Final Exam Study Guide Mathematical Thinking, Fall 2003

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Griselda Anderson
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1 Final Exam Study Guide Mathematical Thinking, Fall 2003 Chapter R Chapter R contains a lot of basic definitions and notations that are used throughout the rest of the book. Most of you are probably comfortable with most of this sort of material since we have been using this vocabulary and notation throughout the semester. The important mathematical topics covered are: Arithmetic with Real Numbers You should be comfortable with all basic operations on real numbers. These include adding, subtracting, multiplying, dividing, negation (multiplication by 1), reciprocals, exponents, and absolute values. In particular, you should know how to use the proper order of operations when simplifying an expression involving the operations mentioned. You should also be familiar with the properties of addition and multiplication (associativity, commutativity, distributivity, etc.... ). Exponents You should know the definition of a r where r is an integer. You should also know the properties of exponents, and be comfortable using them to simplify expressions. Polynomials and Factoring You should know how to add, subtract, and multiply polynomials. The two basic rules for doing this are to multiply using the distributive law, and to add like terms. You should also be able to factor polynomials. In particular you should know how to factor out the greatest common factor of a sum, you should know how to factor a quadratic polynomial. You should also know how to recognize and use special factorizations like perfect square trinomials, differences of squares, and sums and differences of cubes. 1

2 Chapter 1 Solving equations You should know how to solve linear and quadratic equations in one variable. You should also be able to solve an equation with several variables for some designated variable in terms of the others. You should also know how to solve equations involving absolute values, and you should be able to solve word problems that lead to equations of these sorts. Linear inequalities You should be able to solve inequalities involving one variable raised to the first power. The same stategies used for solving linear equations are used here. The main thing that one needs to be careful about is that when multiplying both sides of an inequality by a negative number, it is necessary to swith the direction of the inequality, i.e. a < b a > b and similarly when < is replaced by >,, or. Chapter 2 In this chapter we introduced graphing. You should know how to graph a point (x, y) in the x y coordinate system. Lines You should know how to graph a line given its equation in standard form, or slope-intercept form. You should understand the slope of a line, and what a slope tells us geometrically about a line, i.e. whether it rises or falls as we move left to right, and how steep the line is. You should also be able to find the equation of a line given either two points on the line, or one point and the slope. Functions You should know the definition of a function, and understand what it means to graph a function. You should be comfortable using function notation (f(x) =... ). You should know how to perform operations on functions, like addition/subtraction, multiplication/division, composition (given f(x) and g(x) compute f(x) + 5g(x), f(x)g(x), f(g(x)), etc.). Chapter 3 You should know how to solve 2 2 and 3 3 linear systems, and be able to solve word problems that lead to such systems. 2

3 Chapter 4 Rational Expressions A rational expression is any expression that can be written as a polynomial divided by a polynomial. You should know how to reduce a rational expression to lowest terms, and how to add, subtract, multiply, and divide rational expressions. You should also be able to use these operations to solve equations involving rational expressions. Polynomial Long Division You should know how to divide two single variable polynomials using polynomial long division. Complex Fractions The book defines a complex fraction to be a fraction with rational expressions in the numerator or denominator, for example 3x 4 2 x x 3 + x. 1+x 2 You should know how to turn such an expression into a rational expression (i.e. a fraction involving a polynomial over a polynomial) and be able to solve equations involving such expressions. Chapter 5 Radicals You should know what the symbol n a means and how to compute it for numbers or expressions that are a perfect n-th power. You should know the basic properties of radicals ( n ab = n a n b, and n a b = n a n b ), and know how to add, subtract, multipy and divide expressions involving radicals. You should be able to put a radical expression in simplified form, and should know how to rationalize the denominator of certain fractions containing radicals in the denominator. Finally you should know how to solve equations involving radicals. Rational Exponents You should know the definition of rational exponents a p q = ( q a) p = q a p and should know that all the rules that were true for integer exponents are still true for rational exponents. You should be able to use the definition of 3

4 rational exponents to switch a radical expression to one involving an exponent and vice versa. You should be comfortable using the rules of exponents to add/subtract/multiply/divide expressions involving rational exponents. Chapter 6 Solving Quadratics You should know how to solve a quadratic equation by completing the square, and by using the quadratic formula. (You will be required to know how to complete square on the exam) Graphing Quadratics The graph of an equation of the form y = ax 2 + bx + c a 0 always has the shape of a parabola. Given such an equation, you should know how to find the coordinates of the y-intercept, the x-intercepts (if any), the vertex, and also be able to determine whether the parabola opens up or opens down. You should be able to use all this information to draw the graph. Inequalities Involving Rational Expressions You should know how to solve inequalites involving rational expressions and polynomials in one variable. The basic strategy in all of these cases is to rearrange the inequality so that one side is zero so that we have rational expression <, >, or 0. The rational expression can only change sign at points where it is zero or undefined, so find all the points where the expression is zero or undefined, plot these points on the number line, and test values in between each point to determine whether the inequality is true or false in that interval. Chapter 7 Exponential Functions An exponential function is one of the form f(x) = b x for b > 0. You should know roughly what the graph of one of these functions looks like, and how it differs when b > 1 and when 0 < b < 1. 4

5 Basics of Logarithms The symbol log b y (for b > 0, y > 0) is defined to be equal to the power to which we need to raise b in order to get y, e.g. log 3 81 = 4 because 3 4 = 81. In general we have that y = b x and log b y = x are equivalent statements. You should be comfortable changing an equation involving a logarithm into one involving an exponent, and vice versa. You should also be able to use this equivalence to compute some logarithms by hand. For example, if we are asked to compute log we write down the equation x = log and switch this to the exponential equation 25 x = 125. Now, observing that 25 = 5 2, and 125 = 5 3, this equation becomes (5 2 ) x = 5 3 and using the properties of exponents this becomes 5 2x = 5 3. This will only be true when 2x = 3 or x = 3 2. Hence we get log = x = 3 2. Using a calculator to evaluate logarithms Not every logarithm can be evaluated by hand, so we will need to use a calculator in some cases. Many calculators are capable of computing log 10, also called the common log, and denoted just log with no subscript. To compute other logs on a calculator, we use the change of base formula log b a = log 10 a log 10 b where we know how to compute the right hand side of this on our calculators. For example, to compute log 6 11 on a calculator, we compute log and log so we conclude log 6 11 = log 11 log Solving equations with variable in the exponents We can use the fact that log b b r = r to get approximate solutions to equations with variables in the exponents. For example, to solve the equation 3 2x 1 = 6 5

6 take log 3 of both sides to obtain log 3 3 2x 1 = log 3 6. On the left hand side, we get log 3 3 2x 1 = 2x 1. For the right hand side we use a calculator to compute log 3 6 log 6 log , so the equation becomes which we can solve for x to get 2x x One case where such equations arise is in compound interest problems. The formula for compound interest is ( A = P 1 + r ) nt n where A is the amount you end up with, P is the amount that you started with, n is the number of times per year that interest compounds, r is the annual rate, and t is the time measured in years. A typical problem might say: How long will it take for $500 to double if invested at 8% compounded 4 times per year? In this case, we have A = 1000 (double of $500), P = 500, r =.08, n = 4, and we want to find what t is. Substituting these values in the equation we get ( 1000 = ) 4t 4 Dividing both sides by 500 gives and taking log 1.02 on both sides gives Solving for t gives = 500 (1.02) 4t. 2 = (1.02) 4t log = log t = 4t. t = 1 4 log = 1 log 2 4 log = so it will take approximately 8.75 years for the money to double. 6

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