Sum and Product of the Roots. Nature of the Roots. Radical Equations. Quadratic-Type Equations. Principles of Inequalities.
Absolute Value Inequalities. Higher Degree Inequalities. Linear Inequalities in Two Variables. Systems of Linear Inequalities. Rational Functions. Vertical Asymptotes. Horizontal Asymptotes. Graphing Rational Functions. Zeros of Polynomial Equations. Solving Polynomial Equations. Approximating Real Zeros. Chapter Simple Interest.
Compound Interest. Applications of Logarithms. Applications of Exponents. Laws of Logarithms. Common Logarithms. Using a Common Logarithm Table. Natural Logarithms. Using a Natural Logarithm Table. Finding Logarithms Using a Calculator. Fundamental Counting Principle Permutations. Using a Calculator. Chapter 27 Chapter Simple Probability.
Compound Probability. Mathematical Expectation. Binomial Probability. Conditional Probability. Determinants of Order n. Properties of Determinants. Value of a Determinant.
Homogeneous Linear Equations. Chapter 30 Operations with Matrices. Elementary Row Operations. Inverse of a Matrix. Matrix Equations. Matrix Solution of a System of Equations. Chapter 31 Chapter 32 Proper Fractions. Partial Fractions. Identically Equal Polynomials. Fundamental Theorem. Finding the Partial Fraction Decomposition. To identify those items that are placed adjacent available in the Electronic Tutor software, please look for the Mathcad icons, to a problem number.
A complete list of these Mathcad entries follows below. For more information about the software, including the sample screens, see Appendix C on page These are addition, subtraction, multiplication, and division When two numbers a and b are added, their sum is indicated by a b. When a number b is subtracted from a number a, the difference is indicated by a - b.
Subtraction may be defined in terms of addition. For example, 8 - 3 is that number x which when added to 3 yields 8, i. The operation of multiplication may be indicated by a cross, a dot or parentheses. When letters are used, as in algebra, the notation p x q is usually avoided since X may be confused with a letter representing a number.
The expression alb is also called a fraction, having numerator a and denominator b. Division by zero is not defined. See Problems l. Division may be defined in terms of multiplication.
Natural numbers 1 , 2 , 3 , 4 ,. If two such numbers are added or multiplied, the result is always a natural number. The positive rational numbers include the set of natural numbers.
Positive irrational numbers are numbers which are not rational, such as fi,n. Zero, written 0, arose in order to enlarge the number system so as to permit such operations as 6 - 6 or 10 - Zero has the property that any number multiplied by zero is zero.
Zero divided by any number 0 i. Negative integers, negative rational numbers and negative irrational numbers such as -3, - 1 3 , and -fi, arose in order to enlarge the number system so as to permit such operations as 2 - 8, n- 3 n or 2 - 2 f i. Zero is considered a rational number without sign. The real number system consists of the collection of positive and negative rational and irrational numbers and zero. Unless otherwise specified we shall deal with real numbers.
To do this, we choose a point on the line to represent the real number zero and call this point the origin. It can be proved that corresponding to each real number there is one and only one point on the line; and conversely, to every point on the line there corresponds one and only one real number.
The position of real numbers on a line establishes an order to the real number system. If a point A lies to the right of another point B on the line we say that the number corresponding to A is greater or larger than the number corresponding to B, or that the number corresponding to B is less o r smaller than the number corresponding to A. By the absolute value or numerical value of a number is meant the distance of the number from the origin on a number line. Absolute value is indicated by two vertical lines surrounding the number.
Similarly, in multiplication we may write ab c de or a bc de , the result being independent of order or grouping. To add two numbers with unlike signs, find the difference between their absolute values and prefix the sign of the number with greater absolute value.
Thus the reciprocal of 3 i. Clearly there is no such number, since any number multiplied by 0 must yield 0. From b and e it is seen that division by zero is an undefined operation.
