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All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)

Factoring Completely Factor the expression completely. 97. 12x3 + 18x 98E 99E 100E 101E 102E 103E 104E Factoring Completely Factor the expression completely. 105. 49 4y2 106E 107E 108E 109E 110E 111E 112E Factoring Completely Factor the expression completely. 113. x2(x2 1) 9(x2 1)114E 115E 116E 117E 118E 119E 120E 121E 122E 123E 124E 125E 126E 127E 128E Factoring Completely Factor the expression completely. (This type of expression arises in calculus when using the Product Rule) 129. (x2+3)1/323x2(x2+3)4/3 130E 131E 132E 133E 134E Volume of Concrete A culvert is constructed out of large cylindrical shells cast in concrete, as shown in the figure. Using the formula for the volume of a cylinder given on the inside front cover of this book, explain why the volume of the cylindrical shell is V=R2hr2h Factor to show that V = 2 average radius height thickness Use the unrolled diagram to explain why this makes sense geometrically Mowing a Field A square field in a certain state park is mowed around the edges every week. The rest of the field is kept unmowed to serve as a habitat for birds and small animals (see the figure). The field measures b feet by b feet, and the mowed strip is x feet wide. (a) Explain why the area of the mowed portion is b2 (b 2x)2. (b) Factor the expression in part (a) to show that the area of the mowed portion is also 4x(b x).137E 138E 139E 140E 141E 142E Which of the following are rational expressions? (a) 3xx21 (b) x+12x+3 (c) x(x21)x+3 To simplify a rational expression, we cancel factors that are common to the __________ and __________. So the expression (x+1)(x+2)(x+3)(x+2) simplifies to __________.3E 4E 5E 6E Domain Find the domain of the expression. 7. 4x2 10x + 3 Domain Find the domain of the expression. 8. x4 + x3 + 9x Domain Find the domain of the expression. 9. x21x3 10E Domain Find the domain of the expression. 11. x+3 12E 13E 14E 15E 16E 17E 18E Simplify Simplify the rational expression. 19. x2+5x+6x2+8x+15 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E Multiply or Divide Perform the multiplication or division and simplify. 31.x2+7x+12x2+3x+2x2+5x+6x2+6x+9 32E 33E 34E Multiply or Divide Perform the multiplication or division and simplify. 35. x3x+1xx2+2x+1 36E 37E 38E 39E 40E 41E 42E Add or Subtract Perform the addition or subtraction and simplify. 43. 3x+11x+2 44E 45E 46E Add or Subtract Perform the addition or subtraction and simplify. 47. u+1+uu+1 48E 49E 50E 51E 52E 53E 54E 55E 56E 57E 58E 59E 60E 61E 62E Compound Fractions Simplify the compound fractional expression. 63. 1x1+1x+3x+1 64E 65E 66E Compound Fractions Simplify the compound fractional expression. 67. xyyx1x21y2 68E 69E 70E 71E 72E Expressions Found in Calculus Simplify the fractional expression. (Expressions like these arise in calculus.) 73. 11+x+h11+xh 74E 75E Expressions Found in Calculus Simplify the fractional expression. (Expressions like these arise in calculus.) 76.(x+h)37(x+h)(x37x)h Expressions Found in Calculus Simplify the fractional expression. (Expressions like these arise in calculus.) 77. 1+(x1+x2)2 78E 79E 80E 81E Expressions Found in Calculus Simplify the expression. (This type of expression arises in calculus when using the quotient rule.) 82. (1x2)1/2+x2(1x2)1/21x2 83E 84E 85E 86E 87E 88E 89E 90E 91E 92E 93E 94E 95E 96E 97E Average Cost A clothing manufacturer finds that the cost of producing x shirts is 500 + 6x + 0.01x2 dollars. (a) Explain why the average cost per shirt is given by the rational expression A=500+6x+0.01x2x (b) Complete the table by calculating the average cost per shirt for the given values of x. x Average cost 10 20 50 100 200 500 1000 99E 100E DISCUSS: Algebraic Errors The left-hand column of the table lists some common algebraic errors. In each case, give an example using numbers that shows that the formula is not valid. An example of this type, which shows that a statement is false, is called a counterexample.DISCUSS: Algebraic Errors Determine whether the given equation is true for all values of the variables. If not, give a counterexample. (Disregard any value that makes a denominator zero.) (a) 5+a5=1+a5 (b) x+1y+1=xy (c) xx+y=11+y (d) 2(ab)=2a2b (e) ab=ab (f) 1+x+x2x=1x+1+x 103E Yes or No? If No, give a reason. (a) When you add the same number to each side of an equation, do you always get an equivalent equation? (b) When you multiply each side of an equation by the same nonzero number, do you always get an equivalent