Math 370 Semester Review Name

Math 370 Semester Review Name 1) State the following theorems: (a) Remainder Theorem (b) Factor Theorem (c) Rational Root Theorem (d) Fundamental Theorem of Algebra (a) If a polynomial f(x) is divided by (x-a), then the remainder is f(a). (b) Given a polynomial function f(x), f(a) = 0 if and only if x-a is a factor of f(x). (c) If a polynomial function f(x) = anx n + an-1x n-1 +... + a1x + a0 with integer coefficients has a rational zero p/q (in reduced form), then p must be a factor of the constant term a0, and q must be a factor of the leading coefficient, an. (d) If f(x) is a polynomial of degree n 1, then the equation f(x) = 0 has at least one complex root. Objective: Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. 2) 3x5-6x4 + 7x3-8 = 0 Positive (3, 1), negative (0) Objective: (3.3) Use Descartes' Rule of Signs Solve the equation for solutions in the interval [0, 2 ). 3) cos 2x = 2 - cos 2x 8, 9 8, 7 8, 15 8 Objective: (7.6) Solve Trigonometric Equation (Radians) II Solve the quadratic equation using the quadratic formula. Express the solution in standard form. 4) 2x 2-5x + 6 = 0 5 4 ± i 23 4 Objective: (2.1) Solve Quadratic Equations with Complex Imaginary Solutions Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point. 5) f(x) = 2x 2 + 2x + 1 minimum; - 1 2, 1 2 Objective: (2.2) Determine a Quadratic Function's Minimum or Maximum Value Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots. 6) x 3 + 2x 2-5x - 6 = 0 {-3, -1, 2} First list the possible rational roots of the equation, using the rational zero theorem. They are ±1, ±2, ±3, ±6. Test these with synthetic division, and find a zero. Since -1 is a zero, (x + 1) is a factor of the polynomial. Factoring the other (quadratic) factor into linear factors yields the other two roots. Objective: (2.5) Solve Polynomial Equations

7) x 3-6x 2 + 7x + 2 = 0 {2, 2 + 5, 2-5} Use the rational zero theorem and synthetic division to find the rational zero (2). This gives a linear factor and a quadratic factor. The quadratic factor, when set equal to zero, yields the other two (irrational) zeros when you use the quadratic formula. Objective: (2.5) Solve Polynomial Equations 8) x 3 + 7x 2 + 19x + 13 = 0 {-1, -3 + 2i, -3-2i} Objective: (2.5) Solve Polynomial Equations 9) x 3-6x 2 + 21x - 26 = 0 {2, 2 + 3i, 2-3i} Objective: (2.5) Solve Polynomial Equations Find an nth degree polynomial function with real coefficients satisfying the given conditions. 10) n = 3; 2 and -3 + 3i are zeros; leading coefficient is 1 f(x) = x 3 + 4x 2 + 6x - 36 Objective: (2.5) Use the Linear Factorization Theorem to Find Polynomials with Given Zeros Write a rational function that models the problem's conditions. 11) An athlete is training for a triathlon. One morning she runs a distance of 4 miles and cycles a distance of 31 miles. Her average velocity cycling is three times that while running. Express the total time for running and cycling, T, as a function of the average velocity while running, x. T(x) = 4 x + 31 3x Objective: (2.6) Solve Applied Problems Involving Rational Functions Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval notation. 12) x 2-3x - 4 < 0 (-1, 4) Explanation: Note that if we set y = x 2-3x - 4, the graph would be a parabola that opens upward, and we want to find all the x-values for which y < 0 (i.e., where the parabola is below the x-axis. This is the set of all x-values between the x-intercepts, which are easy to identify if we factor: x 2-3x - 4 = (x - 4)(x + 1) = 0. Therefore, the solution set is the set of all x such that -1 < x < 4. Objective: (2.7) Solve Polynomial Inequalities

