 Click to see sample problems for Math:  50  51  71  71A  100  110  120  130  140  150  160  180

# Math 180 Calculus and Analytic Geometry

• ###### 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    2. Let 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 , possible rational roots are 3.  Use induction to show is divisible by 6.
• Solution
For n=1, and 0 is divisible by 6.  Assume the statement is true for n=k; that is, is divisible by 6.  Show the statement is true for n=k+1, that is, is divisible by 6.
But,    is divisible by 6.  To show is divisible by 6,   is divisible by 2 since either k is even or k+1 is even.  Thus is divisible by 6 since it is divisible by both 2 and 3.  Therefore is divisible by 6.
4. Solve: • Solution
Let . .
Thus  and  .
Picking k=0 and k=1, we get 5. Identify the curve: • Solution   Next add, then subtract in the parentheses.   Now dividing each term by 16 to make 1 on the right side, we have This is a hyperbola with = 3 and = 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 is differentiable at x = 0.  Justify your answer carefully.
• Solution   Since and , , , using the Squeeze Theorem, .  Thus f is differentiable at x = 0 and its derivative is 0.
2. Use the Mean Value Theorem to show: • Solution
Let Using the Mean Value Theorem, there is a c between 0 and x such that So, But since c is between 0 and x .
Thus  3. Differentiate: • Solution
Use implicit differentiation to get So, 4. Compute: • Solution
Take ln, compute the limit, then take e:     Taking e 5. Integrate: • Solution
First use u-substitution before using the inverse sine formula.Since , let      