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Texas A&M | MATH 102
ALGEBRA
Professors
- Bkaskar Shodavaram
50 sample documents related to MATH 102
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Texas A&M MATH 102
Fall 2008 SEMESTER, MATH 102 section 501 ALGEBRA (3 credit hours) Instructor: Office: Time: Classroom: Office Hours: Text: Donka Lazarov, Department of Mathematics Blocker 607, 845-1421; e-mail: dlazarov@math.tamu.edu (use this one only !) TR 3:55 5 -
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Texas A&M MATH 102
CHEMISTRY 102: SECTIONS 503 SPRING 2008 Instructor: Dr. Vickie M. Williamson Office: HELD 406 Phone: 845-4634 You can leave a message at the first year chemistry office- 845-2356 Lecture: M, W, & F (12:40 to 1:30 PM Heldenfels Room 100) Office Hour -
Texas A&M MATH 102
2.1 Equations Denition. stituted for An equation in x is a statement of equality involving the variable x. b that yields a true statement when sub- A solution or root for the equation is a number x. Equivalent equations are equations that hav -
Texas A&M MATH 102
MATH 102 SPRING 2008 Instructor: Lidia Smith Phone: 979-862-2188 Oce: Blocker 610B Email: lsmith@math.tamu.edu Oce hours: MW 9-10 a.m., Website: http : /www.math.tamu.edu/ lsmith T 11-12 a.m. or by appt. Textbook: Fundamentals of College Algebra, -
Texas A&M MATH 102
Review problems for the final exam (1) Solve the equations a) 4 2 32 + = 2 x1 x+1 x 1 b) x3 x+4 = . 2x 5 3x + 1 (2) Solve the equation using the quadratic formula: x2 4x + 20 = 0. (3) Solve the equation x 3 + x 3 20 = 0. 2 1 (4) Solve the -
Texas A&M MATH 102
Part I - Multiple Choice Each of the 8 multiple choice questions is worth 6 points. Please clearly mark your answers on your scantron AND on your exam. Partial credit will not be given for these questions. 1. Use properties of exponents to simplify -
Texas A&M MATH 102
Each of the 8 multiple choice questions is worth 6 points. Please clearly mark your answers on your scantron AND on your exam. Partial credit will not be given for these questions. Part I - Multiple Choice 1. Find the constant of proportionality k f -
Texas A&M MATH 102
Part I - Multiple Choice Each of the 8 multiple choice questions is worth 6 points. Please clearly mark your answers on your scantron AND on your exam. Partial credit will not be given for these questions. 1. The domain of f (x) = ( x2 9 is 6x + 7 -
Texas A&M MATH 102
Math 102 Quiz 2 Solutions Show all your work. 1. Spring 2008 5 3 2 3 (2 points ) Solve the equation x 1 = 4 + x. 2 5 2 3 3 3 3 Thus x = 5. So x = 5. 3 5 3 (2 points ) Solve the equation 8 = 2 + . x x Solution. Multiply both sides by x to obtain 8 -
Texas A&M MATH 102
Math 102 Quiz 8 Spring 2008 1. Solve the equation: Solution. log2 (3x + 1) = log2 5 + log2 2. log2 (3x + 1) = log2 (5 2) log2 (3x + 1) = log2 10 Using law 1 of logarithms we can rewrite the equation as Since logarithmic functions are one-to-on -
Texas A&M MATH 102
Name Math 102 Quiz 7 Sec Spring 2008 (1) Find the inverse of the function f (x) = 1 + 3x . 5 - 2x (2) Graph the function y = ex-3 + 4 by starting with the graph of f (x) = ex and applying transformations. Label three points on the graph of your f -
Texas A&M MATH 102
5.1 Inverse Functions Denition. A function f with domain D and range R is a one-to-one function if whenever x1 = x2 in D, then f (x1 ) = f (x2 ). Remark: An equivalent way of writing the condition for a one-to-one function is this: if f (x1 ) = f -
Texas A&M MATH 102
APPLICATIONS OF EXPONENTIAL FUNCTIONS Suppose that it is observed experimentally that the number of bacteria in a culture doubles every day. Assume 100 bacteria are present at the start. If t denotes a variable that measures time in days, and if f (t -
Texas A&M MATH 102
6.6 The algebra of matrices mn Denition. Let m and n be positive integers. An matrix A is an array of the following form a11 a21 a31 . . . where each a12 a22 a32 . . . a13 a23 a23 . . . . . . a1n a2n a3n , . . . amn rows o -
Texas A&M MATH 102
2.6 Inequalities Denition 6.1. An inequality is a statement that two quantities or expressions are not equal. An inequality can be expressed using one of the relations <, , >, . relation reads as inequality example < less than a<b less than or equ -
Texas A&M MATH 102
1.2 Exponents and Radicals Exponential Notation If a is any real number and n is a positive integer, then the nth power of a is an = a a a The number a is called the base and n is called the exponent. Zero and Negative Exponents n factors -
