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Math 160 Precalculus Mathematics

  • If you are considering taking a lower-level class than Math 160:
    • We encourage you to consider taking the corequisite course—Math 16—which provides additional support for Math 160.
    • The next level lower is Math 150 (Trigonometry).
  • If you are considering taking a higher-level class than Math 160:
 
  • Sample of what you should know before taking Math 160:
    1. Solve the rational equation: start fraction 2 x over x plus 5 end fraction plus start fraction 69 over 4 x squared plus 17 x minus 15 end fraction equals start fraction 12 over 4 x minus 3 end fraction
    2. Given that z equals 3 plus 2 i and w equals 2 minus 3 i, evaluate start fraction z over w end fraction plus z w
    3. Solve the logarithmic equation:  log base 2 of open parenthesis 3 x minus 1 close parenthesis minus log base 2 of open parenthesis 2 x plus 1 close parenthesis equals negative 2
    4. Sketch the graphs of f of x equals cosine x semicolon g of x equals sine x for 0 less than or equal to x less than or equal to 2 pi
    5. Solve the following trigonometric equations:  2 sine x plus 1 equals 0 square root of 3 tangent x minus 3 equals 0for 0 less than or equal to x less than or equal to 2 pi
  • Sample of what you will learn in Math 160:
    1. Solving equations of n-degree polynomial:  How many complex and real zeros does the function  f of x equals 2 x to the fourth minus 3 x cubed minus x squared plus 8 x minus 6  have?
      • Solution
        Possible rational solutions:plus or minus 1 comma plus or minus 1 half comma plus or minus 2 comma plus or minus 3 comma plus or minus 3 halves comma plus or minus 6
        Check x = 1.  Synthetic Division:
        Synthetic division table with root 1 in the upper left corner with a horizontal line under the 1 and a vertical line to the right of the 1 comma top row 2 negative 3 negative 1 8 negative 6 comma second row blank space 2 negative 1 negative 2 negative 6 comma a horizontal line to separate the second and third rows comma third row 2 negative 1 negative 2 6 0
        Check x = 3/2.  Synthetic Division:
        Synthetic division table with root negative 3 halves in the upper left corner with a horizontal line under the negative 3 halves and a vertical line to the right of the negative 3 halves comma top row 2 negative 1 negative 2 6 comma second row blank space negative 3 6 negative 6 comma a horizontal line to separate the second and third rows comma third row 2 negative 4 4 0
        So,
         2 x squared minus 4 x plus 4 equals 0 right arrow x squared minus 2 x plus 2 equals 0
        right arrow x equals start fraction 2 plus or minus start square root 4 minus 8 end square root over 2 end fraction equals start fraction 2 plus or minus 2 i over 2 end fraction equals 1 plus or minus i
        Zeros of f(x) are: x equals 1 semicolon negative 3 halves semicolon and 1 plus or minus i
    2. Solve the trigonometric equation: 6 cosine squared x minus 7 cosine x equals 5
    3. Graph the rational function:  f of x equals start fraction 2 x squared plus 5 x minus 3 over x squared minus 2 x minus 8 end fraction
      • Solution
        f of x equals start fraction 2 x squared plus 5 x minus 3 over x squared minus 2 x minus 8 end fraction equals start fraction open parenthesis 2 x minus 1 close parenthesis open parenthesis x plus 3 close parenthesis over open parenthesis x minus 4 close parenthesis open parenthesis x plus 2 close parenthesis end fraction
         
        Domain colon capital d equals start set x such that x is not equal to 4 comma negative 2 end set
         
        x-intercept: f of x equals 0 right arrow open parenthesis 2 x minus 1 close parenthesis open parenthesis x plus 3 close parenthesis equals 0 right arrow x equals 1 half comma negative 3
        y-intercept:
         f of 0 equals start fraction negative 3 over negative 8 end fraction equals 3 eighths
         
        Vertical asymptotes: 
        open parenthesis x minus 4 close parenthesis open parenthesis x plus 2 close parenthesis equals 0 right arrow x equals 4 comma negative 2
        Horizontal asymptote: y equals 2 over 1 equals 2
         
        graph with x scale from negative 10 to 10 and y scale from negative 10 to 10 comma vertical asymptotes at x equals negative 2 and x equals 4 and horizontal asymptote at y equals 2 comma from left to right the graph starts close to y equals 2 but below it and curves downward towards a y value of negative infinity as the x values get close to negative 2 comma on the right side of x equals negative 2 the graph starts close to a y value of positive infinity and curves downward crossing the y axis at 3 eighths and the x axis at 1 half and then continuing down towards a y value of negative infinity as the x values get close to 4 comma on the right side of x equals 4 the graph starts close to a y value of positive infinity and curves downward towards a y value of 2 as x approaches infinity
         
    4. An airplane is flying on bearing of south 10 degrees east at 500 mph.  A wind is blowing with bearing of  south 50 degrees west at 40 mph.  Find the actual ground speed and direction of the plane.
      • Solution
        airplane colon vector a equals 500 open parenthesis cosine of 280 degrees vector i plus sine of 280 degrees vector j close parenthesis
        equals 500 open parenthesis 0 point 1 7 4 vector i minus 0 point 9 8 5 vector j close parenthesis
        equals 86 point 8 3 vector i minus 492 point 4 vector j
         
        wind colon vector w equals 40 open parenthesis cosine of 220 degrees vector i plus sine of 220 degrees vector j close parenthesis
        equals 40 open parenthesis negative 0 point 7 6 6 vector i minus 0 point 6 4 3 vector j close parenthesis
        equals negative 30 point 6 4 vector i minus 25 point 7 1 vector j
         
        The result vector with respect to the ground:
        vector g equals vector a plus vector w
        equals open parenthesis 86 point 8 3 minus 30 point 6 4 close parenthesis vector i plus open parenthesis negative 492 point 4 minus 25 point 7 1 close parenthesis vector j
        equals 56 point 1 9 vector i minus 520 point 1 1 vector j
         
        ground speed:
        magnitude of vector g equals start square root 56 point 1 9 squared plus open parenthesis negative 520 point 1 1 close parenthesis squared end square root equals 523 point 1 4 miles per hour
         
        direction:
        theta equals inverse tangent of open parenthesis negative 520 point 11 over 56 point 1 9 close parenthesis equals negative 83 point 8 3 degrees
         
        True direction:
        right arrow true direction colon south open parenthesis 90 minus 83 point 8 3 close parenthesis east equals south 6 point 1 7 degrees east
    5. Determine whether the infinite geometric series converges. If it does, find its sum. 
    the sum from n equals 0 to infinity of 2 open parenthesis negative 1 third close parenthesis to the n
      • Solution
        the sum from n equals 0 to infinity of 2 open parenthesis negative 1 third close parenthesis to the n equals start fraction a over 1 minus r end fraction start set first row a equals 2 second row r equals negative 1 third semicolon absolute value of r equals 1 third less than 1 end set right
        The series is convergent, and its sum: start fraction 2 over 1 plus 1 third end fraction equals start fraction 2 over 4 thirds end fraction equals 3 halves