This illustrates the commutative law for addition. This illustrates the associative law for addition. Thus 6. Thus 4 7. This illustrates the associative law for multiplication. Note: 3lr is a proximately 3 3. Write the equation of each parabola in standard form.
Central ellipses have their center at the origin, vertices and foci lie on one axis, and the covertices lie on the other axis. We will denote the distance from a vertex to the center by a, the distance from a covertex to the center by b, and the distance from a focus to the center by c. We call the line segment between the vertices the major axis and the line segment between the covertices the minor axis.
If the numerator for a2 is x 2 , then the major axis lies on the x axis. If the numerator for a2 is y 2 , then the major axis lies on the y axis. In 4 the major axis is parallel to the y axis and the minor axis is parallel to the x axis. Determine the center, foci, vertices, and covertices for each ellipse. The center is at 0,O. The covertices are at 0, b and 0, - 6 , so B 0, 3 and B' 0, Since y2 is over the larger denominator, the vertices and foci are on the y axis. The center is m, Write the equation of the ellipse having the given characteristics.
Central hyperbolas have their center at the origin and their vertices and foci on one axis, and are symmetric with respect to the other axis. The line segment between the vertices is called the transverse axis.
The denominator of the positive fraction for the standard form is always a2. When lines are drawn through the points R and C and the points S and C, we have the asymptotes of the hyperbola. The asymptote is a line that the graph of the hyperbola approaches but does not reach. Find the coordinates of the center, vertices, and foci for each hyperbola. From c? Write the equation of the hyperbola that has the given characteristics. Write the equation of each hyperbola in standard form.
Graph either yl or y l and y 2 simultaneously. Using the y scale to be 0. For the circle, ellipse, and hyperbola, it is usually necessary to center the graphing window at the point h, k the center of the conic section. However, the parabola is viewed better if the vertex h , k is at one end of the viewing window. Solved Problems Mfhcd I a Ellipse b Hyperbola c Parabola Fig.
Note that if x is any real number except zero, y is real. The graph is a hyperbola see Fig. The graph consists of two intersecting lines see Fig. Thus y is imaginary for all real values of x and the graph does not exist. The midpoint M of the line segment having endpoints xl,yl and x 2 , y 2 is Thus, the center is The radius of a circle is the distance from the center to the endpoint of the diameter.
If the graphs do not intersect, the simultaneous solutions are imaginary. One linear and one quadratic equation Solve the linear equation for one of the unknowns and substitute in the quadratic equation. Thus the simultaneous solutions are 3,4 and 4,3.
Eliminate the constant term between both equations. Thus the four solutions are: Method 2. The solution proceeds as in Method 1. Miscellaneous methods 1 Some systems of equations may be solved by replacing them by equivalent and simpler systems see Problems The solutions are: 3,2 ; -3,2 ; 3, -2 ; -3, The four solutions are: J,T ; 4v3 4v3 The solutions are 2,l and -2, The four solutions are: 6,2 ; -6, -2 ; 2,6 ; -2, Equation 2 indicates that those solutions in which the product xy is negative e.
The solutions are 3,2 and 2,3. What are the numbers? Hence the required numbers are 9, Find the numbers. Hence the required positive numbers are 7, 4. Find its dimensions. Solving 1 and 2 simultaneously, the required sides are 12 and 18ft. Find the lengths of the two legs. Solving 1 and 2 simultaneously, we find the legs have lengths 9 and 40ft. Supplementary Problems Find the numbers if the sum of their squares is If the short side is increased by 11ft and the long side decreased by 7ft, the length of the diagonal remains the same.
Find the dimensions of the original rectangle. The following indicate the meaning of inequality signs. A conditional inequalify holds only for particular values of the letters involved. Hence, x - a is between b units below 0, -b, and b units above 0. Now we select a test such as 0, 5 that is not on either line, determine which side of each line to shade and shade only the common region. If the objective is a linear function and the constraints are linear inequalities, the values, if any, that maximize o r minimize the objective occur at the corners of the region determined by the constraints.
