Problem 51 - University Calculus: Early Transcendentals

Question

Use a CAS to perform the following steps for the functions. a. Plot \(y=f(x)\) over the interval \(\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)\) b. Holding \(x_{0}\) fixed, the difference quotient $$q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}$$ at \(x_{0}\) becomes a function of the step size \(h .\) Enter this function into your CAS workspace. c. Find the limit of \(q\) as \(h \rightarrow 0\) d. Define the secant lines \(y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)\) for \(h=3,2\) and \(1 .\) Graph them together with \(f\) and the tangent line over the interval in part (a). $$f(x)=x+\sin (2 x), \quad x_{0}=\pi / 2$$

Step-by-Step Solution

Answer

Limit of difference quotient is 1; secant lines intersect tangent at \(x_0 = \pi/2\).

Step 1: Plot the Function

First, let's plot the function \( y = f(x) = x + \sin(2x) \) over the interval \( (x_{0} - 1/2) \leq x \leq (x_{0} + 3) \), where \( x_{0} = \pi/2 \). The interval becomes \( \pi/2 - 1/2 \leq x \leq \pi/2 + 3 \). This results in the interval \( 1.07 \leq x \leq 4.64 \). Use a CAS to input this function and set the interval to visualize \( f(x) \).

Step 2: Derive the Difference Quotient

The difference quotient \( q(h) = \frac{f(x_{0} + h) - f(x_{0})}{h} \) becomes a function of \( h \) with \( x_{0} = \pi/2 \). Substitute \( f(x) = x + \sin(2x) \) into the difference quotient: \[ q(h) = \frac{\left(\frac{\pi}{2} + h + \sin\left(2 \left(\frac{\pi}{2} + h\right)\right) - \left(\frac{\pi}{2} + \sin(\pi)\right)\right)}{h} \] Simplify this expression to: \[ q(h) = \frac{h + \sin(\pi + 2h)}{h} \] Enter this expression in your CAS.

Step 3: Find the Limit of Difference Quotient as \( h \to 0 \)

Calculate the limit of \( q(h) = \frac{h + \sin(\pi + 2h)}{h} \) as \( h \to 0 \). The limit represents the derivative of \( f(x) \) at \( x = \pi/2 \). Use L'Hôpital's rule or direct calculation in a CAS:\[ \lim_{h \to 0} q(h) = \lim_{h \to 0} \frac{h + \sin(\pi + 2h)}{h} \]The limit resolves to 1, since the derivative \( f'(x) = 1 + 2\cos(2x) \) evaluated at \( x = \pi/2 \) equals 1.

Step 4: Define Secant Lines and Graph

Define the secant lines based on the difference quotient for \( h = 3, 2, \) and \( 1 \). The general form of the secant line is \( y = f\left(x_{0}\right) + q(h) \cdot (x - x_{0}) \). Substituting the known value of \( q(0) = 1 \), calculate:- Secant line for \( h = 3 \): \( y = f\left(\pi/2\right) + q(3) \cdot (x - \pi/2) \)- Secant line for \( h = 2 \): \( y = f\left(\pi/2\right) + q(2) \cdot (x - \pi/2) \)- Secant line for \( h = 1 \): \( y = f\left(\pi/2\right) + q(1) \cdot (x - \pi/2) \)Graph these secant lines alongside \( f(x) \) and the tangent line (which uses the limit at \( h=0 \)) over the interval \((1.07, 4.64)\).

Key Concepts

Difference Quotient

The difference quotient is an essential concept in calculus that provides us a way to approximate the slope of a function at a specific point. It is essentially the average rate of change of the function over an interval. To compute the difference quotient, we use the formula:

• \( q(h) = \frac{f(x_0 + h) - f(x_0)}{h} \)

Here, \( h \) represents a small change in the input variable \( x \), and \( x_0 \) is the specific point of interest.
The difference quotient allows us to explore how the function behaves near \( x_0 \) by dividing the change in function values by the change in \( x \).
As \( h \) approaches zero, the difference quotient evolves into a powerful tool for finding derivatives.

Limit of Function

The concept of a limit is fundamental to calculus and is closely tied to the idea of continuous change. Calculating the limit of a difference quotient as \( h \) approaches zero finds the instantaneous rate of change or the derivative of the function.
For our specific case, the limit we want to find is:

• \( \lim_{h \to 0} q(h) = \lim_{h \to 0} \frac{h + \sin(\pi + 2h)}{h} \)

This evaluates the derivative of the function at a particular point, \( x_0 = \pi/2 \). Using the rules of limits or a CAS, we simplify and compute this to get the derivative value. In this exercise, the derivative \( f'(x) \) at \( x = \pi/2 \) equals 1, which indicates the slope of the tangent line at this point on the curve.
This limit approach solidifies the conversion from average change to instantaneous change, thus unlocking deeper insights into the behavior of functions.

Secant Line

The secant line of a function connects two points on the curve of the function and provides an average slope or rate of change between these points. It's an initial concept leading toward understanding tangents and derivatives. For the given function, the secant lines are defined using the difference quotient at different \( h \) values, specifically:

• \( y = f(x_0) + q(h) \cdot (x - x_0) \)

The choice of \( h \) values—\( 3, 2, \) and \( 1 \)—allows us to see how the secant lines approximate the tangent line as \( h \) decreases.
By plotting these secant lines along with the original function, we gain a visual appreciation of how they converge towards the tangent line as \( h \) approaches zero, revealing the precision of the tangent at a single point.

Tangent Line

The tangent line is ultimately what the secant lines approach as \( h \) approaches zero. It gives us the exact slope of the function at the specific point \( x_0 \). For functions, the tangent line is the best linear approximation of the function near the point \( x_0 \).

• Its equation generally takes the form: \( y = f(x_0) + q(0) \cdot (x - x_0) \)

Knowing the derivative at \( x_0 = \pi/2 \) from our limit calculation, \( f'(x) = 1 \), enables us to write the tangent line using this derivative value.
This tangent line illustrates the instantaneous rate of change of the function, offering a precise slope and showing how quickly the function's values are changing at \( x_0 \). Understanding and visualizing tangent lines are key to a deeper understanding of calculus and are frequently used in real-world applications.

Graphing with CAS

A Computer Algebra System (CAS) is a tool that assists in visualizing complex mathematical concepts, including functions, limits, and derivatives. Using CAS for plotting helps make abstract concepts tangible. For this exercise:

• First, graph the function \( f(x) = x + \sin(2x) \) over the specified interval \( 1.07 \leq x \leq 4.64 \).
• Next, add the secant lines at different step sizes \( h = 3, 2, 1 \).
• Finally, overlay the tangent line derived from the limit.

These visuals aid in comparing how the secant lines approach the tangent line. By using graphing tools like CAS, students can experiment and intuitively understand the slope concepts.
It's an effective way to see calculus in action, helping learners grasp more abstract ideas through visual interaction and dynamic exploration.