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. $$ f(x)=-x^{4}+4 x^{3}+8 x^{2}+5,[0,5] $$

Short answer

After graphing the function and the secant line, find the slope of the secant line. The slope of the secant line is also the slope of the tangent line. Use this slope, set it to equal to the derivative of the function, and solve for x to get the points where the tangent line meets the graph. After finding these points, graph the tangent lines. Now, you should have graphs of the function, secant line, and tangent lines all on one graph.

Step-by-step solution

Part A: Graph the Function

First, graph the function \(f(x)=-x^{4}+4 x^{3}+8 x^{2}+5\). This can be done using a graphing utility. Simply input the provided function and the interval [0,5] into the graphing calculator or software.

Part B: Find and Graph the Secant Line

Next, the secant line passing through the points on the function at the interval endpoints needs to be found. That means the secant line passes through the points (0,f(0)) and (5,f(5)). Calculate the function value at these points and find the slope of the line connecting them. The slope can be found with the formula \((y_2 - y_1) / (x_2 - x_1)\). After finding the slope, apply it in the equation of the line, y = mx + c, where m is the slope, and c is the y-intercept. Graph this line on the graph.

Part C: Find and Graph the Tangent Lines

Lastly, identify and graph the tangent lines parallel to the secant line calculated in Part B. As the slopes of parallel lines are equal, the slope of the tangent line will be the same as the slope of the secant line. To find the points where this parallel tangent line touches the curve, we need to find the derivative (rate of change) of the function f, and set it equal to the slope of the secant line. Solve for x in this equation to find the points where the tangent line(s) touch the curve. This might require calculus, specifically the application of derivatives and solving equations. Finally, graph these tangent lines on the graph of the function.

Key concepts

Secant Line Calculation

Understanding the concept of a secant line is fundamental in calculus, and it's a crucial step when graphing functions and their properties. A secant line intersects a curve at two distinct points, essentially providing an average rate of change between those points.

To calculate a secant line, you need two points on a curve. In the exercise, these points are given by the endpoints of the interval [0,5] on the function.The mathematical step to find the secant line involves computing the slope using the points (0,f(0)) and (5,f(5)). This is achieved using the formula for slope, which is \[\begin{equation}m = \frac{y_2 - y_1}{x_2 - x_1}\end{equation}\]

1. Calculate the y-values by substituting the x-values of the points into the function.
2. Use the slope formula to determine the slope 'm' of the secant line.
3. With the slope and one point known, use the point-slope form, or y-intercept form, to write the equation of the secant line.

Once the equation of the secant line is formed, it can be graphed alongside the original function to visually represent this average rate of change. Remember, the y-intercept 'c' can be derived if needed, ensuring that the equation fits the graphical points accurately.

Tangent Lines Parallel to Secant

Once we have the secant line, we might be interested in exploring tangent lines—lines that lightly touch the curve at exactly one point without crossing it. Particularly, we may want to find tangent lines that are parallel to the previously determined secant line. Since parallel lines have the same slope, we're looking for tangent lines with the same slope as the secant line.

To find such lines, we use calculus and the concept of derivatives. A derivative represents the slope of a tangent line at any point on a function. Here's how the process unfolds:

• Take the derivative of the function, which gives us a formula for the slope of the tangent at any point along the function.
• Set the derivative equal to the slope of the secant line.
• Solve for 'x' to find the specific points where the slope of the tangent line (the derivative) matches the slope of the secant line.

These x-values correspond to the points on the curve where the tangent lines will be parallel to the secant line. Once we find these points, we can use the original function to get the corresponding y-values, giving us the points of tangency. From here, we can construct the equations of the tangent lines using the point-slope form, ensuring the slope remains equal to that of the secant line.

Using Derivatives in Calculus

Derivatives are the cornerstone of calculus and are crucial for analyzing the behavior of functions. They provide a way to determine the rate at which a function's output changes with respect to its input, which corresponds to the slope of a tangent line at any point. In the context of our exercise, we utilized derivatives to find tangent lines that are parallel to a given secant line. This application is just one example of the power of derivatives.

In a broader sense, derivatives can be used to solve a myriad of problems, such as:

1. Finding local maxima and minima of a function, which are points where a curve peaks or dips.
2. Understanding the concavity of functions, which indicates the direction and manner in which a curve bends.
3. Analyzing real-life situations like calculating velocity and acceleration in physics.

To use derivatives effectively, one must understand how to differentiate various functions, apply the chain rule, product rule, and quotient rule for composite functions, and even tackle implicit differentiation for more complex scenarios. With derivatives being such a significant concept in calculus, mastering their use opens a door to a deeper understanding of the mathematical principles that govern change and motion in the natural world.