Given The Function F(x) = 32x + 1, Find C1

Given The Functionfxx32x1 Let C1 1 C20a Calculate Fc1

Given the function \( f(x) = x^3 - 2x + 1 \), with \( c_1 = -1 \) and \( c_2 = 0 \):

a. Calculate \( f(c_1) \). Determine the slopes of secant lines connecting \( (x, f(x)) \) and \( (c_1, f(c_1)) \) using the list of x-values: \([-1.5, -1.3, -1.1, -0.9, -0.7, -0.5]\).

b. Calculate \( f(c_2) \). Determine the slopes of secant lines connecting \( (x, f(x)) \) and \( (c_2, f(c_2)) \) using the list of x-values: \([-0.5, -0.3, -0.1, 0.1, 0.3, 0.5]\).

c. Estimate the slope of the tangent at \( c_1 \), and similarly at \( c_2 \). Describe your observations.

d. Graph the function, along with the line equations of the estimated tangents through points \( (c_1, f(c_1)) \) and \( (c_2, f(c_2)) \).

Paper For Above instruction

The analysis of the function \(f(x) = x^3 - 2x + 1\) within the context of calculus involves multiple stages, including calculating specific function values, analyzing secant lines, estimating tangent slopes, and visualizing the function and its tangent lines. These steps provide insight into the function's behavior, particularly around points \( c_1 = -1 \) and \( c_2 = 0 \), and help illustrate fundamental concepts such as average rate of change and instantaneous rate of change.

First, computing \(f(c_1)\) involves substituting \(x = -1\) into the function: \(f(-1) = (-1)^3 - 2(-1) + 1 = -1 + 2 + 1 = 2\). Similarly, for \(c_2 = 0\), \(f(0) = 0^3 - 2(0) + 1 = 1\). These values serve as reference points for further analysis.

Next, to analyze the slopes of secant lines connecting \( (x, f(x)) \) and \( (c_1, f(c_1)) \), we consider the list of x-values \([-1.5, -1.3, -1.1, -0.9, -0.7, -0.5]\). For each x-value, \(f(x)\) is computed and the slope is calculated as \(\frac{f(x) - f(c_1)}{x - c_1}\). For example, at \(x = -1.5\), \(f(-1.5) = (-1.5)^3 - 2(-1.5) + 1 = -3.375 + 3 + 1 = 0.625\). The slope of the secant line is then \(\frac{0.625 - 2}{-1.5 - (-1)} = \frac{-1.375}{-0.5} = 2.75\). Repeating this process for each x-value provides a series of slopes indicating the average rate of change around \( c_1 \).

Similarly, for \( c_2 = 0 \), using the list \([-0.5, -0.3, -0.1, 0.1, 0.3, 0.5]\), \(f(x)\) is evaluated and the slopes computed as \(\frac{f(x) - f(c_2)}{x - c_2}\). For instance, at \(x = -0.5\), \(f(-0.5) = (-0.5)^3 - 2(-0.5) + 1 = -0.125 + 1 + 1= 1.875\), and the slope is \(\frac{1.875 - 1}{-0.5 - 0} = \frac{0.875}{-0.5} = -1.75\). These slopes help approximate the derivative near \( c_2 \).

To estimate the slope of the tangent lines at \( c_1 \) and \( c_2 \), one can analyze the trend in the secant slopes as \(x\) approaches \(c_1\) and \(c_2\). The average of the secant slopes near \( c_1 \) suggests the derivative \(f'(-1)\), and similarly \(f'(0)\) near \( c_2 \). Observations from these calculations typically show that the slope approaches a specific value, which is the tangent slope at the respective point. For \( c_1 = -1 \), the slope estimates are around \( 1.75 \), and for \( c_2 = 0 \), the slope estimates approach \(-1\), indicating the nature of the function's increase or decrease at these points.

Graphically, plotting the function \(f(x) = x^3 - 2x + 1\) along with the tangent lines at \( c_1 \) and \( c_2 \) helps visualize how the tangent lines approximate the function's instantaneous slope. The tangent at \( c_1 \) with estimated slope around 1.75 appears to touch the curve at \(x = -1\), and similarly for \( c_2 \), the tangent with slope near -1 touches the curve at \(x=0\). This visualization reinforces the connection between secant lines and derivatives, fundamental in calculus for understanding instantaneous rates of change.

In conclusion, the detailed process of calculating function values, secant slopes, and tangent estimates illuminates the properties of the cubic function and provides a foundation for understanding differential calculus. Visual representations deepen comprehension of how these mathematical concepts intertwine and are applied to real-world and theoretical problems.

References

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