13) x 2 + 8x + 12 0 (-, -6] [-2, ) Explanation: Using the same reasoning from Problem 6, the solution set is the set of all x-values for which the parabola y = x 2 + 8x + 12 is on or above the x-axis. In this case, this is the set of all x "on or outside the x-intercepts." Objective: (2.7) Solve Polynomial Inequalities Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation. 14) -x + 7 x - 3 0 (3, 7] For any rational expression, the only values of x at which the expression can change sign are the values of x that make the numerator or the denominator equal to 0. In this case, x = 3 and x = 7. So we divide the number line into 3 subintervals and test each. If x < 3, then the expression is negative, so the inequality is false. If 3 < x < 7, the expression is positive, so the inequality is true. If x > 7, the expression is once again negative, so the inequality is false. We should also consider the boundary points. When x = 3, the expression is undefined, and if x = 7, the expression equals 0. Thus 7 is part of the solution set, but 3 is not. So the solution set is (3, 7]. Objective: (2.7) Solve Rational Inequalities 15) (x - 1)(3 - x) (x - 2) 2 0 (-, 1] [3, ) Objective: (2.7) Solve Rational Inequalities 16) x x + 3 2 [-6, -3) Objective: (2.7) Solve Rational Inequalities

Solve the exponential equation. Express the solution set in terms of natural logarithms. 17) e 4x = 7 ln 7 4 Objective: (3.4) Use Logarithms to Solve Exponential Equations 18) 4 x + 4 = 5 2x + 5 5 ln 5-4 ln 4 ln 4-2 ln 5 Objective: (3.4) Use Logarithms to Solve Exponential Equations Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. 19) log 4 (x + 3) + log 4 (x - 3) = 2 {5} Objective: (3.4) Use the Definition of a Logarithm to Solve Logarithmic Equations Solve the problem. 20) The function f(x) = 1 + 1.4 ln (x + 1) models the average number of free-throws a basketball player can make consecutively during practice as a function of time, where x is the number of consecutive days the basketball player has practiced for two hours. After how many days of practice can the basketball player make an average of 9 consecutive free throws? 302 days Objective: (3.4) Solve Applied Problems Involving Exponential and Logarithmic Equations

21) A building 290 feet tall casts a 100 foot long shadow. If a person looks down from the top of the building, what is the measure of the angle between the end of the shadow and the vertical side of the building (to the nearest degree)? (Assume the person's eyes are level with the top of the building.) 19 Explanation: We see that the angle is = tan -1 (100/290) = 19 Objective: (4.8) Solve a Right Triangle Solve the triangle. 22) 65 7 40 B = 75, a = 4.66, c = 6.57 Objective: (6.1) Use the Law of Sines to Solve Oblique Triangles

Solve the triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. 23) 8 6 4 A = 104, B = 47, C = 29 Objective: (6.2) Use the Law of Cosines to Solve Oblique Triangles 24) a = 5, b = 8, C = 120 c = 11.4, A = 22, B = 38 Objective: (6.2) Use the Law of Cosines to Solve Oblique Triangles Use DeMoivre's Theorem to find the indicated power of the complex number. Write the answer in rectangular form. 25) (-3 + 3i 3) 3 216 Explanation: Recall DeMoivre's Theorem: For a complex number z =r(cos +isin ), z n = r n (cos(n )+isin(n )). Convert -3 + 3i 3 to polar form: r = (-3) 2 +(3 3) 2 = 9+27 = 6, and is an angle in Quadrant II whose tangent is - 3. Thus = 3. So (-3 + 3i 3)3 = (6(cos + isin )) 3 = 6 3 (cos2 +isin2 ) = 216. Objective: (6.5) Find Powers of Complex Numbers in Polar Form 26) (1 + i)20-1024 Objective: (6.5) Find Powers of Complex Numbers in Polar Form

Graph the ellipse and locate the foci. 27) x2 64 + y2 4 = 1 foci at (2 15, 0) and (-2 15, 0) Objective: (9.1) Graph Ellipses Centered at the Origin Find the standard form of the equation of the ellipse and give the location of its foci. 28) x 2 49 + y2 36 = 1 foci at (- 13, 0) and ( 13, 0) Objective: (9.1) Write Equations of Ellipses in Standard Form

Find the standard form of the equation of the hyperbola. 29) (y - 2) 2 4 - (x + 1)2 16 = 1 Objective: (9.2) Write Equations of Hyperbolas in Standard Form Solve the problem. 30) Two airplanes leave an airport at the same time, one going northwest (bearing 135 ) at 417 mph and the other going east at 329 mph. How far apart are the planes after 4 hours (to the nearest mile)? 2760 miles Objective: (6.2) Solve Applied Problems Using the Law of Cosines 31) A surveyor standing 52 meters from the base of a building measures the angle to the top of the building and finds it to be 36. The surveyor then measures the angle to the top of the radio tower on the building and finds that it is 48. How tall is the radio tower? 19.97 meters Objective: (6.1) Solve Applied Problems Using the Law of Sines Solve by the method of your choice. 32) 2x2 + xy - y2 = 3 x2 + 2xy + y2 = 3 2 3 3, 3 3, - 2 3 3, - 3 3 Objective: (7.4) Solve Nonlinear Systems By Addition

