6 3 ) Problems Expand and simplify each of the following algebraic expressions as much as possible. 8. ( x + 2)( x + 2) (x 2 1x ) 2

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1 MA 143 Precalculus Essentials - Fall 2013 Assignment 1. Review of Exponents, Radicals and Fractions Dr. E. Jacobs Read Section 1.1, 1.2, 1.3 and 1.4 Section 1.1/Problems 25-30, Section 1.2/Problems 15-26, Section 1.3/Problems 13-60, Section 1.4/Problems No problems from this assignment will be collected. However, you will get a quiz on problems similar to the problems below: Problems 1-6. Express each of the following in simplest form. I want the exact answers. Decimal approximations are not sufficient. ( ) /4 Problems Expand and simplify each of the following algebraic expressions as much as possible. 7. ( x + 2)( x 2) 8. ( x + 2)( x + 2) 9. ( x ) 2 x 2 (x 2 1x ) y 1 x x y y x Note: A quiz on this material will be given on Friday, August 30. Your score on the quiz will be counted as your score on Assignment 1.

2 Assignment 2. Introduction to Functions Read Section 2.1 Section 2.1/Problems Note: For each of these problems, express each answer in its simplest form. 1. If f(x) = x 2 + 2x, calculate f(1) and f(a 1). 2. If F (x) = 3x + 1, calculate F (2x) and F (x + 2) 3. If G(x) = x 1 3, calculate and simplify the expression G(3x + 1) 4. For each of the following functions, calculate and simplify the expression f(x+h) f(x) h a. f(x) = 4x + 3 b. f(x) = 1 x 2 5. Let f(x) be defined as follows: Graph y = f(x) for 2 x 2. f(x) = { x 2 for x < 0 x + 1 for x 0

3 Assignment 3. Linear Functions Read Section 1.10 Section 1.10/Problems 5-12, 19-38, Find the equation of each of the following lines: a. The line through (0, 3) and (2, 5) b. The line through (2, 4) and (3, 0) 2. Find the equation of the line that passes through the point (2, 8) and is parallel to the line y + 2x = 3 3. The relationship between temperature measured in degrees Celsius (x) and degrees Fahrenheit (y) is known to be linear - that is, it has the form y = mx + b. Water boils at 100 C and 212 F and freezes at 0 C and 32 F. Calculate m and b 4. Suppose your salary was $28,500 in 2003 and $32,900 in Assume your salary follows a linear growth pattern. a) Write a linear equation giving your salary S in terms of the year t where t = 0 corresponds to the year b) Use the linear equation to predict your salary in A small business purchases a piece of equipment for $875. After 5 years the equipment will be outdated, having no value. Write a linear equation giving the value of the equipment y in terms of the time x. Assume that 0 x 5.

4 Assignment 4. Domains, Graphs, Asymptotes, Average Rate of Change Read Section 2.1, 2.2, 2.4, 3.7 Section 2.1/Problems 35-64, 71, 75-78, Section 2.2/Problems 4-28, 33-38, Section 2.4/Problems 1-22, Section 3.7/Problems 21-72, An internet electronics company charges $20 shipping for orders under $80 but provides free shipping for orders that are $80 or more. Thus, if P (x) is the total price for an order of x dollars, P (x) is given by the formula: P (x) = Graph this function on the interval 0 x 100. { x + 20 for x < 80 x for x Find the domain for each of the following functions: a. f(x) = 1 x 6 1 d. G(x) = (x + 1) b. g(x) = 2x 4 c. F (x) = x2 2 x e. H(x) = x The average rate of change of a function f between x and x+h is given by the expression f(x+h) f(x) h. Calculate the average rate of change for each of the following functions. Your answer in each case will be an expression involving both x and h. Simplify each answer as much as possible. a. f(x) = 4x b. f(x) = 4x 2 c. f(x) = 10x 3 d. f(x) = 4x x 3 e. f(x) = 2 x 4. Graph each of the following functions. Make sure that your graph shows all the vertical and horizontal asymptotes, if any. a. y = 2 x 2 b. y = 1 x c. y = 8x2 +x 2x 2 +1 d. y = x (x 2) 2 e. y = x2 x 2 4

