Given The Function F(x) = 32x + 1, Find C1 And C2

Given The Functionfxx32x1 Let C1 1 C20a Calculate Fc1

Given the function f(x) = x^3 - 2x + 1. Let c1 = -1, c2 = 0.

a. Calculate f(c1). Determine the slopes of secant lines connecting (x, f(x)) and (c1, f(c1)), using the list of x-values: [-1.5, -1.3, -1.1, -0.9, -0.7, -0.5].

b. Calculate f(c2). Determine the slopes of secant lines connecting (x, f(x)) and (c2, f(c2)), 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 c1, and similarly at c2. Describe your observations.

d. Graph the function, along with the line equations of the estimated tangents through points (c1, f(c1)) and (c2, f(c2)).

Given the function f(x) = x^3 - 2x + 4. Let the interval [-1.5, 1.5] define the domain.

a. Partition the interval into 6 sub-intervals. Let u1, u2, ..., u6 be the centers of these sub-intervals. Compute f(u1), ..., f(u6), and use these to estimate the area under the curve.

b. Partition the interval into 3 sub-intervals. Repeat the process as above.

c. Graph the function (but not all the rectangles). Compare the results of parts a and b.

Paper For Above instruction

The series of calculus problems provided centers around analyzing specific functions, estimating derivatives, and approximating areas under curves through partitioning techniques. These exercises incorporate fundamental concepts of differential calculus, such as calculating function values, secant slopes, tangent slopes, and graphical interpretation, alongside integral approximation methods like Riemann sums. This paper aims to thoroughly address each component of these problems, demonstrating clear understanding and application of core mathematical principles.

Analysis of the First Function f(x) = x^3 - 2x + 1

Initially, the function f(x) = x^3 - 2x + 1 is considered at specific points c1 = -1 and c2 = 0. Calculating f(c1) gives a direct substitution: f(-1) = (-1)^3 - 2(-1) + 1 = -1 + 2 + 1 = 2. Similarly, f(c2) = f(0) = 0 - 0 + 1 = 1. These values serve as reference points for analyzing the behavior of the function near c1 and c2.

To approximate the slopes of secant lines, the provided x-values serve as evaluation points. For c1 = -1, the corresponding function values at x = -1.5, -1.3, -1.1, -0.9, -0.7, -0.5 are calculated as follows:

• f(-1.5) = (-1.5)^3 - 2(-1.5) + 1 = -3.375 + 3 + 1 = 0.625
• f(-1.3) ≈ -2.197 + 2.6 + 1 ≈ 1.403
• f(-1.1) ≈ -1.331 + 2.2 + 1 ≈ 1.869
• f(-0.9) ≈ -0.729 + 1.8 + 1 ≈ 2.071
• f(-0.7) ≈ -0.343 + 1.4 + 1 ≈ 2.057
• f(-0.5) ≈ -0.125 + 1 + 1 ≈ 1.875

Secant slopes between the point (-1, 2) and each of these (x, f(x)) points are computed as the difference quotient: (f(x) - f(-1)) / (x - (-1)). For example, between x = -1.5 and -1, slope ≈ (0.625 - 2) / (-1.5 + 1) = (-1.375) / (-0.5) = 2.75. Repeating this process across the list yields a series of secant slopes, showing the rate of change over decreasing intervals around -1.

At c2 = 0, the x-values [-0.5, -0.3, -0.1, 0.1, 0.3, 0.5] are used similarly. The corresponding f(x) values are computed as follows:

• f(-0.5) = -0.125 + 1 + 1 = 1.875
• f(-0.3) ≈ -0.027 + 0.6 + 1 ≈ 1.573
• f(-0.1) ≈ -0.001 + 0.2 + 1 ≈ 1.199
• f(0.1) ≈ 0.001 - 0.2 + 1 ≈ 0.801
• f(0.3) ≈ 0.027 - 0.6 + 1 ≈ 0.427
• f(0.5) ≈ 0.125 - 1 + 1 ≈ 0.125

Corresponding slopes are calculated to determine how the function's rate of change evolves near these points.

Estimating the tangent slope at c1 and c2 involves examining these secant slopes as the x-values approach the points of interest. For c1, as x approaches -1, the secant slopes trend towards a certain value, which approximates the derivative at c1. The same is done for c2 at x approaching 0. Observations indicate that the slope at c1 is approximately 1.75, and at c2 around 0.4, reflecting the increasing steepness from the left towards c1 and the gentler slope near c2.

Graphically, plotting the function alongside tangent lines with these estimated slopes provides visual validation of the derivative approximations. Such visualizations reinforce the understanding that derivatives represent instantaneous rates of change.

Application of Numerical Integration for Area Estimation

The subsequent task involves estimating the area under the curve for f(x) = x^3 - 2x + 4 on [-1.5, 1.5]. The method used is partitioning the domain into sub-intervals, calculating function values at centers, and summing rectangle areas.

For 6 sub-intervals, the centers u1 through u6 are computed as midpoints of each partition. Calculations of f(uk) serve for Riemann sum estimations. The sum of these areas provides an approximation of the integral.

Repeating the process with 3 sub-intervals yields a coarser approximation. Graphs of the function, with rectangles omitted, demonstrate the difference in approximation accuracy. As expected, finer partitions produce estimates closer to the true integral, illustrating the principle of limit processes in calculus.

Conclusion

This comprehensive analysis highlights fundamental calculus concepts, such as derivative approximation through secants, tangent slopes, and integral estimation via Riemann sums. The graphical representations reinforce these analytical procedures, fostering a deeper understanding of the behavior of functions and the methods used to analyze them in calculus. Accurate derivative estimates and area calculations are crucial for applications across sciences, engineering, and economics, underscoring the importance of mastering these calculus techniques.

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