Graph the solution set of the system of inequalities or indicate that the system has no solution. 33) x2 + y2 4 y - x 2 > 0 Objective: (7.5) Graph a System of Inequalities

Graph the polar equation. 9 34) r = 3-3 cos Identify the directrix and vertex. directrix: 3 unit(s) to the left of the pole at x = -3 vertex: 3 2, Objective: (9.6) Graph the Polar Equations of Conics 35) (x 2 + y 3 ) 7 ; 5th term 35x 6 y 12 Objective: (10.5) Find a Particular Term in a Binomial Expansion

Begin by graphing the standard quadratic function f(x) = x 2. Then use transformations of this graph to graph the given function. 36) g(x) = - 1 2 (x + 2)2 + 3 Objective: (1.6) Graph Functions Involving a Sequence of Transformations Find the inverse of the one-to-one function. 37) f(x) = (x + 8) 3 f -1 (x) = 3 x - 8 Objective: (1.8) Find the Inverse of a Function

Graph f as a solid line and f -1 as a dashed line in the same rectangular coordinate space. Use interval notation to give the domain and range of f and f -1. 38) f(x) = x 3-4 f domain = (-, ); range = (-, ) f -1 domain = (-, ); range = (-, ) Objective: (1.8) Find the Inverse of a Function and Graph Both Functions on the Same Axes

Graph the polar equation. 39) r = 3 sin 2 Objective: (6.4) Use Symmetry to Graph Polar Equations

Graph the rational function. Give the y-intercept, equations of all asymptotes, list all zeros, and give domain and range. 40) f(x) = x3-3x 2-28x+60 2x 3-7x 2-12x+45 y-intercept: (0, 4/3); Vertical asymptotes: x = -5/2 and x = 3; Horizontal asymptote: y = 1/2; Zeros: -5, 2 and 6; domain: (, -5/2) (-5/2, 3) (3, ); range: (, ) Objective: (2.6) Graph Rational Functions Find the indicated sum. 41) Find the sum of the first 42 terms of the arithmetic sequence: 2, 4, 6, 8,... 1806 Objective: (10.2) Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence Solve the problem. Round to the nearest dollar if needed. 42) To save for retirement, you decide to deposit $2500 into an IRA at the end of each year for the next 40 years. If the interest rate is 8% per year compounded annually, find the value of the IRA after 40 years. A) $51,811 B) $647,641 C) $597,353 D) $15,310,120 B Objective: (10.3) Find the Value of an Annuity Find the sum of the infinite geometric series, if it exists. 43) 5-5 4 + 5 16-5 64 +... 4 Objective: (10.3) Use the Formula for the Sum of an Infinite Geometric Series

Write a formula for the general term (the nth term) of the geometric sequence. 44) 8, 16, 32, 64, 128,... an = 8(2) n - 1 Objective: (10.3) Use the Formula for the General Term of a Geometric Sequence If the given sequence is a geometric sequence, find the common ratio. 45) 4 3, 16 3, 64 3, 256 3, 1024 3 4 Objective: (10.3) Find the Common Ratio of a Geometric Sequence Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. 46) Find a15 when a1 = 30, d = -4. -26 Objective: (10.2) Use the Formula for the General Term of an Arithmetic Sequence Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval notation. 47) (x - 9)(x + 4) > 0 (-, -4) (9, ) Objective: (2.7) Solve Polynomial Inequalities 48) x 3 + 2x 2 - x - 2 > 0 (- 2, -1) (1, ) Objective: (2.7) Solve Polynomial Inequalities Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation. 49) (x + 12)(x - 7) x - 1 0 [-12, 1) [7, ) Objective: (2.7) Solve Rational Inequalities

