Problem 6 - 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 \(h .\). $$ f(x)=\frac{1}{\sqrt{x}} \text { . Approximate } f^{\prime}(4) $$

Step-by-Step Solution

Answer

By repeating the process for smaller and smaller values of \(h\), both positive and negative, you get the successive approximations of the derivative \(f^{\prime}(4)\). As \(h\) gets closer to zero, these approximations should get closer and closer to the exact value of the derivative at \(x = 4\). The exact value is not given, but the process should provide a good approximation.

Step 1: Evaluate the Function

Firstly, evaluate the function, \(f(x) = \frac{1}{\sqrt{x}}\), at the given point \(x = 4\) to get \(f(4) = \frac{1}{2}\).

Step 2: Compute the Difference Quotient for Positive h

Now, use a positive \(h\) value and compute the difference quotient \(\frac{f(c+h)-f(c)}{h}\). For \(h = 1\), we get\(\frac{f(4+1)-f(4)}{1}= \frac{\frac{1}{\sqrt{5}} - \frac{1}{2}}{1} = -\frac{2\sqrt{5} - 4}{20}\).You should repeat this step for smaller values of \(h\) - such as \(h = 0.1, 0.01, 0.001\), etc.

Step 3: Compute the Difference Quotient for Negative h

Now, use a negative \(h\) value and compute the difference quotient \(\frac{f(c+h)-f(c)}{h}\). For \(h = -1\), we get\(\frac{f(4-1)-f(4)}{-1}= \frac{\frac{1}{\sqrt{3}} - \frac{1}{2}}{-1} = \frac{2\sqrt{3} - 4}{24}\).Again repeat this for smaller values of \(h\) - such as \(h = -0.1, -0.01, -0.001\), etc.

Step 4: Approximate the Derivative

Now that the difference quotients have been computed for both positive and negative \(h\) values, you should look at the behavior of these values as \(h\) approaches zero. These values will converge to an approximation of the derivative of the function at the point \(x = 4\).

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept in calculus used to estimate the derivative of a function at a particular point. It involves calculating the average rate of change of the function over an interval. The expression for the difference quotient is \( \frac{f(c+h)-f(c)}{h} \), where \( c \) is the point where we want to find the derivative, and \( h \) is a small increment.
In practice, the difference quotient provides a way to approximate the slope of the tangent line to the curve at a specific point. This is done by finding the slope of the secant line connecting two points on the curve: \( (c, f(c)) \) and \( (c+h, f(c+h)) \).
Some key points about the difference quotient:

• It approximates the derivative by looking at the change in the function values over \( h \).
• As \( h \) decreases, the secant line better approximates the tangent line, providing a closer estimate of the derivative.
• Both positive and negative values of \( h \) should be used to ensure a more accurate approximation.

Understanding the difference quotient is essential for comprehending the concept of derivatives, especially when analytically finding derivatives is complex.

Limits

Limits play a crucial role when using the difference quotient to estimate derivatives. The concept of a limit is all about understanding what value a function approaches as the input approaches a certain point. In the context of derivatives, we are interested in what happens to the difference quotient as \( h \) approaches zero.
Mathematically, a derivative is defined as the limit of the difference quotient as \( h \) tends to zero:\[f'(c) = \lim_{{h \to 0}} \frac{f(c+h)-f(c)}{h}\]
Why are limits important?

• As \( h \) gets smaller, the approximation of the derivative becomes more accurate, converging to the actual value of the derivative.
• The limit helps bridge the notation of a `difference' (a finite change) to something `infinitesimally small' (differential calculus).
• Using both positive and negative \( h \) provides insight into the directional behavior of the function's rate of change.

Hence, limits help refine our understanding and provide the foundation for the analytical definition of derivatives.

Numerical Approximation

Numerical approximation is a method to estimate values that are difficult to obtain analytically. In calculus, when it comes to finding derivatives, numerical methods often come in handy, especially for complex functions or when exact solutions are not feasible.
Using the difference quotient with various values of \( h \) serves as a numerical approach. Here's how:

• Start with larger values of \( h \) to get an initial estimate.
• Gradually decrease \( h \) to see how the estimated derivative changes, refining the approximation.
• Calculate the difference quotient for both positive and negative \( h \) values to get a comprehensive view.

Numerical approximation helps in:

• Understanding the behavior of the function's rate of change at a specific point.
• Developing intuition around derivatives if you're just starting with calculus.
• Providing practical solutions when symbolic differentiation is not possible or when dealing with empirical data.

By implementing these numerical techniques, students can extract meaningful conclusions about a function's behavior at the given point, enhancing their calculus acumen.