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About Brian Davidson
Expertise
I can answer questions ranging from Pre-Algebra through AP Calculus BC (first year college calculus), as well as some questions in Discreet Mathematics (probability, matrix theory, graph theory, and combinatorics).

Experience
From my earliest years as a math student, I have dedicated myself to learning the concepts and discipline of mathematics. In addition to a rigorous mathematics course sequence throughout junior high and then high school, I have served for four years on my school's math team and won several first place awards in both state and national competitions. I earned a perfect score on the AP Calculus BC exam, and (in addition) have done in-depth course work in discrete mathematics and differential calculus. Of course, I more commonly find myself offering aid in less advanced forms of math, which is why "Math for Kids" is perfect for me, also given my education credentials and previous experience.

Education/Credentials
Although I am just a college student, my educational experiences are broad. I have served as a certified student tutor for mathematics and physics both inside and outside of high school (2005 to 2009), served as a student leader by helping teach an integrated science/math class at my high school, where I was able to develop my teaching style through lessons and group interaction. Although I have much to learn before beginning to teach professionally (as I plan to do), I am proud to have helped over 80 students achieve success in mathematics over a four year period, all while expanding my own knowledge and appreciation for the discipline. And although I have studied many advanced forms of mathematics, some of my most successful students came to me for help with pre-algebra, algebra, or elementary geometry.

Awards and Honors
Outstanding Student Teacher Award (2009), Senior Student Tutor/Seminar Lecturer (2008), 3 First-place mathematical olympiad finishes, 4-second place.

Past/Present Clients
I have tutored between a 80 and 90 fellow students (both same age and younger) during high school before my graduation, although for privacy reasons, I'd rather not give specific names in such an online context. If more information is needed, please feel free to contact me through private email at bldavidson1990@hotmail.com

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Math for Kids - ap calculus ab

Expert: Brian Davidson - 10/30/2009

Question
My Problems (set 2):
[DN1] Find the slope of y=3x² at the point (-2, 12).
[DN2] How many vertical asymptotes exist for the function 1/(2sin²x)-(sinx)-1?
[DN3] The function f is a continuous function. Consider the function (this is a piecewise function) f(x)=(x²+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

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