Question

Use a graphing utility to (a) graph the function \(f\) on the given interval, (b) find and graph the secant line through points on the graph of \(f\) at the endpoints of the given interval, and (c) find and graph any tangent lines to the graph of \(f\) that are parallel to the secant line. $$ f(x)=2 e^{x / 4} \cos \frac{\pi x}{4},[0,2] $$

Short answer

The steps include graphing the function, finding and graphing the secant line by identifying the function values at the endpoints of the interval, and finding and graphing tangent lines parallel to the secant line by setting the derivative of the function equal to the slope of the secant line and solving for x.

Step-by-step solution

Graphing the Function

Firstly, the function \(f(x)=2e^{x / 4} \cos \left(\frac{\pi x}{4}\right)\) should be graphed using a graphing utility on the interval [0,2]. This involves inputting the function into the graphing utility and setting the x-axis limits to 0 and 2.

Find and Graph the Secant Line

The next step is to find and graph the secant line through the points on the graph at the endpoints of the interval [0,2]. This can be done by finding \(f(0)\) and \(f(2)\) by plugging in these values into the function f. Then, find the slope of the secant line by using the two-point formula \(m = \frac{f(2)-f(0)}{2-0}\). The secant line can then be graphed on the interval [0,2].

Find and Graph Tangent Lines

The final step is to find and graph the tangent lines parallel to the secant line. The slope of these tangent lines will be the same as it is for the secant line. To find these lines, find the derivative \(f'(x)\) of the function and set it equal to the slope of the secant line to solve for x. The tangent lines can then be graphed on the graph. Depending on the function and the slope of the secant line, there may be no, one, or possibly more solutions.

Key concepts

Secant Lines

Understanding secant lines in calculus is crucial when assessing the behavior of curves represented by functions. A secant line can be thought of as a straight line that intersects two or more points on the curve of a given function. In the context of the exercise, to graph a secant line using a graphing utility, we need to identify two specific points on the function's graph within the given interval, which is \[0,2\] for the function \(f(x)=2e^{x / 4} \cos \frac{\pi x}{4}\).

After determining the y-values of the function at these endpoints by calculation, \(f(0)\) and \(f(2)\), we calculate the slope \(m\) using the formula \(m = \frac{f(2)-f(0)}{2-0}\). This formula gives us a numerical value that represents the rate at which the function changes between the two points. This slope is then used to construct the secant line over the interval, providing a visual representation that depicts how the function progresses from one point to the other. Students should ensure that the graphed secant line extends between the x-values of the chosen points to correctly display the average rate of change of the function over that interval.

Tangent Lines

Tangent lines are a fundamental concept in calculus that describe lines that touch a curve at a single point without crossing it. These lines represent instantaneous rates of change, akin to the concept of speed at a specific moment in time. Unlike secant lines, which cross two or more points, tangent lines are all about local behavior at a single point.

In the given exercise, the challenge involves finding tangent lines to the function that are parallel to the previously determined secant line. This means these tangent lines will have the same slope as that of the secant line. To find possible points where the tangent lines are parallel, we differentiate the function to get \(f'(x)\) and set it equal to the slope of the secant line. Solving this equation may give us one or multiple values of \(x\), for which there are corresponding tangent lines. These tangent lines are crucial for understanding the behavior of the function at specific points and are pivotal in optimizing problems, curve sketching, and motion analysis.

Graphing Utility

A graphing utility is an invaluable tool for students and educators alike, especially when dealing with complex functions like \(f(x)=2e^{x / 4} \cos \frac{\pi x}{4}\). It simplifies the process of graphing such functions and aids in visualizing concepts like secant and tangent lines. When using a graphing utility, the first step often involves entering the function's equation and defining the range for the variable \(x\).

For our exercise, we would graph the function on the interval \[0,2\]. Upon entering the function details, we can easily plot the graph and use features within the utility to mark points, draw secant and tangent lines, and explore the relationship between different graphical elements. Graphing utilities often include capabilities for performing calculations like derivatives and slope computations, which are particularly useful when finding tangent lines. The visual output helps in reinforcing the concepts and gives students a graphical representation to support the algebraic work. By leveraging a graphing utility, one gains both numerical and visual insights into the behavior of functions, making it easier to understand and apply calculus principles.