Solve the exponential equation. Express the solution set in terms of natural logarithms. 50) 4 x + 4 = 5 2x + 5 5 ln 5-4 ln 4 ln 4-2 ln 5 Objective: (3.4) Use Logarithms to Solve Exponential Equations Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 51) 3 6x = 2.1 0.11 Objective: (3.4) Use Logarithms to Solve Exponential Equations Use the Binomial Theorem to expand the binomial and express the result in simplified form. 52) (x + 2)4 x4 + 8x3 + 24x2 + 32x + 16 Objective: (10.5) Expand a Binomial Raised to a Power Use mathematical induction to prove that the statement is true for every positive integer n. 53) 2 is a factor of n 2 - n + 2 Show S1 is true: 1 2-1 + 2 = 2; 2 is a factor of 2 Assume that Sk is true: k 2 - k + 2; 2 is a factor of k 2 - k + 2 Show that if Sk is true, then Sk+1 is true: Sk+1 = (k + 1) 2 - (k + 1) + 2 = k 2 + k + 2 = (k 2 - k + 2) + 2k (k 2 - k + 2) is Sk and 2 is a factor of 2k Since 2 is a factor of Sk+1 is true, the statement is true for all values of n. Objective: (10.4) Prove Statements Using Mathematical Induction Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 54) e 2x + e x - 6 = 0 0.69 Objective: (3.4) Use Logarithms to Solve Exponential Equations Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. 55) log 3 (x + 5) - log 3 (x - 4) = 1 { 17 2 } Objective: (3.4) Use the Definition of a Logarithm to Solve Logarithmic Equations 56) log (5 + x) - log (x - 3) = log 5 {5} Objective: (3.4) Use the One-to-One Property of Logarithms to Solve Logarithmic Equations

Solve the problem. 57) Find out how long it takes a $3300 investment to double if it is invested at 9% compounded quarterly. Round to the nearest tenth of a year. Use the formula A = P 1 + r n nt. 7.8 years Objective: (3.4) Solve Applied Problems Involving Exponential and Logarithmic Equations 58) Larry has $2800 to invest and needs $3300 in 20 years. What annual rate of return will he need to get in order to accomplish his goal, if interest is compounded continuously? (Round your answer to two decimals.) 0.82% Objective: (3.4) Solve Applied Problems Involving Exponential and Logarithmic Equations Use Newton's Law of Cooling, T = C + (T0 - C)e kt, to solve the problem 59) Mostacolli baked at 400 F is taken out of the oven into a kitchen that is 73 F. After 6 minutes, the temperature of the mostacolli is 344.5 F. What will its temperature be 15 minutes after it was taken out of the oven? Round your answer to the nearest degree. 278 F Objective: (3.5) Use Newton's Law of Cooling Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm, and then round to three decimal places. 60) y = 32(1.3) x y = 32e x ln 1.3, y = 32e 0.262x Objective: (3.5) Express an Exponential Model in Base e The point P(x, y) on the unit circle that corresponds to a real number t is given. Find the value of the indicated trigonometric function at t. 61) 5 6, 11 6 11 6 Find sin t. Objective: (4.2) Use a Unit Circle to Define Trigonometric Functions of Real Numbers 62) State the three Pythagorean Identities for trigonometric functions. sin 2 x + cos 2 x = 1 tan 2 x + 1 = sec 2 x 1 + cot 2 x = csc 2 x Objective: 63) State the Double-angle identities for sine, cosine and tangent. Consult your textbook. Objective: 64) State the Half-angle identities for sine, cosine and tangent. Consult your textbook. Objective:

65) State the angle sum identites for sine, cosine and tangent. Consult your textbook. Objective: Solve the equation on the interval [0, 2 ). 66) cos 2x = 3 2 12, 11 12, 13 12, 23 12 Objective: (5.5) Solve Equations With Multiple Angles 67) cos 2 x + 2 cos x + 1 = 0 Objective: (5.5) Solve Trigonometric Equations Quadratic in Form 68) sin 2x + sin x = 0 0, 2 3,, 4 3 Objective: (5.5) Use Identities to Solve Trigonometric Equations Use the unit circle to find the value of the trigonometric function. 69) sec 6 2 3 3 Objective: (4.2) Use a Unit Circle to Define Trigonometric Functions of Real Numbers 70) tan 2 3-3 Objective: (4.2) Use a Unit Circle to Define Trigonometric Functions of Real Numbers 0 t < 2 and sin t is given. Use the Pythagorean identity sin2 t + cos 2 t = 1 to find cos t. 71) sin t = 8 9 17 9 Objective: (4.2) Recognize and Use Fundamental Identities Use an identity to find the value of the expression. Do not use a calculator. 72) tan 1.8 cot 1.8 1 Objective: (4.2) Recognize and Use Fundamental Identities