equation? (c) When you square each side of an equation, do you always get an equivalent equation?What is a logical first step in solving the equation? (a) (x + 5)2 = 64 (b) (x + 5)2 + 5 = 64 (c) x2 + x = 2 Explain how you would use each method to solve the equation x2 4x 5 = 0. (a) By factoring: __________ (b) By completing the square: __________ (c) By using the Quadratic Formula: __________4E 5E The equation (x + 1)2 5(x + 1) + 6 = 0 is __________ of type. To solve the equation, we set W = __________. The resulting quadratic equation is __________.To eliminate the denominators in the equation 3x+5x+2=2, we multiply each side by the lowest common denominator __________ to get the equivalent equation __________To eliminate the square root in the equation 2x+1=x+1, we __________ each side to get the equation __________.9E Solution? Determine whether the given value is a solution of the equation. 10. 1 [2 (3 x)] = 4x (6 + x) (a) x = 2 (b) x = 4 11E 12E 13E 14E 15E 16E 17E 18E Linear Equations The given equation is either linear or equivalent to a linear equation. Solve the equation. 19. x32=53x+7 20E 21E 22E 23E 24E 25E 26E Linear Equations The given equation is either linear or equivalent to a linear equation. Solve the equation. 27. 3x+112=13x+3 28E Linear Equations The given equation is either linear or equivalent to a linear equation. Solve the equation. 29(t 4)2 = (t + 4)2 + 32 30E 31E 32E Solving for a Variable Solve the equation for the indicated variable. 33. P = 2l + 2w; for w 34E Solving for a Variable Solve the equation for the indicated variable. 35. ax+bcx+d=2; for x 36E 37E 38E Solving for a Variable Solve the equation for the indicated variable. 39. V=13r2h; for r 40E 41E Solving for a Variable Solve the equation for the indicated variable. 42. A=P(1+i100)2; for i Solving for a Variable Solve the equation for the indicated variable. 43. h=12gt2+v0t; for t Solving for a Variable Solve the equation for the indicated variable. 44. Sn(n+1)2; for n Solving by Factoring Find all real solutions of the equation by factoring. 45. x2 + x 12 = 0 46E 47E 48E Solving by Factoring Find all real solutions of the equation by factoring. 49. 4x2 4x 15 = 0 50E 51E 52E 53E 54E 55E 56E 57E 58E 59E 60E 61E 62E 63E 64E 65E 66E 67E 68E 69E 70E Quadratic Equations Find all real solutions of the quadratic equation. 71. 3x2 + 6x 5 = 0 Quadratic Equations Find all real solutions of the quadratic equation. 72. x2 6x + 1 = 0 73E 74E 75E 76E 77E 78E 79E 80E 81E 82E 83E Discriminant Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. 84. x2 + 2.21x + 1.21 = 0 85E 86E Other Equations Find all real solutions of the equation. 87. x2x+100=50 88E 89E 90E Other Equations Find all real solutions of the equation. 91. 10x12x3+4=0 92E 93E 94E 95E 96E 97E 98E Other Equations Find all real solutions of the equation. 99. 2x+x+1=8 100E 101E 102E Other Equations Find all real solutions of the equation. 103. x4 13x2 + 40 = 0 104E 105E 106E Other Equations Find all real solutions of the equation. 107. x4/3 5x2/3 + 6 = 0 108E 109E 110E 111E 112E Other Equations Find all real solutions of the equation. 113. |3x + 5| = 1 114E 115E 116E 117E 118E 119E 120E 121E 122E More on Solving Equations Solve the equation for the variable x. The constants a and b represent positive real numbers. 123. x4 5ax2 + 4a2 = 0 124E 125E 126E Falling-Body Problems Suppose an object is dropped from a height h0 above the ground. Then its height after t seconds is given by h = 16t2 + h0, where h is measured in feet. Use this information to solve the problem. 127. If a ball is dropped from 288 ft above the ground, how long does it take to reach ground level?Falling-Body Problems Suppose an object is dropped from a height h0 above the ground. Then its height after t seconds is given by h = 16t2 + h0, where h is measured in feet. Use this information to solve the problem. 128. A ball is dropped from the top of a building 96 ft tall. (a) How long will it take to fall half the distance to ground level? (b) How long will it take to fall to ground level?Falling-Body Problems Use the formula h = 16t2 + v0t discussed in Example 9. 129. A ball is thrown straight upward at an initial speed of v0 = 40 ft/s. (a) When does the ball reach a height of 24 ft? (b) When does it reach a height of 48 ft? (c) What is the greatest height reached by the ball? (d) When does the ball reach the highest point of its path? (e) When does the ball hit the ground?130E Shrinkage in Concrete Beams As concrete dries, it shrinksthe higher the water content, the greater the shrinkage. If a concrete beam has a water content of w kg/m3, then it will shrink by a factor S=0.032w2.510,000 where S is the fraction of the original beam length that disappears due to shrinkage. (a) A beam 12.025 m long is cast in concrete that contains 250 kg/m3 water. What is the shrinkage factor S? How long will the beam be when it has dried? (b) A beam is 10.014 m long when wet. We want it to shrink to 10.009 m, so the shrinkage factor should be S = 0.00050. What water content will provide this amount of shrinkage?The Lens Equation If F is the focal length of a convex lens and an object is placed at a distance x from the lens, then its image will be at a distance y from the lens, where F, x, and y are related by the lens equation 1F=1x+1y Suppose that a lens has a focal length of 4.8 cm and that the image of an object is 4 cm closer to the lens than the object itself. How far from the lens is the object?Fish Population The fish population in a certain lake rises and falls according to the formula F=1000(30+17tt2) Here F is the number of fish at time t, where t is measured in years since January 1, 2002, when the fish population was first estimated. (a) On what date will the fish population again be the same as it was on January 1, 2002? (b) By what date will all the fish in the lake have died?Fish Population A large pond is stocked with fish. The fish population P is modeled by the formula P=3t+10t+140, where t is the number of days since the fish were first introduced into the pond. How many days will it take for the fish population to reach 500?Profit A small-appliance manufacturer finds that the profit P (in dollars) generated by producing x microwave ovens per week is given by the formula P=110x(300x), provided that 0 x 200. How many ovens must be manufactured in a given week to generate a profit of 1250?Gravity If an imaginary line segment is drawn between the centers of the earth and the moon, then the net gravitational force F acting on an object situated on this line segment is F=Kx2+0.012K(239x)2 where K 0 is a constant and x is the distance of the object from the center of the earth, measured in thousands of miles. How far from the center of the earth is the dead spot where no net gravitational force acts upon the object? (Express your answer to the nearest thousand miles.)Depth of a Well One method for determining the depth of a well is to drop a stone into it and then measure the time it takes until the splash is heard. If d is the depth of the well (in feet) and t1, the time (in seconds) it takes for the stone to fall, then d=16t12, so t1=d/4. Now if t2 is the time it takes for the sound to travel back up, then d = 1090t2 because the speed of sound is 1090 ft/s. So t2 = d/1090. Thus the total time elapsed between dropping the stone and hearing the splash is t1+t2=d4+d1090 How deep is the well if this total time is 3 s?DISCUSS: A Family of Equations The equation 3x+k5=kxk+1 is really a family of equations, because for each value of k, we get a different equation with the unknown x. The letter k is called a parameter for this family. What value should we pick for k to make the given value of x a solution of the resulting equation? (a) x = 0 (b) x = 1 (c) x = 2 DISCUSS: Proof That 0 = 1? The following steps appear to give equivalent equations, which seem to prove that 1 = 0. Find the error. x=1Givenx2=xMultiplybyxx2x=0Subtractxx(x1)=0Factorx(x1)x1=0x1Dividebyx1x=0Simplify1=0Givenx=1 DISCOVER PROVE: Relationship Between Solutions and Coefficients The Quadratic Formula gives us the solutions of a quadratic equation from its coefficients. We can also obtain the coefficients from the solutions. (a) Find the solutions of the equation x2 9x + 20 = 0, and show that the product of the solutions is the constant term 20 and the sum of the solutions is 9, the negative of the coefficient of x. (b) Show that the same relationship between solutions and coefficients holds for the following equations: x22x8=0x2+4x+2=0 (c) Use the Quadratic Formula to prove that in general, if the equation x2 + bx + c = 0 has solutions r1, and r2, then c = r1r2 and b = (r1 + r2).141E The imaginary number i has the property that i2 = __________.2E 3E If 3 + 4i is a solution of a quadratic equation with real coefficients, then __________ is also a solution of the equation.5E 6E Real and Imaginary Parts Find the real and imaginary parts of the complex number. 7. 5 7i 8E Real and Imaginary Parts Find the real and imaginary parts of the complex number. 9. 25i3 10E

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