MATHEMATICAL SKILLS REVIEW

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1 Chapter 0 GOALS MATHEMATICAL SKILLS REVIEW When you have mastered the contents of the textbook Appendix you will be able to successfully demonstrate your mathematical skills in each of the following eight areas: 1. Powers of Ten Notation 2. Significant Figures 3. Cartesian Graphs 4. Dimensional Analysis 5. Right Triangles 6. Exponents 7. Logarithms 8. Exponential Function 0

2 OVERVIEW MATHEMATICAL SKILLS REVIEW This review of basic mathematics is a good place to begin your study of introductory physics. In the textbook Appendix you will find a helpful review of the eight basic mathematical skills which you will need during your study of this text. SUGGESTED STUDY PROCEDURE Please note that the math review is broken into eight parts: Powers of Ten Notation, Significant Figures Cartesian Graphs, Dimensional Analysis, Right Triangles, Exponents, Logarithms, and The Exponential Function. As you begin your study of this mathematical review, attempt each of the selfcheck exercises which begin on the next page of this Study Guide. Check your answers against the answers given. If you do not score 100% on any individual part, refer to the text Appendix for help. The outline below will provide a quick reference to each of the mathematical skills. Next, turn to the Practice Test in this Study Guide. Complete the practice test and check your answers with those given at the end of the test. In any of the eight mathematical areas where you did not score 100%, again refer to that particular section in the text Appendix and work through additional Example Problems and/or Exercises Chapter Goals Self- Suggested Text Check Readings (Appendix) Powers of Ten Notation 1 A.2 Significant Figures 2 A.3 Cartesian Graphs 3 A.4 Dimensional Analysis 4 A.5 Right Triangles 5 A.6 Exponents 6 A.7 Logarithms 7 A.8 Exponential Function 8 A.9 1

3 SELF-CHECK EXERCISES Powers of Ten Self-Check Solve each of the following problems, and give the answer in powers of ten notation x / = x 10 3 x 5.00 x 10 4 /2.50 x 10 5 = x x x 10 5 =. Powers of Ten Self-Check Answers If you had difficulty in correctly solving these problems, please study the textbook Section Appendix 2 on the powers of ten notation x x x 10 7 Significant Figures Self-Check Solve each of the following problems, and give the correct answer in the appropriate number of significant figures x 1.03/.025 = x 10 3 x 2.5 x 10-2 /2.15 x 10-4 =. 3. Solve for A; 3x 10-4 A/4.5 x 10 3 = 1. A =. Significant Figures Self-Check Answers If you had difficulty in correctly solving these problems, please study the textbook Appendix Section A.3 on Significant Figures (3 significant figures) x 10 5 (2 significant figures) 3. 2 x 10 7 (1 significant figure) Graphing and Dimensional Analysis Self- Check 1. The following table is taken from a drivers manual and shows data for stopping an automobile on dry pavement Velocity Thinking Total Stopping (m/sec) Distance (m) Distance (m) a. Draw a graph of thinking distance (y-axis) versus velocity (x-axis), and find the slope of the curve at the point on the curve where x = 15 m/sec. b. Draw a graph of the total stopping distance (y axis) versus the velocity (x axis), and find the slope of the curve at the point where x = 20 m/sec. 2

4 2. We can define length, mass, and time as fundamental dimensions in a system of measurement. What are the SI (System International) units for a. Length b. Mass c. Time The SI units are related to each other by multiples of ten, and the units are represented by the fundamental unit with the proper prefix. What are the relationships between the fundamental unit and the following common prefixes? d. The prefix centi- means, so one tesla = centiteslas. [10-2 ; 10 2 ] e. The prefix milli- means, so one liter = milliliters. [10-3 ; 10 3 ] f. The prefix kilo- means, so one watt = kilowatts. [10 3 ; 10-3 ] Graphing and Dimensional Analysis Self-Check Answers If you had difficulty in correctly solving these problems, please study the textbook Section A.4, Cartesian Graphs, and A.5, Dimensional Analysis. 1. a sec; b. 4.8 sec 2. a. meter; b. kilogram; c. second; d. 10-2, 10 2 ; e. 10-3, 10 3 ; f. 10 3, 10-3 Right Triangles Self-Check A surveyor wishes to determine the distance between two points A and B, but he cannot make a direct measurement because a river intervenes. He steps off a line AC at a 90º angle to AB and 264 meters long. With his transit, at point C he measures the angle between line AB and the line formed by C and B. Angle BCA is measured to be 62º. What is the distance from A to B? Right Triangles Self-Check Answers Distance AB = 497 meters. If you had difficulty getting this answer, you will find additional information in the textbook Section A.6, Right Triangles, of the appendix. TRIGONOMETRIC RELATIONSHIPS While nearly all of the problems in the book can be worked using the definitions of the sine, cosine, and tangent you have seen derived for right triangles, there are some relationships between these functions that it will be useful for you to know. We will derive them below. From the 3

