Math 180 Calculus and Analytic Geometry

If you are considering taking a lower-level class than Math 180:
- We encourage you to consider taking the corequisite course—Math 18—which provides additional support for Math 180.
- The next level lower is Math 160 (Precalculus Mathematics).
If you are considering taking a higher-level class than Math 180:
- If you have taken an AP or IB Calculus exam, please take your exam results to the Assessment Center.

Sample of what you should know before taking Math 180:

1. Find the unit vector in the direction of <-3,4>. Then use a dot product to find the angle between <1,2> and <-3,1>.
- - Solution
    
    Since $the magnitude of start vector negative 3 comma 4 end vector equals start square root 9 plus 16 end square root equals 5$ , $start fraction start vector negative 3 comma 4 end vector over the magnitude of start vector negative 3 comma 4 end vector end fraction equals start vector negative 3 fifths comma 4 fifths end vector$
    
    $theta equals the inverse cosine of open parenthesis start fraction start vector 1 comma 2 end vector dotted with start vector negative 3 comma 1 end vector over start square root 1 plus 4 end square root times start square root 9 plus 1 end square root end fraction close parenthesis$
    
    $equals the inverse cosine of open parenthesis negative start fraction 4 over the square root of 50 end fraction close parenthesis equals 124.5 degrees$
2. Let $f of x equals x to the fourth minus x cubed minus 2 x squared minus 4 x minus 24$ .

Find all possible rational zeros.
- - Solution
    
    Use the rational zero theorem. p, the divisors of the last term -24 are 1, 2, 4, 6, 8, 12, 24 and q, the divisors of the leading coefficient 1 is 1. Forming all possible quotients of $plus or minus p over q$ , possible rational roots are $plus or minus 1 comma plus or minus 2 comma plus or minus 4 comma plus or minus 6 comma plus or minus 8 comma plus or minus 12 comma plus or minus 24$
3. Use induction to show $n cubed minus n$ is divisible by 6.
- - Solution
    
    For n=1, $1 cubed minus 1$ and 0 is divisible by 6. Assume the statement is true for n=k; that is, $k cubed minus k$ is divisible by 6. Show the statement is true for n=k+1, that is, $open parenthesis k plus 1 close parenthesis cubed minus open parenthesis k plus 1 close parenthesis$ is divisible by 6.
    
    But,
    
    $open parenthesis k plus 1 close parenthesis cubed minus open parenthesis k plus 1 close parenthesis$
    
    $k cubed plus 3 k squared plus 3 k plus 1 minus k minus 1$
    
    $equals open parenthesis 3 k squared plus 3 k close parenthesis plus open parenthesis k cubed minus k close parenthesis$
    
    $k cubed minus k$ is divisible by 6. To show $3 k squared plus 3 k$ is divisible by 6, $3 k squared plus 3 k$ $3 k open parenthesis k plus 1 close parenthesis$ . $k times open parenthesis k plus 1$ is divisible by 2 since either k is even or k+1 is even. Thus $3 k times open parenthesis k plus 1$ is divisible by 6 since it is divisible by both 2 and 3. Therefore $open parenthesis k plus 1 close parenthesis cubed minus open parenthesis k plus 1 close parenthesis$ is divisible by 6.
4. Solve: $sine of 2 x equals one half comma 0 less than or equal to x less than 2 pi$
- - Solution
    
    Let $capital a equals 2 x$ . $sine capital A equals one half right arrow capital a equals pi over 6 comma 5 pi over 6$ .
    
    Thus $2 x equals start fraction pi over 6 end fraction plus 2 pi k$ $right arrow x equals start fraction pi over 12 end fraction plus pi k$ and $2 x equals start fraction 5 pi over 6 end fraction plus 2 pi k$ $right arrow x equals start fraction 5 pi over 12 end fraction plus pi k$ .
    
    Picking k=0 and k=1, we get $pi over 12 comma 13 pi over 12 comma 5 pi over 12 comma 17 pi over 12$
5. Identify the curve: $16 x squared minus 9 y squared plus 64 x minus 90 y equals 305$
- - Solution
    
    $16 x squared minus 9 y squared plus 64 x minus 90 y equals 305$
    
    $16 x squared plus 64 x minus 9 y squared minus 90 y equals 305$
    
    $16 open parenthesis x squared plus 4 x close parenthesis minus 9 open parenthesis y squared plus 10 y close parenthesis equals 305$
    
    Next add, then subtract $Open parenthesis start fraction b over 2 end fraction close parenthesis squared$ in the parentheses.
    