5 Assignment 5. Preview of Calculus - Limits and Derivatives Read Section Section 13.2/Problems 11-22, Section 13.3/Problems 3-7, 9-10, 15-22, Section 13.4/Problems 5-16 Problems 1-4. Calculate each of the following limits. I suggest that you refer to your graphs of the functions in problem 4 on Assignment 4 to verify your answers. 1 8x 2 + x 1. lim 2. lim x x x 2x lim x x (x 2) 2 4. lim x x 2 x 2 4 Problems 5-8. If we take h to be closer and closer to 0, then the average rate of change gets closer and closer to the instantaneous rate of change. In calculus notation, this is described as: f f(x + h) f(x) (x) = lim h 0 h Calculate the instantaneous rate of change for each of the following functions using this limit formula. Your answer in each case will be an expression involving only x. Simplify each answer as much as possible. 5. f(x) = 4x 2 6. f(x) = 10x 3 7. f(x) = 4x x 3 8. f(x) = 2 x 9. For the function f(x) = 4x 2, find the equation of the line that is tangent to the curve at x = If c and n are constants, then derivative of cx n can be found immediately with the formula cnx n 1. Use this formula to find the derivatives of each of the following: a. y = x4 2 b. y = 2 x c. y = 2x x d. y = 1 x

6 Assignment 6. Quadratic Functions Read Section 3.1 Section 3.1/Problems 33-42, 45-48, 55, 56, 61, 64, 69, 75, It is always possible to use the method of completing the square to write a quadratic function in the standard form y = a(x h) 2 + k. Write each of the following quadratic functions in standard form and find the value of x that either maximizes or minimizes the function. a. f(x) = x 2 + 6x + 10 b. g(x) = 3 x 1 2 x2 2. A rocket is shot straight up into the air with an initial velocity of v 0 feet per second, and its height h(t) in feet above the ground after t seconds is given by h(t) = 16t 2 + v 0 t a. The rocket hits the ground after 4 seconds. Calculate v 0. b. What is the maximum height attained by the rocket? 3. The number of miles M that an automobile can travel on one gallon of gasoline is a function of its speed v (in miles per hour). If M = 1 30 v2 + 5 v for 0 < v < 70 2 find the value of v that maximizes M 4. There are 2,400 feet of fencing available to make a rectangular horse corral. a. Find a function that models the area of the corral in terms of the width x of the corral. b. Find the dimensions of the rectangle that maximize the area of the corral. 5. A rain gutter is formed by bending up the sides of a 30-inch wide rectangular sheet of metal, as shown below. a. Find a function that models the cross-sectional area of the gutter in terms of x. b. Find the maximum cross-sectional area of the gutter.

7 Assignment 7. Radian Measure Read Section 6.1 Section 6.1/Problems 3-26, 51-67, Suppose s denotes the length of the arc intercepted on a circle of radius r by a central angle of θ radians. If s = 4π kilometers and θ = π/2 radians, find the radius r. 2. Convert each degree measure to radians. Do not use a calculator. Write your answers as rational multiples of π. a) 30 b) 45 c) 90 d) 120 e) Convert each radian measure to degrees. Do not use a calculator. 3π π π π 2π a) 5 b) 3 c) 4 d) 6 e) 3 4. What radius should be used for a circular monorail track if the track is to change its direction of 7 in a distance of 24 meters (measured along the arc of the track)? 5. A motorcycle is moving at a speed of 88 kilometers per hour. If the radius of its wheels is 0.38 meters, find the angle in degrees through whch a spoke of a wheel turns in 3 seconds. 6. A satellite is orbiting a certain planet in a perfectly circular orbit of radius 7680 kilometers. If it makes it two-thirds of a revolution every hour, find : a) its angular speed ω b) its linear speed v 7. A nautical mile may be defined as the arc length intercepted on the surface of the earth by a central angle of measure 1 minute (1/60 of a degree). The radius of the earth is feet. How many feet are there in a nautical mile? 8. A girl rides her bicycle to school at a speed of 12 miles per hour. If the wheel diameter is 26 inches, what is the angular speed of the wheels? 9. A rpm (revolutions per minute) phonograph record has a radius of 14.6 centimeters. What is the linear speed (in centimeters per second) of a point on the rim of the record? 10. The earth, which is approximately 93,000,000 miles from the sun, revolves about the sun in a nearly circular orbit in approximately 365 days. Find the approximate linear speed (in miles per hour) of the earth in its orbit.

8 Assignment 8. Trigonometric Functions Read Section 5.1, 5.2, 6.2 Section 6.2/Problems 3-8, Problems 1-3. In each of the following three problems, the angle θ is an acute angle in a right triangle. Find the values of the six trigonometric functions for each case. 1. The side opposite θ has length 3 and the side adjacent to θ has length The side opposite θ has length 2 and the hypotenuse has length The side opposite θ has length 2 3 and the side adjacent to θ has length A 30-foot ladder leaning against a vertical wall just reaches a window sill. If the ladder makes an angle of 47 with the level ground, how high is the window sill? (Round off your answer to the nearest foot). 5. A guy wire 8 meters long helps support a CB base antenna mounted on top of a flat roof. If the wire makes an angle of 49.5 with the horizontal roof, how far above the roof is it attached to the antenna? (Round off your answer to the nearest tenth of a meter).

