Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 5 through 8 , estimate \(f^{\prime}(c)\) by calculating the difference quotient \(\frac{f(c+h)-f(c)}{h}\) for successively smaller values of \(|h| .\) Use both positive and negative values of \(h .\). $$ f(x)=\sqrt[3]{x} \text { . Approximate } f^{\prime}(8) $$

Step-by-Step Solution

Answer

The approximate value of \(f'(8)\) using this method of difference quotients is approximately 0.167.

Step 1: Understand the problem and the derivative

The first step is to understand what is required. In this case, we are asked to calculate the derivative of the function \(f(x) = \sqrt[3]{x}\) at the point \(x = 8\). The derivative of a function at a point is the instantaneous rate of change of the function at that point. Each value of \(h\) we calculate the difference quotient for will give us a different approximation of this rate of change.

Step 2: Substitution of the function in difference quotient

To solve \(f^{\prime}(c)\) by calculation, substitute \(\sqrt[3]{x}\) into the difference quotient formula. This yields \(\frac{f(8+h) - f(8)}{h}\), which simplifies to \(\frac{\sqrt[3]{(8+h)} - \sqrt[3]{8}}{h}\).

Step 3: Estimation using different values of h

Estimate the derivative by calculating this difference quotient for various positive and negative values of \(h\), approaching zero. As the value of \(h\) gets smaller, the result gets closer to the true value of the derivative. This process can continue until a sufficiently accurate estimate is obtained, for example until the difference from one estimate to the next is less than a specified tolerance level.

Step 4: Approximate derivative at x = 8

For instance, substituting \(h = 0.1\) and \(h = -0.1\) gives \( \approx 0.166\) and \( \approx 0.168\) respectively. As \(h\) becomes smaller, the calculated values converge to the true value of \(f'(8)\), which is approximately \( \approx 0.167\). This indicates that the rate of change of the function \(f(x) = \sqrt[3]{x}\) at \(x = 8\) is approximately 0.167.

Key Concepts

Derivative Approximation

Understanding the concept of derivative approximation is critical for students grappling with calculus. In essence, when we speak of approximating a derivative, we're referring to finding a value that comes close to the actual derivative at a certain point of a function. This is typically achieved using the difference quotient \[ \frac{f(c+h)-f(c)}{h} \] for values of h that approach zero.

Consider our current example, where you want to approximate the derivative of \(f(x) = \sqrt[3]{x}\) at \(x = 8\). By inserting small positive and negative values of h, such as 0.1 and -0.1, we can calculate that \(f^\prime(8) \approx 0.167\). The smaller the values of h we use, the closer we get to the true derivative. This can be imagined as zooming in on the curve of the function at \(x = 8\), where the slope of the tangent line represents the derivative.

As we continue to use smaller values of h, the concept of limits comes into play. The limit of the difference quotient as h approaches zero is the precise definition of the derivative. Therefore, when you cannot compute the derivative analytically, derivative approximation acts as a bridge to understanding the nature of the function's instantaneous rate of change at a specific point.

Instantaneous Rate of Change

The term instantaneous rate of change appears frequently in calculus and is intimately related to the concept of the derivative. It measures how fast a function's output changes at an exact point. Think of it as the speedometer of a car showing how fast you're going at one instant, rather than over a longer trip where we would measure average speed.

When you calculate the difference quotient \[ \frac{f(c+h)-f(c)}{h} \], for various values of h, this process simulates the function's behavior very close to the point of interest — in our textbook example, the point \(x = 8\). Hence, although the values derived from this method are not the exact instantaneous rate of change, they provide an approximation that can often be sufficient for practical purposes. The true instantaneous rate of change, or the exact derivative, at \(x = 8\) for our function \(f(x) = \sqrt[3]{x}\) can be determined by evaluating the limit of the difference quotient as h approaches zero, which refines the approximation until it represents the exact rate of change. This connections between limit and rate provides the foundational framework for many more complex concepts in differential calculus.

Functional Notations

The topic of functional notations in calculus is another cornerstone that enables clear communication of mathematical ideas. In the context of derivatives and calculus, functional notation such as \(f(x)\), \(f^\prime(x)\), and \(f^\prime(c)\) serves as a shorthand to represent various aspects of functions and their derivatives. For example, \(f(x)\) refers to the value of the function at any general input \(x\), while \(f^\prime(x)\) denotes the derivative of the function at \(x\), which is the same as the instantaneous rate of change at that point.

In our problem, when we discuss \(f^\prime(8)\), we refer specifically to the derivative of the function \(f\) when the input is 8. Knowing how to read and interpret these notations is crucial for working through calculus problems since they provide a concise way to express complex operations. Furthermore, they become indispensable when we deal with higher-order derivatives, where notations like \(f^\prime\prime(x)\) indicate the second derivative, or the rate at which the rate is changing — a concept known as concavity.