Expert: Brian Davidson - 10/30/2009

Question
My Problems (set 2):   
[DN1] Find the slope of y=3x˛ at the point (-2, 12).
[DN2] How many vertical asymptotes exist for the function 1/(2sin˛x)-(sinx)-1?
[DN3] The function f is a continuous function. Consider the function (this is a piecewise function) f(x)=(x˛+bx) if x≤5 or (5sin(π/2*x)) if x>5. What does b equal?

Don’t say that these are the same problems b/c they’re not. I get very pissed off when any of you experts do not answer the questions b/c you “assume” these are the same problems I sent before.

Answer
Asad,

For problem #1, we recall the formula for the derivative of an exponential function.  The formula says that if we have a function of the form ax^b, the derivative (slope) function will be abx^(b-1).  Applying this formula, we find that the slope function (derivative) is 6x.  At the point (-2,12), the slope is therefore 6*(-2) = -12


For problem #2, we are asked to find vertical asymptotes.  If we didn't know calculus, this would be a tough question to answer...  We'd have to use guess and check.  But think about the 'definition' of a vertical asymptote....  At any vertical asymptote, doesn't the SLOPE (i.e. derivative) approach infinity (or negative infinity)?  If we accept this fact, then we can use derivatives to find the vertical asymptotes.
Specifically, lets   a) find the slope function (derivative)  and b) find where this NEW function approaches infinity or negative infinity to find any vertical asymptotes.

We take the derivative of the function given above.  Unfortunately, we get an ugly answer for the slope function.  It is:  (-(sin(x)^3+1)*cosx)/(sinx)^3    The good news is that all we care about is the denominator.  Remember that if a function's denominator approaches zero (and its numerator does NOT), the function will generally approach +/-infinity (this is a broad generalization, and math teachers wouldn't like me putting it so simply, but it works here).    If we look at the denominator, we see that it is (sinx)^3.  If the more simple sin(x) is to approach zero, we have only two possibilities (x approaches zero or it approaches pi).    Since we have (sinx)^3, we are essentially tripling the number of possibilities (the explanation of exactly why this works is very in-depth, and I won't bore you with it unless you want to know and follow up).  So technically, we have six places where our graph approaches infinity.  But for each assymptote, the graph approaches infinity from BOTH the POSITIVE and NEGATIVE direction, so the number of UNIQUE assymptotes is only half this (THREE).
That explanation had some 'smoke and mirrors' to avoid a much more in-depth analysis, but please feel free to follow up if you'd like more nitty-gritty details on WHY I could make the assumptions I did.

For problem #3, we recall the definition of continuity.  For a function to be continuous, its value at a particular "x" must be equal for both parts of the piecewise function AND its derivative must exist and be equal there.  To find "b", we must set the two piecewise functions equal to one another AND set their derivative (slope) functions equal to one another.  This will give us a system of equations (2 equations) for two variables (n and b), so we can solve it!  I'll leave it to you to try and solve this one numerically, using derivatives.  If you can't get an answer, please feel free to email me back, and I'll help you out with the specifics.  

Hope that helps!

-Brian