9 Assignment 9. Applications to Right Triangle Problems Read Section 6.2 Section 6.2/Problems 31-38, A jetliner is climbing so that its path is a straight line that makes an angle of 8.5 with the horizontal. How many meters does the jetliner rise while traveling 300 meters along its path? (Round off your answer to the nearest meter) 2. To measure the height of a cloud cover at night, a spotlight is aimed straight upward from the ground. The resulting spot of light on the clouds is viewed from a point on the level ground 850 meters from the spotlight, and the angle of elevation is measured at Find the height of the cloud cover to the nearest meter. 3. A lifeguard is seated on a high platform so that her eyes are 7 meters above sea level. Suddenly she spots the dorsal fin of a great white shark at a 4 angle of depression. Estimate, to the nearest meter, the horizontal distance between the platform and the shark. 4. Biologists studying the migration of birds are following a migrating flock in a light plane. The birds are flying at a constant altitude of 1200 feet and the plane is following at a constant altitude of 1700 feet. The biologists must maintain a distance of at least 600 feet between the plane and the flock in order to avoid disturbing the birds; therefore, they must monitor the angle of depression of the flock from the plane. Find the maximum allowable angle of depression, rounded off to the nearest angle. 5. A nature photographer using a telephoto lens photographs a rare bird roosting on a high branch of a tree at an angle of elevation of The distance between the lens and the bird is 330 feet. In order to obtain a more detailed photograph of the bird, the photographer cautiously moves closer to the base of the tree. The angle of elevation of the bird is now Find the new distance between the photographer s lens and the bird.

10 Assignment 10. Nonacute Angles Read Section 5.2, 6.3 Section 5.2/Problems 5-24, 65-72, Section 6.3/Problems Evaluate the six trigonometric functions of the angle θ in the standard position if the terminal side of θ contains the given point (x, y). Do not use a calculator - leave all answers in the form of a fraction or an integer. a. (2, 7) b. ( 2, 4) 2. If sec θ = 5 4 and csc θ = 5 3, use trigonometric identities to find sin θ, cos θ, tan θ and cot θ. 3. Find the values of the remaining five trigonometric functions if cos θ = θ in Q IV (Quadrant 4) 4. Find the quadrant containing θ for the given conditions: a. cos θ > 0 and sin θ < 0 b. sin θ > 0 and cot θ < 0 5. Let θ be the angle from the positive x-axis to the line 3y + 5x = 0, as shown in the diagram below. Find the values of sin θ and cos θ.

11 Assignment 11. Graphs of Trigonometric Functions, Vertical and Horizontal Shifts Read Section 2.5, 2.7, 5.3, 5.4 Section 2.5/Problems 4-33, Section 5.3/Problems 3-14, 17-38, 79, Section 5.4/Problems 4-7, 57 Problems 1-3. Refer to the graph of y = x and draw the graphs of each of the following functions by shifting the graph of y = x appropriately. 1. y = x 2 2. y = x y = x + 2 Problems 4-6. Sketch the graph of the function defined by each equation, find the amplitude and the period, and indicate one cycle on your graph. Start the cycle at a node for the sine functions and at a crest for the cosine function. You may use a graphing calculator or computer to help you draw the graph. 4. y = 2 sin x 5. y = 1 + cos πx 6. y = 2 sin x 3 Problems Write the equations of each of the following sine curves. Problem 7. Problem 8. Problem 9. Problem 10.

12 Assignment 12. Trigonometric Identities Read Section 7.1 Section 7.1/Problems 3-90 Problems 1-4. Use the fundamental identities to simplify each expression: 1. cot v sec v cot 2 y 1 + tan 2 y 3. (sec γ 1)(sec γ + 1) tan γ 4. cos γ 1 sin γ + cos γ 1 + sin γ 5. Rewrite the following trigonometric expression in terms of sines and cosines and simplify the result: sin y + tan y 1 + sec y Problems Show that each trigonometric equation is an identity. sin β csc β + cos β sec β = 1 7. (cos 2 t)(1 + tan 2 t) = 1 8. sin 2 v + tan 2 v + cos 2 v = sec 2 v 9. (cot β + csc β) 2 = sec β + 1 sec β cos( α) 1 + tan( α) sin( α) 1 + cot( α) = sin α + cos α

