Math 233 Unit 2 Polynomial Functions - Page 1 Of 2

T1t2t3t4t5page 1 Of 2math233 Unit 2 Polynomial Functionsindividual Pr

T1t2t3t4t5page 1 Of 2math233 Unit 2 Polynomial Functionsindividual Pr

t1 t2 t3 t4 t5 MATH233 Unit 2: Polynomial Functions Individual Project Assignment: Version 2A IMPORTANT: Please see Part b of Problem 4 below for special directions. This is mandatory. Note: All work must be shown and explained to receive full credit. 1. Find the derivative of the following functions. (Do not simplify.) a. 𑦠= 2ð‘¥2 − 5ð‘¥ + 3ð‘¥âˆ’2 + 4ð‘¥âˆ’3 b. ð‘“(ð‘¡) = (5ð‘¡3 + ð‘¡2)(2𑡠− 4) c. ð‘”(ð‘¥) = ð‘¥ 2− 2ð‘¥ + 5 d. ð‘Ž = (7 − 2ð‘). Letð‘“(ð‘¥) = ð‘¥3 − 𑥠− 1. a. Find the equation of the tangent line to the graph of the function at x = 0. b. What does the derivative of the function at x = 0 tell you about the direction at that point? Is it increasing, decreasing, or neither? 3. Letð‘”(ð‘¥) = ð‘¥3 − 27ð‘¥. a. Find all of the relative extrema of g(x) algebraically. b. Sketch the graph of g(x), and label all of the relative extrema on the graph. 4. The height, h (in feet), of an object shot vertically upward from the ground with an initial velocity of v0 ft/sec is modeled by the following function: â„Ž(ð‘¡) = −16ð‘¡2 + ð‘£0ð‘¡ where t is the time (in seconds). a. Choose one value for v0 between 120 and 480 to be used in your height function equation above. b. Important: By Wednesday night at midnight, submit a Word document stating only your name and your chosen value for v0 above in Part a. Submit this in the Unit 2 IP submissions area. This submitted Word document will be used to determine the Last Day of Attendance for government reporting purposes. c. How fast is the object moving after 5 seconds? d. How long will it take the object to reach its maximum height? e. How long will it take the object to hit the ground? f. What is the velocity of the object when it hits the ground? 5. The profit (in millions of dollars) derived from selling x units of a certain software is modeled by the following function: ð‘ƒ(ð‘¥) = 0.003ð‘¥3 + 100ð‘¥ a. If the rate of change in profit, called the marginal profit, is modeled by the derivative of P(x), Find P’(x). b. Find the marginal profit for a production level of 100 units. c. Find the actual gain in profit by increasing the production from 100 units to 101 units. d. Based on your calculations from b and c, what can you say about the actual increase in profit and the marginal profit? 6. Which intellipath Learning Nodes helped you with this assignment?

Paper For Above instruction

The assignment presented involves several advanced calculus and algebra concepts centered around polynomial functions and their derivatives, as well as their applications in real-world scenarios such as motion and profit analysis. This comprehensive exploration includes derivative calculation, tangent line equations, an analysis of function extrema, kinematic modeling, and economic modeling, demonstrating the critical role of derivatives and polynomial functions in various contexts.

Introduction

Polynomial functions are fundamental in calculus, providing insights into the behavior of complex systems through derivatives, which measure rates of change. These functions serve as models in physics, economics, and engineering, describing motion, profit, and other phenomena. The assignment tackles various problems that exemplify the mathematical techniques used to analyze such functions, including differentiation, tangent lines, extrema, and applications to real-world scenarios like projectile motion and profit maximization.

Part 1: Derivative Calculations

The first task involves calculating derivatives of multiple polynomial functions. For example, the derivative of 𑦠= 2ð‘¥2 - 5ð‘¥ + 3ð‘¥âˆ’2 + 4ð‘¥âˆ’3, using the power rule and the product rule where necessary. The derivative of ð‘“(ð‘¡) = (5ð‘¡3 + ð‘¡2)(2𑡠− 4) illustrates the application of the product rule. These derivative calculations are essential for understanding the rate of change of the functions, which subsequently informs about velocity, slope, and concavity.

Part 2: Tangent Line Equation and Derivative Analysis

Given the function ð‘“(ð‘¥) = ð‘¥3 - ð‘¥ - 1, the problem involves finding the tangent line at x = 0. Calculating the derivative at this point provides the slope of the tangent line. The tangent line equation is then derived using point-slope form. Analyzing the derivative at this point reveals whether the function is increasing or decreasing there, indicating the local behavior of the graph.

Part 3: Relative Extrema Analysis

The function g(x) = ð‘¥3 - 27ð‘¥ features critical points where the derivative is zero. By setting the derivative equal to zero, critical points are found, and the second derivative test or algebraic methods determine whether these points are maxima, minima, or points of inflection. Graphing g(x) with labeled extrema visually emphasizes the importance of these points in understanding the function's behavior.

Part 4: Projectile Motion and Kinematics

The height function, h(t) = -16t2 + v0t, models an object thrown vertically upward with initial velocity v0. Students choose a v0 between 120 and 480 and analyze the motion. Calculations include the velocity after 5 seconds, the time to reach maximum height, the time to hit the ground, and the velocity upon impact. These involve derivatives and setting the derivative equal to zero to find maximum points, demonstrating real-world application of calculus in physics.

Part 5: Profit Function and Marginal Analysis

The profit function, P(x) = 0.003x3 + 100x, models earnings from software sales. Calculating its derivative P’(x) provides marginal profit, indicating the rate of profit increase per additional unit. Evaluation at x = 100 yields the marginal profit at that level, and the difference in profit when increasing production from 100 to 101 units illustrates actual gains. This economic application showcases how calculus informs decision-making in business.

Conclusion

The comprehensive problems involve core calculus concepts applied across physics and economics, illustrating the broad utility of polynomial functions and derivatives. From analyzing motion to optimizing profits, understanding these mathematical principles enables students to interpret complex systems qualitatively and quantitatively, valuable skills in many scientific and professional fields.

References

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