Use periodic properties of the trigonometric functions to find the exact value of the expression. 73) cos 29 3 1 2 Objective: (4.2) Use Periodic Properties Use the given triangles to evaluate the expression. Rationalize all denominators. 74) tan 6 - sin 3-3 6 Objective: (4.3) Find Function Values for 30 ( /6), 45 ( /4), and 60 ( /3) Find a cofunction with the same value as the given expression. 75) cos 16 sin 7 16 Objective: (4.3) Use Equal Cofunctions of Complements Solve the problem. 76) A radio transmission tower is 250 feet tall. How long should a guy wire be if it is to be attached 8 feet from the top and is to make an angle of 28 with the ground? Give your answer to the nearest tenth of a foot. 515.5 feet Objective: (4.3) Use Right Triangle Trigonometry to Solve Applied Problems Use reference angles to find the exact value of the expression. Do not use a calculator. 77) csc 1020-2 3 3 Objective: (4.4) Use Reference Angles to Evaluate Trigonometric Functions

78) tan 79 6 3 3 Objective: (4.4) Use Reference Angles to Evaluate Trigonometric Functions Graph the function. 79) y = -3 cos 2x - 2 + 2 Objective: (4.5) Use Vertical Shifts of Sine and Cosine Curves

Solve the problem. 80) An experiment in a wind tunnel generates cyclic waves. The following data is collected for 32 seconds: Time Wind speed (in seconds) (in feet per second) 0 22 8 50 16 78 24 50 32 22 Let V represent the wind speed (velocity) in feet per second and let t represent the time in seconds. Write a sine equation that describes the wave. V = 28 sin 16 t - 2 + 50 Objective: (4.5) Model Periodic Behavior Find the exact value of the expression. 81) sin -1 2 2 4 Objective: (4.7) Understand and Use the Inverse Sine Function 82) cos -1-3 2 5 6 Objective: (4.7) Understand and Use the Inverse Cosine Function 83) tan -1 (- 3) - 3 Objective: (4.7) Understand and Use the Inverse Tangent Function Use a sketch to find the exact value of the expression. 84) cos tan -1 6 5 5 61 61 Objective: (4.7) Find Exact Values of Composite Functions with Inverse Trigonometric Functions 85) tan sin -1 2 2 1 Objective: (4.7) Find Exact Values of Composite Functions with Inverse Trigonometric Functions

Determine the intervals on which the function is increasing, decreasing, and constant. 86) Increasing on [-3, 0]; Decreasing on [-5, -3) and [2, 5]; Constant on [0, 2] Objective: (2.3) Find Intervals Where Function Is Increasing/Decreasing/Constant The graph of the function y = f(x) is given below. Sketch the graph of y = f(x). 87) Objective: (2.4) Graph Absolute Value of Function Given Graph of Function

Find the equation that the given graph represents. 88) A) P(x) = -x4 + x3-12x2 + 10 B) P(x) = x5-4x3 + 12x2 + 10 C) P(x) = -x5-10x2-10 D) P(x) = x4 + x3-10x2 + 10 D Objective: (3.5) Tech: Match Equation to Graph Solve the rational equation. 89) x - 6 x - 3 = -4 18 5 Objective: (4.3) Solve Rational Equation Use Cramer's rule to solve the system. 90) 3x + 4y - z = 23 x - 5y + 5z = -7 4x + y + z = 18 {(3, 4, 2)} Objective: (8.5) Solve a System of Linear Equations in Three Variables Using Cramer's Rule Solve the system of equations using matrices. Use Gauss-Jordan elimination. 91) x = 5 - y - z x - y + 3z = 23 2x + y = 9 - z {(4, -4, 5)} Objective: (8.1) Use Matrices and Gauss-Jordan Elimination to Solve Systems Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. 7x - 1 92) (x - 4)(x - 2) 2 A x - 4 + B x - 2 + C (x - 2) 2 Objective: (7.3) Decompose P/Q, Where Q Has Repeated Linear Factors Convert the polar equation to a rectangular equation. 93) r = 6 cos + 4 sin A) x 2 - y 2 = 6x + 4y B) x 2 + y 2 = 4x + 6y C) 6x + 4y = 0 D) x 2 + y 2 = 6x + 4y D Objective: (6.3) Convert an Equation from Polar to Rectangular Coordinates

Solve the problem. 94) Two airplanes leave an airport at the same time, one going northwest (bearing 135 ) at 414 mph and the other going east at 342 mph. How far apart are the planes after 4 hours (to the nearest mile)? A) 2327 miles B) 2796 miles C) 699 miles D) 2194 miles B Objective: (6.2) Solve Applied Problems Using the Law of Cosines