13 Assignment 13. Inverse Functions Read Section 2.7, 5.5, 6.4 Section 2.7/Problems 5-23, 37-60, Section 5.5/Problems 3-10, Section 6.4/Problems 3-6, Problems 1-3. Find the inverse of each of the following functions. 1. f(x) = 2 + 4x 2. f(x) = 4 x 2 (where x > 0) 3. f(x) = x x 2 Problems 4-6 answers in radians. Evaluate each expression without using a calculator or tables. Give ( 4. cos 1 1 ) 2 5. arccos tan 1 ( 1) 7. Use a calculator to evaluate each expression. Give answers in radians. a. arcsin b. sin 1 ( ) Problems 8-9. tables. Find the exact value of each expression without using a calculator or 8. tan(tan 1 3) 9. cos[arctan( 2)] 10. Rewrite as an algebraic expression in terms of x: sin(arctan x)

14 Assignment 14. Sine and Cosine of Sums and Differences, Double Angle Formulas Read Section 7.2, 7.3 Section 7.2/Problems 3-42, Section 7.3/Problems 3-16, Problems 1-5. Simplify each expression as much as possible. 1. cos(2π γ) ( ) 3π 2. csc 2 + β 3. sin(α β) cos β + cos(α β) sin β ( 4. sin t π ) ( + cos t π ) sin 2 t cos 2 t + cos 4t

15 Assignment 15. Trigonometric Equations Read Section 7.4, 7.5 Section 7.4/Problems 17-44, Section 7.5/Problems 3-34, 43, 45, 47, 51, 52 Problems 1-5. Solve each trigonometric equation with the side condition 0 θ < 360 or 0 t < 2π. Assume that when the variable is given as θ that the angle is measured in degrees and when the variable is given as t that the angle is measured in radians. Do not use a calculator or tables. 1. sin 2θ = cos 2t = 2 sin 2 t 3. cos θ + 3 sin θ = 1 4. sin θ + cos θ = 1 5. cot 2 t + csc 2 t = 3

16 Assignment 16. Exponential and Logarithm Functions Read Section 4.1, 4.2, 4.3 Section 4.1/Problems 9-16, Section 4.2/Problems 7-16, 21, 23, 31, Section 4.3/Problems 7-36, 86, Change each equation to its exponential form: a. log = 3 b. log 2 16 = 4 c. log 6 1 = 0 d. ln e = 1 2. Change each equation to its logarithmic form: a. 2 3 = 8 b = 10, 000 c = d. e 0 = 1 3. Evaluate each logarithm a. log 10 (1, 000, 000) b. log 2 64 c. log e 1 d. log Sketch the graph of each of the following equations: a. y = ( ) x 3 b. y = 2 ( ) x A drug is eliminated from the body through urine. The initial dose is 10 mg and the amount A(t) in the body t hours later is given by: A(t) = 10(0.8) t Determine when only 2 mg of the drug are left in the body.

17 Assignment 17. Properties of Logarithms, Applications of Logarithms and Exponentials Read Section 4.4, 4.5, 4.6 Section 4.4/Problems 7-52, 72, Section 4.5/Problems 3-28, 37-58, Section 4.6/Problems 1-33 Problems 1-6. Solve for x. Decimal approximations from a calculator are acceptable x = e x = log 10 ( x ) = log 2 x + log 2 ( 1 x 2 ) = 1 5. ln(ln x) = 0 6. ln ( e 2x) = 6 7. If the pollution of Lake Erie were stopped suddenly, it has been estimated that the level y of pollutants would decrease according to the formula: y = y 0 e t with time t in years and y 0 is the initial pollutant level (the pollutant level at which further pollution ceased). How many years would it take to clear 50% of the pollutants? 8. Chemists use a number denoted by ph to describe quantitatively the acidity or basicity of solutions. By definition, ph = log 10 [ H + ] where [H + ] denotes the hydrogen ion concentration (in moles per liter). In vinegar, the hydrogen ion concentration is [H + ] Approximate the ph of vinegar. 9. The population N(t) (in millions) of India t years after 1985 may be approximated by the formula: N(t) = 762e 0.022t How many years does it take for the population to triple? 10. If the interest is compounded continuously at the rate of 6% per year, the compound interest formula takes the form: P (t) = P 0 e 0.06t where P 0 denotes the initial deposit. Approximate the number of years it takes an initial deposit of $5,000 to grow to $40,000.

18 Assignment 18. Additional Topics Read Section 12.1, 13.5 and 9.1 Section 12.1/Problems 55-60, 41-48, Section 13.5/Problems 13-14, Section 9.1/Problems 1-74 This assignment should be completed by the last day of the course. At this point, it will be too late for your instructor to grade any homework, so no problems will be collected. However, you will be responsible for this material on the final exam.