Problem 58 - Calculus Single Variable

Question

A function \(f,\) a point \(c,\) an increment \(\Delta x,\) and a positive integer \(n\) are given. Use the method of increments to estimate \(f(c+\Delta x)\). Then let \(h=\Delta x\) / \(N\). Use the method of increments to obtain an estimate \(y_{1}\) of \(f(c+h) .\) Now, with \(c+h\) as the base point and \(y_{1}\) as the value of \(f(c+h),\) use the method of increments to obtain an estimate \(y_{2}\) of \(f(c+2 h)\). Continue this process until you obtain an estimate \(y_{N}\) of \(f(c+N \cdot h)=f(c+\Delta x) .\) We say that we have taken \(N\) steps to obtain the approximation. The number \(h\) is said to be the step size. Use a calculator or computer to evaluate \(f(c+\Delta x)\) directly. Compare the accuracy of the one step and \(N\) -step approximations. $$ f(x) 1 / \sqrt[3]{x}, c=-8, \Delta x=1, N=4 $$

Step-by-Step Solution

Answer

Both the one-step and N-step approximations involve calculating \(\frac{1}{(-7)^{1/3}}\), but may vary slightly depending on the calculation approach. Compare precision.

Step 1: Define Parameters

Start by identifying the parameters given in the problem:- The function, \(f(x) = \frac{1}{\sqrt[3]{x}}\)- The base point, \(c = -8\)- The increment, \(\Delta x = 1\)- The number of steps, \(N = 4\).

Step 2: Calculate Step Size

The step size \(h\) is calculated using the formula: \[h = \frac{\Delta x}{N} = \frac{1}{4} = 0.25.\]

Step 3: Estimate f(c + Δx) using Method of Increments

The original increment method gives one estimate:\[f(c + \Delta x) = f(-8 + 1) = f(-7).\]Using the function, calculate:\[f(-7) = \frac{1}{\sqrt[3]{-7}} = \frac{1}{(-7)^{1/3}}.\]

Step 4: Calculate y1 using Initial Base Point

Calculate \(y_1\) with \(h = 0.25\):\[f(c+h) = f(-8 + 0.25) = f(-7.75).\]Substituting into the function, we have:\[y_1 = \frac{1}{(-7.75)^{1/3}}.\]

Step 5: Calculate y2 using Updated Base Point

Use \(c + h = -7.75\) as the base and \(y_1\) as \(f(c+h)\):\[f(c+2h) = f(-7.75 + 0.25) = f(-7.5).\]Calculate:\[y_2 = \frac{1}{(-7.5)^{1/3}}.\]

Step 6: Calculate y3 using Updated Base Point

Use \(c + 2h = -7.5\) as the base:\[f(c+3h) = f(-7.5 + 0.25) = f(-7.25).\]Calculate:\[y_3 = \frac{1}{(-7.25)^{1/3}}.\]

Step 7: Calculate y4 using Updated Base Point

Finally, use \(c + 3h = -7.25\) as the base:\[f(c+4h) = f(-7.25 + 0.25) = f(-7).\]Calculate:\[y_4 = \frac{1}{(-7)^{1/3}}.\]

Step 8: Compare Approximations and Direct Value

Compute the direct value of \(f(-7)\) using a calculator:\[f(-7) = \frac{1}{(-7)^{1/3}}.\]Compare the direct one-step estimate and the \(N\)-step estimate \(y_4\). Evaluate accuracy against the direct computation.

Key Concepts

Method of Increments

The method of increments involves a clever approach to estimate function values incrementally. The primary idea is to begin with a known base point and function value, then take small steps, or increments, to approximate the function's value further away. In our exercise, we start with a specific point, called \( c \), and want to estimate the value of the function \( f \) at a distant point \( c + \Delta x \). The formula works by repeatedly applying increments to small sections of the function, where \( \Delta x \) is the total distance you want to cover.
By breaking this distance into smaller parts, or steps, and estimating the function’s change piece by piece, students can gradually approach the desired point with greater accuracy. This method is particularly useful in fields like calculus, where exact evaluation of complex functions is difficult without digital tools.

Step Size Calculation

Step size calculation is a crucial aspect of numerical approximation with increments. The step size, usually denoted by \( h \), determines how small each increment is. In our exercise, \( h \) is calculated by dividing the total change \( \Delta x \) by the number of steps \( N \). Specifically:
\[ h = \frac{\Delta x}{N} \]
For this problem, where \( \Delta x = 1 \) and \( N = 4 \), the step size becomes \( h = 0.25 \).
This means each incremental step takes a quarter-unit stride towards the final estimate. A smaller step size generally leads to a more accurate approximation since it allows for finer changes along the function's curve.

• With a larger \( N \), the function's path is traced more accurately.
• However, extremely small step sizes can overburden computations, making the process inefficient. It’s about finding the right balance for precision and computation power.

Function Estimation

Function estimation involves calculating the approximate values of a function at various increments. With the method of increments, once we have the step size, we can begin estimating the function at different points.
At each step, from the base point \( c \), consider:\( h = 0.25 \), the first calculation estimates \( f(c+h) \). In this exercise, with \( f(x) = \frac{1}{\sqrt[3]{x}} \), the calculations proceed as follows:

• Start with \( y_1 = f(-7.75) \)
• Then, use the result to derive \( y_2 = f(-7.5) \).
• Continue incrementally until \( y_4 = f(-7) \).

Each \( y_n \) uses the function formula to estimate the value at that particular point. Function estimation through such incremental steps provides better precision by allowing adjustments at each small step, refining the approximation as you advance.

Comparison of Approximations

Comparing approximations is an essential part of understanding the accuracy of your estimates. In this exercise, we obtain two approximations: a one-step approximation and an \( N \)-step approximation. Here's how they differ:

The one-step approximation involves a direct calculation of \( f(c + \Delta x) \), in this case, \( f(-7) \). While fast, it risks being less precise if \( \Delta x \) is large.
The \( N \)-step approximation, however, involves multiple smaller steps using the method of increments, in this case, bringing values from \( y_1 \) to \( y_4 \), ending at \( f(-7) \).

• The \( N \)-step method generally provides a pathway to more accurate results.
• Each smaller step potentially captures nuances in the function's behavior, reducing overall estimation error compared to a single leap.

Both methods should be calculated and compared to the actual values obtained through digital computation to check for accuracy. This understanding helps in determining which method suits best given the specific situation or computational resources.