Problem 29 - Calculus: Early Transcendentals

Question

Tangent lines with zero slope a. Graph the function \(f(x)=x^{2}-4 x+3\) b. Identify the point \((a, f(a))\) at which the function has a tangent line with zero slope. c. Confirm your answer to part (b) by making a table of slopes of secant lines to approximate the slope of the tangent line at this point.

Step-by-Step Solution

Answer

Answer: The function has a tangent line with a zero slope at the point \((2, -1)\).

Step 1: Graph the function

To graph the function \(f(x) = x^2 - 4x + 3\), you should use graphing software or manually create a table of values for different \(x\), substitute them in the function, and plot the corresponding points. Once you plot the points depending on the example values of \(x\), you can visualize a parabola that opens upwards.

Step 2: Find the derivative

To identify the point \((a, f(a))\) at which the function has a tangent line with a zero slope, we have to find the derivative of the function \(f(x)\). The derivative represents the slope of the tangent line at any point on the graph: $$f'(x) = \frac{d}{dx}[x^2 - 4x + 3] = 2x - 4.$$

Step 3: Set the derivative to zero

Since we're looking for a tangent with a zero slope, we set the derivative \(f'(x)\) to zero and solve for \(x\): $$2x - 4 = 0 \Rightarrow x = 2.$$ Now, we've found the \(x\)-coordinate of the point \((a, f(a))\), which is \(a = 2\).

Step 4: Find the y-coordinate

Now that we have the \(x\) value, we need to find the corresponding \(y\) value. Substitute our \(x = 2\) into the original function \(f(x)\): $$f(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1.$$ Now, we have the point \((a, f(a))\) at which the tangent line has a zero slope: \((2, -1)\).

Step 5: Create a table of slopes for secant lines

To confirm our answer and check that \((2, -1)\) is the point at which the tangent line has a zero slope, create a table of slopes of secant lines around that point. Remember that secant lines connect two nearby points on the graph, so the slope of a secant line can be found using the difference quotient: $$\frac{f(x) - f(2)}{x - 2}.$$ Choose several \(x\) values near the point \(x = 2\) and calculate the slope of each secant line. As they become closer to \(2\), the slopes should approach zero. Example: $$\begin{array}{c|c} x & \frac{f(x) - f(2)}{x - 2} \\ \hline 1.9 & -0.1 \\ 1.99 & -0.01 \\ 2 & - \\ 2.01 & 0.02 \\ 2.1 & 0.2 \end{array}$$ The table shows that the slope of the secant lines approaches zero as the \(x\) values get closer to \(2\), which confirms that the tangent line's slope at the point \((2, -1)\) is indeed zero.

Key Concepts

Zero Slope

Understanding the concept of zero slope is crucial when analyzing functions and tangents. Slope indicates the steepness or incline of a line. When a line has a zero slope, it is perfectly horizontal. This means there is no rise or fall as you move along the line.

For a function, the slope of the tangent line at a particular point is found using the derivative. If the derivative equals zero, this indicates a flat tangent, or in other words, a tangent line with zero slope. In practice, identifying such a point involves determining where the derivative of the function is equal to zero. This informs us where the graph has critical points like peaks, valleys, or flat areas.

• Zero slope implies a horizontal tangent line.
• It's found by setting the derivative equal to zero.
• Helps locate critical points on a graph.

Graphing Functions

Graphing functions is a visual way to understand how they behave. It allows us to see the shape of the function, its direction, and identify key features like intercepts and critical points. For the function \( f(x) = x^2 - 4x + 3 \), we can plot its graph to visualize it more clearly.

This particular function is a quadratic, characterized by its parabolic shape. Parabolas open either upwards or downwards depending on the leading coefficient's sign. In this case, since the coefficient \( x^2 \) is positive, our parabola opens upwards.

• Use a table of values to plot points on the graph.
• Notice the symmetry in the graph for quadratic functions.
• The vertex of the parabola can offer insights into maxima or minima.

Derivative Calculus

Derivative calculus is a powerful tool in understanding the behavior and movement of functions. The derivative of a function gives the slope of the tangent line at any given point on the curve. This can tell us how the function is increasing or decreasing at that point.

Calculating the derivative involves differentiating the function with respect to \( x \). For \( f(x) = x^2 - 4x + 3 \), the derivative is \( f'(x) = 2x - 4 \). Solving for when this derivative equals zero gives points where the slope of the tangent is flat.

• The derivative represents instantaneous rate of change.
• Provides critical information about the slope of tangents.
• Is essential for finding extremes and inflection points.

Secant Lines

Secant lines are linear approximations that intersect a curve at two points. They give us a way of approximating the rate of change over a specific interval of a graph. By evaluating secant lines around a point of interest, we can estimate the slope of the tangent line.

In practice, as with the example function, you choose points increasingly closer to the point of interest, and calculate the slope between these points. The key formula for a secant line's slope is the difference quotient, \( \frac{f(x) - f(a)}{x - a} \). As \( x \) approaches \( a \), the secant line's slope approaches the slope of the tangent line.

• Secant lines intersect two points on a curve.
• Useful for approximating tangent line slopes.
• The difference quotient helps find secant slopes.