math homework solution

math homework solution

Advanced Techniques for Solving Challenging Math Problems

1. Introduction to Problem-Solving Strategies

This handout lays out a variety of problem-solving strategies that include logical reasoning, combinations and permutations, time and distance, acronyms, and many more. The handout also contains challenge problems. While more is involved, there is a direct relationship between the level of challenge of a problem and the level of problem-solving competence of the student. That is, stronger problem-solving tools correspond to more difficult problems that can be tackled successfully, ceteris paribus. Consequently, while exposure to challenging problems results in improved problem-solving skills, the reverse effect is even more dramatic. Instructors can teach students to solve many math problems, but to be successful students need problem-solving tools. The best way to learn and decide which tools to use is by spending considerable time learning to solve problems. More effective than reading this handout is this strategy: with pencil and paper, work through as many of the problems contained in the text to learn which strategies to use, interact, learn, and do.

2. Algebraic Manipulations and Simplifications

Summary: Basic algebraic manipulation is a critical step in solving challenging algebra problems. Tools such as factoring and regrouping terms can simplify or transform expressions. Identifying similarities between the expressions in the problem and the solutions can shed light on various hidden symmetry properties such as invariance under a transformation. Also useful are the principles of weighted average and recursive relations in solving certain sequences and series problems.

Basic Techniques: Basic algebraic manipulation is a critical step in solving challenging algebra problems: expand or rewrite expression in a more convenient form, factor expression, regroup terms, use symmetry properties (f(x) = f(a+b−x) etc.), use recursive relation, use identities such as (a + 2)2 = a^2 + 4a.

Advanced Techniques: Beyond the basic techniques listed on the previous page, some unique algebraic techniques can help solve challenging algebra problems. For problems involving sequences and series, it is useful to examine the properties of the common difference. Other tools such as factoring and regrouping terms can simplify or transform expressions. Identifying similarities between the expressions in the problem and the solutions can shed light on various hidden symmetry properties such as invariance under a transformation. The principles of weighted average and recursive relations are also especially useful in solving certain sequence and series problems.

Invariance: Many algebra problems can be solved using the same principle to identify symmetries and patterns, which is the concept of invariance. Some typical invariance principles include: examine properties of expression before and after transformation, if necessary, introduce auxiliary variables, do not overlook trivial possibilities/changes, express problem condition using summation notation, use recursive relation to establish a relationship based on predecessor cases.

3. Trigonometric Identities and Applications

An identity is an equation that is true for all possible values of the variable. Trigonometric identities equate two mathematical quantities. When an identity includes a variable, it is no longer an identity, but a conditional equation.

Angle Measure: Trigonometry compares the angles and side lengths of two problems. The three primary functions of angles are called the sine, cosine, and tangent. The hypotenuse is the longest side of a right angle triangle; it is located opposite of the 90° angle. The opposite side is the side of the triangle that is across from the angle in question. The adjacent side is the outer right angle of the triangle, a horizontal measurement.

Pythagorean Identity: The Pythagorean Identity is an equation that relates to the sine and cosine functions in relation to the cosine function.

Double-Angle Identities: The double angle identities are formulas that allow us to find the sine, cosine, or tangent of a double angle.

Half-Angle Identities: Half angle identities are equalities that involve trigonometric functions. They can be used to find new identities using old ones.

Co-function Identities: The co-function identities are identities of elementary trigonometric identities. They are called co-function identities because the sine, cosine, tangent, cotangent, secant, etc. are called the co-function of cosecant, angles, etc.

Period, Domain, and Range of Trig Functions: The domain of a sine function is always -1 to 1; the range of any sine or cosine function is -1 to 1. The period of a sine and cosine function is 360, meaning that the function repeats every 360 on its graph.

4. Calculus Techniques for Optimization Problems

Optimization deals with selecting the best option among several choices. Optimization problems are very frequently encountered in various fields of science. When computers are in use, people give a table of values which describe the behavior of the function on some small piece of the domain, and the program returns something that must have been on that table. Calculus provides powerful and direct techniques for finding the least or greatest value, helping people avoid that original tedious work. The techniques introduced involve finding where a graph has horizontal tangent lines. The method helps find the most important feature concerning the shape of the graph of a continuous function.

Optimization is a process of finding the best value of a function. This can be either the largest value or the smallest value. Any quantity which must be maximized or minimized satisfies the target of optimization problems. For example: finding the maximum volume of a box as a function of its surface area VI as a function of numbers a and b III depends on the choice of x and y, the amounts of overtime used by x and y, VI = -2(-3x^2 + 12x) = 2(3x^2 – 12x) and II = -3(x^2 – 10x) = 3(10 – x^2).

5. Advanced Topics in Number Theory and Combinatorics

In this chapter, we explore advanced number theory and combinatorics theorems to help students solve increasingly challenging problems. Topics that are covered range from representative topics in number theory, such as the Chinese remainder theorem, to representative combinatorics techniques such as generating functions. This chapter is intended for students who have already explored these topics to learn these advanced methods that often appear on competitions. This chapter should be used as a topic-specific technique reference. Then, after students participate in mock math competitions, they can refer to this chapter to determine if any techniques can be adapted to old problems.

Prime-ization in its most general forms is very effective when trying to apply number theoretic structures to complex numbers, polynomials, and be turned into an inhibitor of some natural finite integer structure, such as zs = or polynomials can use multiplication to combine polynomial structures of two more natural ones into a larger structure or induce modular rites on when none exist, or use nonunits to arrive at a contradiction. The Large Modal Closet and the Many Fashion below it tell us that there are many classes of number sets that have both a lot of structure and a lot of numbers. The maximum of structure (for example, the densest collection of local optimizations) usually occurs locally and not throughout.

Consider a set of integers called a “fashion” if an ultra-strong structural statement applies to that set of integers. For example, a set of integers is called “multiple-representational” if the integers in that set have the property that there exists more than one ordered list of representations in universities or structures (such as triples or quadrants or other naturally grouped categories) that multiply to form all of the integers in the set.