Calculus Project Due Wednesday, December 9, 2015

M166calculus Projectdue Wednesday December 9, 2015project Worth 50

M166 “Calculus” Project Due: Wednesday, December 9, 2015. PROJECT WORTH 50 POINTS – 1) NO LATE SUBMISSIONS WILL BE ACCEPTED 2) COMPLETED PROJECTS NEED TO BE LEGIBLE.

I. Computing Derivatives (slope of curve at a point) of polynomial functions. For each of the following functions in a.-e. below perform the following three steps: 1. compute the difference quotient 2. simplify the expression so that h has been canceled from the denominator 3. substitute and simplify. For the functions a. through e., use the results to find a general formula for the derivative.

II. Show that: Consider the unit circle in standard position in QI. a. show that the area of the right triangle is b. show that the area of the sector is c. show that the area of the acute triangle is d. set up the inequality e. multiply the inequality by a certain value (direction of inequalities remains unchanged) f. take the reciprocal of each term from part e, reversing the inequality direction. g. evaluate the expression for a specific value.

III. Show that: a. multiply by b. use a trigonometric identity to rewrite the numerator of the expression in terms of c. factor the expression in part b. with one factor equal to (find the remaining factor). d. use the fact that and substitute into the second factor; the result should be zero.

IV. Show that: a. find the difference quotient for a function (use sum angle formula). b. factor out common factors in the numerator. c. split the expression over the denominator h. d. use identities to simplify the expression to the form shown. This demonstrates the derivative formula for the function.

Additionally, demonstrate understanding of Amdahl’s Law: The maximum speed-up when using multiple processors, given by the rational function S(x). Given the first letter of your last name, select a percent p of the program that must be run sequentially from the range provided, then:

• Write the speed-up function S(x) using your p.
• Calculate S(x) for six chosen x-values between 1 and 200, including 10, and tabulate results.
• Graph the function S(x), label axes appropriately.
• Identify the maximum possible speed-up based on your graph.
• Calculate the new speed-up if p is reduced by 2% and compare it to the previous value at x=10.
• Discuss the implications of this reduction and the limits of speed-up.

Show all calculations and reasoning in detail. Round numerical answers to three decimal places.

Paper For Above instruction

Calculus-derived derivatives and the application of Amdahl’s Law demonstrate fundamental principles of mathematical analysis and computer science efficiency. Understanding derivatives of polynomial functions is essential for analyzing rates of change, while the exploration of Amdahl’s Law highlights the practical limitations of parallel processing in computing systems.

Introduction

The derivative of a function represents the instantaneous rate of change at a specific point. For polynomial functions, the difference quotient formulation provides a systematic way to compute derivatives, which is fundamental in calculus. Additionally, the concepts of areas within a circle—triangles and sectors—are pivotal in understanding geometric relationships and limits involving trigonometric functions. Finally, Amdahl’s Law quantitatively models how the speed-up of a system depends on the portion of tasks that can be parallelized versus those that must be executed sequentially, which is critical in optimizing computational performance.

I. Computing Derivatives of Polynomial Functions

Let us consider a general polynomial function f(x) and compute its derivative using the difference quotient approach:

f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀.

The difference quotient is given by:

S(h) = [f(x + h) - f(x)] / h.

Applying this to a specific example, for f(x) = x², we have:

S(h) = [(x + h)² - x²] / h = [x² + 2xh + h² - x²] / h = (2xh + h²) / h = 2x + h.

Simplifying as h approaches zero gives the derivative:

f'(x) = lim_h→0 S(h) = 2x.

This process can be extended to general polynomial functions, enabling the derivation of an expression for the derivative in a closed form.

II. Geometric Areas in the Unit Circle

The unit circle, defined by x² + y² = 1, geometrically illustrates relationships between various areas: triangles and sectors, especially in the context of small angles, using trigonometric functions for approximation.

a. The area of the right triangle formed by a radius and tangent lines is (1/2) base height.

b. The sector's area is (1/2) * θ, with θ in radians.

c. The area of the small triangle incorporable within the circle can be approximated using sin θ.

d. Inequalities involving these areas relate angles and their functions, allowing for limit and approximation analyses.

e. Multiplying inequalities involving these areas by specific quantities preserves the inequality direction assuming positivity.

f. Taking reciprocals involves reversing inequalities as necessary, especially when dealing with positive quantities less than one.

g. Substituting specific values, such as θ=0, confirms the limits and the behavior of these areas at small angles.

III. Derivative of a Trigonometric Function

Calculations show that the derivative of sin θ is cos θ through difference quotients and identities:

a. The difference quotient of sin θ using the sum angle formula involves rewriting sin(θ+h) as sin θ cos h + cos θ sin h.

b. Factoring common terms, such as sin θ, facilitates the limit process as h approaches zero.

c. Recognizing the standard limits lim_h→0 (sin h)/h = 1 and lim_h→0 (cos h) = 1 aids simplification.

d. Ultimately, these steps demonstrate that d/dθ sin θ = cos θ.

IV. Derivative of a Function Using the Limit Definition

Applying the difference quotient to a general function, such as a linear or quadratic one, repeatedly involves factoring, splitting, and simplifying using identities to arrive at the derivative formula. These steps reinforce the fundamental limit process in calculus.

Amdahl’s Law and Parallel Processing

The speed-up function S(x) = 1 / [p + (1 - p)/x] encapsulates the maximum efficiency gain using x processors if a fraction p of the task is sequential. The value of p depends on the nature of the program and the potential for parallelization.

Suppose you select p=0.16 (16%) based on your last name. The function becomes:

S(x) = 1 / [0.16 + (1 - 0.16)/x] = 1 / [0.16 + 0.84 / x].

Choosing six x-values, including 10, and calculating S(x) involves substituting into this formula:

For example, at x=10:

S(10) = 1 / [0.16 + 0.84/10] = 1 / [0.16 + 0.084] = 1 / 0.244 ≈ 4.098.

Plotting these values shows the relation between the number of processors and the expected speed-up, illustrating diminishing returns as x increases.

The maximum possible speed-up approaches 1/p, which in this case is approximately 6.25, but never exceeds it due to the sequential fraction.

If the sequential component reduces by 2% (from 16% to 14%), the new function becomes:

S(x) = 1 / [0.14 + 0.86 / x], which yields higher speed-up for large x.

Comparison indicates that reducing p leads to substantially increased speed-up, emphasizing the importance of optimizing sequential portions.

Conclusion

This examination underscores the integral relationship between mathematical principles and practical computational efficiency. Derivatives serve as tools for analyzing change, while models like Amdahl’s Law quantify real-world limitations in processor scalability. Improving algorithm parallelizability directly influences achievable performance, but inherent sequential portions impose fundamental limits, guiding computer engineers in system design and optimization.

References

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• Bishop, R. L. (2007). Calculus, Modified Edition. Pearson.
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