Texas A&M MATH 102
5.4 Logarithmic Functions Every exponential function f (x) = ax with a > 0 and a = 1 is a one-to-one function and therefore, has an inverse function. The inverse function f -1 is called the logarithmic function with base a and is denoted by loga. -
Texas A&M MATH 102
6.3 Systems of Inequalities Inequalities in two variables x and y . Example. a) y2 + x2 < 4 b) y > x2 c) y < 2x + 3 d) 3x - 4y > 12 Denition. A solution of an inequality in x and y is an ordered pair (a, b) that yields a true statement if a a -
Texas A&M MATH 102
6.8 The determinant of a square matrix a11 a12 , a21 a22 the determinant of A is dened by det A = a11 a22 - a21 a12 . A= The determinant of a square matrix A is a number that associated with the matrix and is denoted by |A| or det A. If A is a -
Texas A&M MATH 102
2.4 Complex Numbers Complex numbers are introduced in order to nd solutions for some equations that have no real solutions. For example the equation x2 = 9 does not have real solutions, but it has two complex roots: 3i and 3i. The imaginary unit d -
Texas A&M MATH 102
1.1 Real Numbers (1) The set of natural numbers {1, 2, 3, 4, } denoted by N. (2) The set of integers { , -3, -2, -1, 0, 1, 2, 3, } denoted by Z. (3) The set of rational numbers: Q, consists of the numbers that can be expressed in the form m -
Texas A&M MATH 102
Part I - Multiple Choice Each of the 8 multiple choice questions is worth 6 points. Please clearly mark your answers on your scantron AND on your exam. Partial credit will not be given for these questions. 1. The domain of f (x) = ( x2 9 is 6x 7 -
Texas A&M MATH 102
SOME USEFUL FORMULAS Formulas to use for completing the square (x a)2 = x2 2ax + a2 (x + a)2 = x2 + 2ax + a2 Factoring Formula (also used for rationalizing denominators) x2 a2 = (x a)(x + a) (x + a)(x a) = x2 a2 Laws of Exponents am an = a -
Texas A&M MATH 102
Name Math 102 Extra-credit assignment Sec Spring 2008 Show all your work. Due Friday, April 25. 1. (2 points) Find the constant of proportionality k from the conditions: r if varies directly as s and inversely as t, and s = 3, t = 12, then -
Texas A&M MATH 102
2.2 Applied Equations Problem 2.1. Before the nal exam a student has an average test score of 78. If the nal exam counts as one-third of the nal grade, what score must the student receive inorder to have a nal average of 80? Solution. Let x denot -
Texas A&M MATH 102
Review problems for exam 3 The exam will cover sections 4.5-4.6 and 5.1-5.6. This material is covered by the online homeworks 4-7 and quizzes 5-8. (1) Find all holes, vertical asymptotes or horizontal asymptotes. a) f (x) = 3(x2 - 4x + 4) 2(x2 - 4) -
Texas A&M MATH 102
1.3 Sets If Algebraic Expressions elements b of Denition. the set. A set is a collection of objects, and these objects are called the S is a set, then aS means that a is an element of S, and bS / means that is not an element of De -
Texas A&M MATH 102
Math 102, Section 500 Algebra, Spring 2006 Blocker 160 MWF 8:00-8:50 Instructor: Amy Collins Office: Milner 029 Phone: 845-5324 Office hours: MWF 1:00-3:00 and by appointment Email address: acollins@math.tamu.edu Email is by far the best way to reach -
Texas A&M MATH 102
3.5 Graphs of Functions Denition. A function f is called even if f (x) = f (x) for every x in its domain. A function f is called odd if f (x) = f (x) for every x in its domain. y -axis and an odd function is symmetric with respect to the origin. U -
Texas A&M MATH 102
3.2 Graphs of Equations a true statement if x = a and y = b. Denition 2.1. The solution of an equation in x and y is an ordered pair (a, b) that yields Example 2.2. (2, 3) is a solution of y2 = 5x 1. Substituting x = 2 and y = 3 gives us LS: 32 -
Texas A&M MATH 102
4.1 Polynomial Functions that is, if f (x) = an xn + an1 xn1 + + a1 x + a0 , where the coecients are real numbers and the exponents are non-negative intege -
Texas A&M MATH 102
4.6 Variation (Proportion) Terminology General formula y y varies directly proportional to x x y = kx y= k x varies inversely proportional to The constant k is called constant of variation or constant of proportionality. In general, graphs -
Texas A&M MATH 102
3.4 Functions Denition. The sets under A function f is a rule that assigns to each element x in a set A exactly one element, called f (x), in a set B. of real numbers. and is called the value of The range of that is, A and The symbol -
