \( f(x) = \sin x \) of Degree 5 on [0, π/2]

1. The sixth derivative of \( \sin x \) is \( -\sin x \), maximized at \( \pi/2 \) with absolute value \( 1 \). 2. Using \( M = 1 \), \( R_5(x) = \frac{1 \cdot (\pi/2)^6}{6!} = \frac{\pi^6}{46080} \).

Step 6: Part E: Determine Error Bound for \( f(x) = \ln(1+x) \) of Degree 5 on [0, 1/2]

1. The sixth derivative of \( \ln(1+x) \) is \(-\frac{120}{(1+x)^6} \), which is maximized at \( x = 0 \) with absolute value \( 120 \). 2. Using \( M = 120 \), \( R_5(x) = \frac{120 \cdot (1/2)^6}{6!} = \frac{15}{64} \).

Step 7: Part F: Determine Error Bound for \( f(x) = \sin x \) of Degree 4 on [0, π/4]

1. The fifth derivative for \( \sin x \) is \( \cos x \), maximized at \( 0 \) with absolute value \( 1 \). 2. Using \( M = 1 \), \( R_4(x) = \frac{1 \cdot (\pi/4)^5}{5!} = \frac{\pi^5}{122880} \).

Key Concepts

Error Bound

When we use a Taylor polynomial to approximate a function, we often want to know how close our approximation is to the actual function. This is where the *error bound* comes in. The error bound tells us the worst-case difference between the true value of the function and the value given by the polynomial.

The formula for the error bound of a Taylor polynomial of degree \( n \) is:

• \( R_n(x) = \frac{M|x - a|^{n+1}}{(n+1)!} \)

Here, \( M \) is an upper bound for the \( (n+1) \)-th derivative of the function over the interval we're looking at. Essentially, \( M \) represents the maximum size that the error can get, scaled by the interval's width raised to the power of \( n+1 \) and divided by \( (n+1)! \), which accounts for the factorial growth of terms.

Use this error bound formula to judge how tight or loose your approximation is. The smaller the \( R_n(x) \), the better your Taylor polynomial approximates the actual function.

Derivative

Understanding derivatives is crucial when dealing with Taylor polynomials. A derivative of a function tells us the rate at which the function's values are changing. In Taylor series, derivatives help us understand the behavior of a function locally around a point.

Each term in a Taylor polynomial involves a derivative. For instance, if you have a polynomial approximating at \( a \), its general form is:

• \( f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots \)

Here, \( f'(a) \), \( f''(a) \), etc., are the derivatives of the function evaluated at \( a \). When calculating error bounds, it's usually necessary to determine the \( (n+1)\)-th derivative to estimate the potential error in approximating the function with a polynomial of degree \( n \). This derivative is what you use to find \( M \) in the error bound formula.

Remember, knowing how to find and interpret derivatives is a fundamental part of working with Taylor approximations, as they directly influence how accurate your approximation could be.

Trigonometric Functions

Trigonometric functions like \(\sin x\) and \(\cos x\) are common in Taylor polynomial problems. They come with periodic properties that are helpful for creating approximations over specific intervals.

The Taylor series for \(\sin x\) and \(\cos x\) around \(a=0\) (often referred to as Maclaurin series) converge due to these periodic properties. For example:

• The Taylor series for \(\sin x\) is \(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\)
• The Taylor series for \(\cos x\) is \(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots\)

For trigonometric functions, determining an upper bound for the derivative is commonly straightforward, due to their maximum values being 1 or less, which is particularly useful when estimating their Taylor polynomial error bounds.

These series are applicable over a wide range of values and are quite accurate once you compute enough terms. The repetitive nature of these functions often makes the process of finding \( M \) more predictable, since the derivatives themselves are simple and follow an alternating pattern.

Exponential Functions

Exponential functions, like \( e^x \), are another popular choice for Taylor polynomial approximation. The exponential function is particularly special because each derivative of \( e^x \) is also \( e^x \), retaining the same form.

For exponential functions, their Taylor series, when centered at 0, is:

• \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)

Due to the nature of \( e^x \), the derivatives do not oscillate like trigonometric functions, making them behave in a predictable and consistent manner across various intervals. These qualities often simplify the process of determining the error bound, especially as \( n \) increases, since each higher order derivative is simply \( e^x \), and the constant \( M \) can remain the same within the considered interval.

The exponential function's behavior is smooth and predictable, which makes it an ideal candidate for Taylor series, allowing for accurate approximations. The unchanging nature of its derivatives simplifies both the calculation and interpretation of error bounds.