The Green Company uses three grades of recycled paper, called grades A , B, and C, produced from scrap paper it collects. Companies that produce these grades of recycled paper do so as the result of a single operation, so the proportion of each grade of paper is fixed for each company. The Green Company needs at least units of grade A paper, units of grade B paper, and units of grade C paper. How should the company place its order so that costs are minimized? Since you can not have a company process a negative number of tons of paper, x L 0 and y 2 0.
These last two constraints are called natural or implied constraints, because these conditions are true as a matter of fact and need not be stated in the problem. We graph the inequalities determined from the constraints see Fig. The minimum for C x,y , if it exists, will occur at point A, B, C, or D,so we evaluate the obective function at these points.
Thus x Hence -4 This last statement is true since the square of any real number different from zero is positive. The above provides a clue as to the method of proof. Note that the proof is essentially a reversal of the steps in the first paragraph. To prove the theorem we start with a - 1 2 2 0, which is known to be true. This is true since the factors are both positive or both negative. Reversing the steps, which are reversible, provides the proof. Reversing the steps provides the proof.
Method 2. He uses 10 sheets of plywood and 15 studs in a small building and 15 sheets of plywood and 45 studs in a large building. Ramone has 60 sheets of plywood and studs available for use.
Each wind chime requires 3 hours of work from Jean and 1 hour of work from Wesley. Each bird house requires 4 hours of work from Jean and 2 hours of work from Wesley. Jean cannot work more than 48 hours per week and Wesley cannot work more than 20 hours per week. OOO6 f J 3. To solve equations in which the variable is in the exponent, we generally start by changing the expression from exponential form to logarithmic form.
The interest is usually paid at the ends of specified equal time intervals, such as monthly, quarterly, semiannually, or annually. The sum of the principal and the interest is called the amount. If a principal, P,is invested for t years at an annual interest rate, r, compounded continuously, then the amount, A , or ending balance, is given by:. However, using the logarithm tables and doing interpolation results in some error.
To deal with this problem and to get greater accuracy, we can use five-place logarithm tables, calculators, or computers. Generally, banks and other businesses use computers or calculators to get the accuracy they need. It is possible to compute the answer to the nearest cent here, while we were able to compute the result to the nearest ten dollars in Example The greater accuracy was possible because the calculator computes with more decimal places in each operation and then the answer is rounded.
In our examples, we rounded to hundredths because cents are the smallest units of money that have a general usefulness. Most calculators compute with 8, 10, or 12 significant digits in doing the operations.
Find the loudness of a sound that has an intensity 1OOOO times the threshold of hearing for the average human ear. The CHAP. If the pH of a solution exceeds 7, it is called an acid, but if its pH is less than 7 it is called a base. Find the pH of the solution whose concentration of hydrogen ions is 5.
The magnitude or Richter number of an earthquake depends on the ratio of the intensity, I, of an earthquake to the reference intensity, Zo, which is the smallest earth movement that can be recorded on a seismograph. Richter numbers are usually rounded to the nearest tenth or hundredth. If the intensity of an earthquake is determined to be 50 OOO times the reference intensity, what is its reading on the Richter scale? If the growth is exponential, what will its population be in ?
A piece of wood is found to contain grams of carbon when it was removed from a tree. If the rate of decay of carbon is 0. Find the amount required to repay the loan at the end of 2 years. What interest rate is he actually paying?
Find the total amount she must pay. Without formula. Using formula. What is the discount? The rate of interest i per year compounded a given number of times per year is called the nominal rate.
The rate of interest r which, if compounded annually, would result in the same amount of interest is called the effective rate. If the interest was compounded annually, what was the interest rate? At this rate, how long will it take the population to double? How does the intensity of the earthquake compare with the reference intensity? OOO6 - O. What is the decibel level of this sound? O00 0 - 2 2. OOO t If the growth was exponential, what was the annual growth rate?
What is the Richter number of the Iranian earthquake? What is the intensity of this Find the pH of each substance with the given concentration of hydrogen ions.