Texas A&M MATH 102
4.5 Rational Functions and Their Graphs. Asymptotes Denition. A function f is a rational function if f (x) = g(x) , h(x) h(x). x where g(x) and h(x) are polynomials. The domain of f consists of all real numbers except the zeros of the d -
Texas A&M MATH 102
3.3 Lines Denition 3.1. A linear equation in x and y is an equation of the form ax + by = c, with a, b, c R. There are other equivalent forms that a linear equation can be written in, as we will see in the examples. The graph of a linear equation -
Texas A&M MATH 102
graph of f for a > 1 Exponential function f (x) = ax with base a y graph of f for a < 1 y x x (a) a>1 (b) a<1 5.2 Exponential Functions Previously, we considered power functions such as x2 , x3/2 . Now we turn to functions such as 2x , (0.5) -
Texas A&M MATH 102
3.6 Quadratic Functions Denition. A function f is a quadratic function if f (x) = ax2 + bx + c, where a, b and c are real numbers with a = 0. The standard form of a quadratic function is f (x) = a(x h)2 + k. The graph of a quadratic function is -
Texas A&M MATH 102
4.2 Dividing Polynomials If f (x) and g(x) are polynomials in x and if g(x) is a factor of f (x), we say that f (x) is divisible by g(x). For example, x2 4 is divisible by x + 2 . Long Division of Polynomials: If f (x) and p(x) are polynomials and -
Texas A&M MATH 102
5.6 Exponential and Logarithmic Equations Solving an exponential equation by taking log of both sides. Example. Solve the equation 3x = 21. Solution. x First note that we can write the equation in logarithmic form obtaining Next we will try to -
Texas A&M MATH 102
2.2 Applied Equations Problem 2.1. Before the nal exam a student has an average test score of 78. If the nal exam counts as one-third of the nal grade, what score must the student receive inorder to have a nal average of 80? Solution. Let x deno -
Texas A&M MATH 102
5.5 Properties of Logarithms We have seen that the base a must be raised to get loga x can be interpreted as an exponent (it is the exponent to which x). It seems reasonable to expect that the laws of exponents can be used to obtain correspond -
Texas A&M MATH 102
3.7 Operations on Functions Functions are often dened using sums, dierences, products or quotients of expressions that can be considered themselves as functions. For example if h(x) = x2 + 3x, we may regard h as the sum of the function f and g give -
Texas A&M MATH 102
2.7 More on Inequalities To solve an inequality involving polynomials of degree greater than 1, we will factor each polynomial in linear and irreducible quadratic factors. If any such factor is nonzero in an interval, then it is either positive t -
Texas A&M MATH 102
2.3 Quadratic Equations ax2 + bx + c = 0, where a = 0. Denition. A quadratic equation in x is an equation that can be written in the form Examples: x2 + x = 2, x2 + 6x + 5 = 0. Zero Factor Theorem. If p and q are algebraic expressions, then pq = -
Texas A&M MATH 102
4.3 Zeros of Polynomials The zeros of a polynomial f (x) are the solutions of the equation f (x) = 0. Each real zero is an x-intercept of the graph if f . Theorem. Fundamental Theorem of Algebra. If a polynomial f (x) has positive degree and compl -
Texas A&M MATH 102
6.2 Systems of linear equations in two variables Recall that a linear equation in x and y is a equation that can be written in form ax + by = c, where a, b, c are real numbers, with a and b not both zero. Similarly, the equation ax + by + cz = d -
Texas A&M MATH 102
6.7 The inverse of a matrix The identity matrix of order n is denoted by In and is the square matrix of order n that has 1 on the main diagonal and 0 elsewhere. I2 = 1 0 0 1 , 1 0 0 I3 = 0 1 0 0 0 1 Denition. Let A be a square matrix of order -
Texas A&M MATH 102
Math 102 Quiz 1 Solution Spring 2008 Show all your work. (1) If x>0 and y < 0, and determine the sign of Solution. ( This is Example 3 page x y y x x y + . Justify your y x 11). Since x and y have answer. opposite signs, both have negat -
Texas A&M MATH 102
4.4 Complex and Rational Zeros of Polynomials If a polynomial f (x) of degree n > 1 has real coecients and if z = a + bi with b = 0 is a complex zero of f (x), then the conjugate z = a - bi is also a zero of f (x). Theorem. Conjugate Zeros. Example.
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