Problem 37 - Calculus

Question

Find \(f(c)=1.001^{10}\) in two ways \(-\) by calculator and by \(f(c)-f(a) \approx f^{\prime}(a)(c-a) .\) Choose \(a=1\) and \(f(x)=x^{10}\)

Step-by-Step Solution

Answer

Using linear approximation, \(1.001^{10} \approx 1.01\). Numerically, \(1.001^{10}\) from a calculator yields around 1.0101.

Step 1: Approximate Using the Calculator

Use a calculator to compute \(1.001^{10}\). This simple power calculation can be entered directly into the calculator, providing a straightforward numerical approximation of \(f(c)\).

Step 2: Define the Derivative of the Function

Given \(f(x) = x^{10}\), calculate the derivative \(f'(x)\). By the power rule, \(f'(x) = 10x^9\).

Step 3: Implement the Linear Approximation Formula

Use the formula \(f(c) - f(a) \approx f'(a)(c-a)\) to approximate \(f(c)\). Let \(a = 1\), \(c = 1.001\), and substitute into the formula:\[f(1.001) - f(1) \approx f'(1)(1.001 - 1)\]

Step 4: Compute the Values in the Approximation

Calculate \(f(1) = 1^10 = 1\) and \(f'(1) = 10 \times 1^9 = 10\). Then the expression becomes:\[f(1.001) - 1 \approx 10 \times 0.001 = 0.01\]

Step 5: Solve for f(c)

Add the result from the linear approximation to \(f(1)\) to find \(f(1.001)\):\[f(1.001) \approx 1 + 0.01 = 1.01\]

Key Concepts

Derivative

Derivatives are an essential concept in calculus. They represent the rate at which a function changes at any given point. In simple terms, a derivative helps us understand how a function behaves when its input varies slightly. Derivatives are calculated by using differentiation rules.

The derivative of a function \(f(x)\) is denoted as \(f'(x)\). For the function \(f(x) = x^{10}\), its derivative \(f'(x)\) describes how the output of \(x^{10}\) changes as \(x\) changes.

To find \(f'(1)\), we use the derivative formula. The specific method used here relies on the **Power Rule**, which simplifies the differentiation process of power functions like \(x^n\).

Power Rule

The Power Rule is one of the fundamental rules of differentiation, making the task of finding derivatives of power functions straightforward. If you have a function in the form \(f(x) = x^n\), the Power Rule states that its derivative is \(f'(x) = nx^{n-1}\).

In our given exercise, \(f(x) = x^{10}\). Applying the Power Rule, the derivative we get is \(f'(x) = 10x^9\). This means for every slight increase in \(x\), the function \(x^{10}\) changes by \(10x^9\) times that increase. It's a handy rule for functions involving powers, helping us quickly find the derivative without complex calculations.

Calculator Approximation

When given a function like \(1.001^{10}\), using a calculator provides a straightforward numerical approximation. Calculators are equipped to handle such computations swiftly and accurately by using power functions.

All you need to do is input \(1.001^{10}\) into the calculator, and it will return the approximate value of \(f(c)\). This process bypasses lengthy manual calculations, making it ideal when you need a quick solution.

In this exercise, the calculator directly computes \(1.001^{10}\) to find the value of \(f(1.001)\), ensuring speed and precision in obtaining the result.

Function Evaluation

Function evaluation involves determining specific values of a function given certain inputs. For example, in the exercise when \(x = 1\), we evaluate the function \(f(x) = x^{10}\) by plugging in the value to get \(f(1) = 1^{10} = 1\).

This process is crucial when applying the linear approximation method, where initial and derived values at specific points are required to approximate new values. Here, after finding the derivative \(f'(x)\) and evaluating \(f(x)\) at \(a = 1\), we use this baseline to apply the linear approximation technique for nearby values like \(c = 1.001\).

Function evaluation is a basic yet powerful tool in mathematics for understanding how functions behave at particular points, serving as the foundation for more complex calculus methods.