How many times the hearing threshold intensity is the intensity of a relatively quiet room? If in the growth rate was 1. If world population decline was exponential, what was the annual rate of decline?
If the decline of the forests is exponential, what is the annual rate of deforestation for El Salvador? If the decay of carbon is exponential with an annual rate of decay of 0. How much of 50 grams of strontium will be left after years? If a calculator is used your answers may vary. For example, if there are 3 candidates for governor and 5 for mayor, then the two offices may be filled in 3.
In general, if al can be done in x1 ways, a2 can be done in x2 ways, a3 can be done in x3 ways,. If an outfit consists of a jacket, a shirt, and a pair of slacks, how many different outfits can the man make? For example, the permutations of the three letters a , b , c taken all at a time are abc, acb, bca, bac, cba, cab. The permutations of the three letters a , b , c taken two at a time are ab, ac, ba, bc, ca, cb. For a natural number n , n factorial, denoted by n! That is, n!
Also, n! Zero factorial is defined to be 1:O! Thus 8P3 denotes the number of permutations of 8 things taken 3 at a time, and 5P5denotes the number of permutations of 5 things taken 5 at a time.
The symbol P n , r having the same meaning as nP, is sometimes used. Permutations with some things alike, taken all at a time The number of permutations P of n things taken all at a time, of which n1 are alike, n2 others are alike, n3 others are alike, etc. For example, the number of ways 3 dimes and 7 quarters can be distributed among 10 boys, each to receive one coin, is 10! Circular permutations The number of ways of arranging n different objects around a circle is n - l!
Thus 10 persons may be seated at a round table in 10 - l! Thus the combinations of the three letters a , b , c taken 2 at a time are ab, ac, bc. Note that ab and ba are 1 combination but 2 permutations of the letters a, b. The symbol nCr represents the number of combinations selections, groups of n things taken r at a time. Thus 9C4denotes the number of combinations of 9 things taken 4 at a time.
The symbol C n , r having the same meaning as ,C, is sometimes used. For example, the number of handshakes that may be exchanged among a party of 12 students if each student shakes hands once with each other student is 12! This formula indicates that the number of selections of r out of n things is the same as the number of selections of n - r out of n things.
Combinations of different things taken any number at a time The total number of combinations C of n different things taken 1 , 2 , 3.
For example, a woman has in her pocket a quarter, a dime, a nickel, and a penny. As factorials get larger, the results are displayed in scientific notation. Many calculators have only two digits available for the exponent, which limits the size of the factorial that can be displayed. Thus, 69! When the calculator can perform an operation, but it can not display the result an error message is displayed instead of an answer. The values of nPrand nCrcan often be computed on the calculator when n!
This can be done because the internal procedure does not require the result to be displayed, just used. Solved Problems In how many ways can he choose 1 language and 1 science?
SOLUTION a The first prize can be awarded in 10 different ways and, when it is awarded, the second prize can be given in 9 ways, since both prizes may not be given to the same contestant. In how many ways can these three positions be filled? SOLUTION The first person may take any one of 5 seats, and after the first person is seated, the second person may take any one of the remaining 4 seats, etc. In how many ways can this be done?
How many such arrangements are possible? Each of the chapters 1 through 33 contains a summary of the necessary definitions and theorems followed by a set of solved problems and concluding with a set of supplementary problems with answers.
Chapter 32 introduces three additional procedures for approximating the real zeros of polynomial equations of degree three or more. Chapter 33 is an informal development of the basic calculus concepts of limit, continuity, and conver-gence using the algebra procedures from the earlier chapters. Chapter 34 contains additional solved problems and supplementary problems with answers for each of the prior chapters.
The choice of whether to use a calculator or not is left to the student. A calculator is not required, but it can be used in conjunction with the book.
There are no directions on how to use a graphing calculator to do the problems, but there are several instances of the general procedures to be used and the student needs to consult the manual for the calculator being used to see how to implement the procedures on that particular calculator.
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