5 Pythagorean theorem you learned in high school, you know the relationship between the three sides of a right triangle; i.e., the sum of the squares of the two sides is equal to the square of the hypotenuse x 2 + y 2 = r 2 (1) Look again at Figure (0-1). Note y/r = sin Θ = cos Α (4) x/r = cos Θ = sin α (5) Since Θ + α + 90º = 180º i.e., the sum of the angles of any triangle equals 180º, then α = 90º - Θ The angle α is called the complement of the angle Θ. From equations (4) and (5) above notice that the sine of an angle is equal to the cosine of its complementary angle and the cosine of the angle is equal to the sine of its complementary angle. Example: An arrow shot into the air comes vertically down and sticks in the grass on the side of a hill inclined 68º from vertical. What portions of the arrow point down the hill and perpendicular to the hill? See Figure: Now we divide both sides of equation (1) by the square of the hypotenuse, r 2, x 2 /r 2 + y 2 /r 2 = 1 Then recall the definitions of sine and cosine, sin Θ = y/r; cosine Θ = x/r so x 2 /r 2 cos 2 Θ; y 2 /r 2 sin 2 Θ thus x 2 /r 2 + y 2 /r 2 = cos 2 Θ + sin 2 Θ = 1 (2) The sum of the squares of the cosine and the sine of any angle is equal to one. In a similar way we can derive a relationship between the tangent, sine and cosine of any angle. tan Θ = y/x. Divide both numerator and denominator by the hypotenuse, r tan Θ = (y/r) / (x/r) = sin Θ/cos Θ (3) The tangent of any angle is equal to the ratio of the sine of the angle to the cosine of the angle. 4

6 Exponents Self-Check x 8 3 = = x 10-3 = =. Exponents Self-Check Answers If you had difficulty with any of the above answers, you can find additional assistance in the textbook Appendix, Section A.7. Logarithms Self-Check Use the properties of Logarithms to solve each of the following problems: 1. 3 x 5 =. 2. 3/5 = =. 4. Solve the following equation for X: 16 = 4(10 2X ) Logarithms Self-Check Answers (1/2) log(4) If you had difficulty with any of these problems, please study the textbook Appendix Section A.8. Exponential Function Self-Check Use the properties of exponential functions to solve each of the following problems. 1. If the population of a growing country is governed by the relation, N = 100, t, where t is in years, how long will it take for the population to double? 2. Predict the total population of the country in 10 years. Exponential Function Self-Check Answers years thousand If you had difficulty with either of the problems posed above, please go to the textbook Appendix Section A.9. 5

7 PRACTICE TEST 1. (Powers of Ten Notation and Significant Figures) Solve the following problems using the powers of ten notation. In each case write your answer with the appropriate number of significant figures. a) 731 x x 431/0.005 =. b) 4.7 x 10-7 x 2.51 x 10 5 /2x10-3 =. c) Solve for A: 795 x x 10 3 A = x (Cartesian Graphs and Dimensional Analysis) The following data were taken from a physics laboratory experiment Velocity of a Rolling Coin Time (m/s) (sec) a) Plot a graph showing velocity versus time for this rolling object. b) Calculate the slope of the curve at (4, 15). c) Give the proper dimensions to the slope. 6

8 3. (Right Triangles) To find the height of a tall building, a physics student steps 75 paces (each 1 meter) from the base of the building. Using a ruler at arm's length (1 meter), the student finds that at this distance, the building appears to be 50 centimeters when compared to the ruler. Determine the approximate height of the building. 4. (Exponents) Calculate the following using the laws of exponents. a) =. b) 10 8 x 10-3 =. c) x =. 5. (Logarithms) Use the properties of logarithms to solve each of the following problems. a) 7 x 4 =. b) 20 4 =. c) 9 3 =. d) Solve the following equation for X: 2700 = 300(10 3X ) 6. (Exponential Function) Use the properties of the exponential function to solve the following problem: a) A biologist determines that during the month of June, the number of live frogs in a small pond is governed by the relationship N = 189e. 075t (t is the number of days). How long will it be until the number of frogs is three times its initial population? b) Predict the total population of frogs by the end of the month (30 days). ANSWERS: 1. a) 6.36 x 10 7 b) 6 x 10 1 c) 1.52 x b) 10 c) m/sec m 4. a) 57 b) 10 5 c) a) 28 b) 5 c) 728 d) a) 14.6 days b)

Math 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts

Introduction Math 121: Calculus 1 - Fall 201/2014 Review of Precalculus Concepts Welcome to Math 121 - Calculus 1, Fall 201/2014! This problems in this packet are designed to help you review the topics