    $16 open parenthesis x squared plus 4 x plus 4 minus 4 close parenthesis minus 9 open parenthesis y squared plus 10 y plus 25 minus 25 close parenthesis equals 305$
    
    $16 open parenthesis x squared plus 4 x plus 4 close parenthesis minus 64 minus 9 open parenthesis y squared plus 10 y plus 25 close parenthesis minus 225 equals 305$
    
    $16 open parenthesis x plus 2 close parenthesis squared minus 9 open parenthesis y plus 5 close parenthesis squared equals 144$
    
    Now dividing each term by 16 to make 1 on the right side, we have
    
    $start fraction open parenthesis x plus 2 close parenthesis squared over 9 end fraction minus start fraction open parenthesis y plus 5 close parenthesis squared over 16 end fraction equals 1$
    
    This is a hyperbola with a = 3 and b = 4. Thus, its center is (-2,-5) traversing a line parallel to the x-axis. Thus add/subtract a and c to the x-values.
Sample of what you will learn in Math 180:

1. Determine whether $f of x equals start set first row x cubed cosine start fraction 1 over x end fraction if x is not equal to 0 second row 0 if x is equal to 0$ is differentiable at x = 0. Justify your answer carefully.
- - Solution
    
    $limit as h approaches 0 of start fraction f open parenthesis 0 plus h close parenthesis minus f of 0 over h end fraction$
    
    $equals limit as h approaches 0 of start fraction h cubed cosine start fraction 1 over h end fraction minus 0 over h end fraction$
    
    $equals limit as h approaches 0 of h squared cosine start fraction 1 over h end fraction$
    
    Since $negative h squared less than or equal to h squared cosine start fraction 1 over h end fraction less than or equal to h squared$ and , $limit as h approaches 0 of h squared equals 0$ , $limit as h approaches 0 of negative h squared equals 0$ , using the Squeeze Theorem, $limit as h approaches 0 of h squared cosine start fraction 1 over h end fraction equals 0$ . Thus f is differentiable at x = 0 and its derivative is 0.
2. Use the Mean Value Theorem to show: $start square root 1 plus x end square root less than or equal to 1 plus one half x$
- - Solution
    
    Let $f of x equal start square root 1 plus x end square root comma a equal 0 comma b equal x$
    
    Using the Mean Value Theorem, there is a c between 0 and x such that $f of b minus f of a equals f prime of c times open parenthesis b minus a close parenthesis$
    
    So, $start square root 1 plus x end square root minus 1 equals start fraction 1 over 2 times start square root 1 plus c end square root end fraction open parenthesis x minus 0 close parenthesis$
    
    But since c is between 0 and x, $start fraction 1 over 2 times start square root 1 plus c end square root end fraction less than or equal to one half$ .
    
    Thus $start square root 1 plus x end square root minus 1 equals start fraction 1 over 2 times start square root 1 plus c end square root end fraction open parenthesis x minus 0 close parenthesis less than or equal to one half x$ $right arrow start square root 1 plus x end square root less than or equal to 1 plus one half x$
3. Differentiate: $tangent open parenthesis x y close parenthesis plus x equals y squared$
- - Solution
    
    Use implicit differentiation to get $secant squared open parenthesis x y close parenthesis open parenthesis y plus x start fraction d y over d x end fraction close parenthesis plus 1 equals 2 y start fraction d y over d x end fraction$
    
    So, $start fraction d y over d x end fraction equals start fraction negative 1 minus y secant squared open parenthesis x y close parenthesis over x secant squared open parenthesis x y close parenthesis minus 2 y end fraction$
4. Compute: $the limit as x approaches infinity of open parenthesis 1 plus start fraction 3 over x end fraction close parenthesis superscript 2x baseline$
- - Solution
    
    Take ln, compute the limit, then take e:
    
    $the limit as x approaches infinity of the natural log of open parenthesis open parenthesis 1 plus 3 over x close parenthesis to the 2 x close parenthesis$
    
    $equals the limit as x approaches infinity of 2 x times the natural log of open parenthesis 1 plus 3 over x close parenthesis$
    
    $equals 2 times the limit as x approaches infinity of start fraction the natural log of open parenthesis 1 plus 3 over x close parenthesis over start fraction 1 over x end fraction end fraction$
    
    $equals 2 times the limit as x approaches infinity of start fraction start fraction 1 over 1 plus start fraction 3 over x end fraction end fraction times negative start fraction 3 over x squared end fraction over negative start fraction 1 over x squared end fraction end fraction$
    
    $equals 2 times the limit as x approaches infinity of start fraction 3 over 1 plus start fraction 3 over x end fraction end fraction equals 6$
    
    Taking e, $limit as x approaches infinity of open parenthesis 1 plus start fraction 3 over x end fraction close parenthesis superscript 2x baseline equals e to the 6$
5. Integrate: $integral of start fraction x over start square root 4 minus x to the 4 end square root end fraction d x$
- - Solution
    First use u-substitution before using the inverse sine formula.Since $x to the fourth equals open parenthesis x squared close parenthesis squared$ , let $u equals x squared right arrow d u equals 2 x d x$ $integral of start fraction x over start square root 4 minus x to the 4 end square root end fraction d x$ $equals integral of start fraction x over start square root 4 minus open parenthesis x squared close parenthesis squared end square root end fraction d x$ $equals one half integral of start fraction d u over start square root 4 minus u squared end square root end fraction$ $equals one half inverse sine of start fraction u over 2 end fraction plus capital c$ $equals one half inverse sine of start fraction x squared over 2 end fraction plus capital c$

Math 180 Calculus and Analytic Geometry

If you are considering taking a lower-level class than Math 180:

If you are considering taking a higher-level class than Math 180:

Sample of what you should know before taking Math 180:

Sample of what you will learn in Math 180: