Expert: Sherry Wallin - 9/21/2009

Question
Here are 4 more questions (you just answered one on graphing an equation for me!) I need "step-by-step" instructions again, if you don't mind. And if you cannot do 4 questions, I completely understand, and any help at all will be greatly appreciated. I'm trying very hard to understand this class!  Thank you so much!

Question #1:
Given: Y = x^2-2x+2 at f(x,y), let Q (x+Δx, y+Δy) be another point on the curve. Differentiate by definition.

Question #2:
Given:  E(a, be, c, d) = ac^2 + cb^2 + cb2 + 5d + 5a.  Partially differentiate to find:
       ∂E  and  ∂E
       ∂b       ∂c   (the top ones are "over" the bottom ones)

Question #3:
Integrate:   a) ∫ 3t^2 + 4t + 8  dt
            b)  2∫^5  t^3 + 9   dt (the 2 in front of the ∫ is at the bottom of the ∫, I didn't know how to express it in here)

Question #4:
Find the maximum or minimum of   x^3 + 3x^2 + 3x

Again, I thank you so much, and I understand if you cannot get to so many questions all at once. I truly appreciate your time.

- Lorri E.


Answer
Question #2:
Given:  E(a, be, c, d) = ac^2 + cb^2 + cb2 + 5d + 5a.  Partially differentiate to find:
      ∂E  and  ∂E
      ∂b       ∂c   (the top ones are "over" the bottom ones)
∂E this means to take the partial derivative of the function E with respect to the variable b.
∂b

If you haven't had calculus I don't see how you can do these problems. Do you  know anything about derivatives???

Anyway, since you are taking the derivative with respect to b this means you are 'holding' the other variables as constants so the partial of ac^2 = 0 since the derivative of a constant is 0. The derivative of b^2 is 2b so the derivative of cb^2 is c(2b) = 2bc. The derivative of b is 1 so the derivative of cb2 is 2c. Like the first term the derivative of 5d = 0 and 5a = 0 since these are being considered constants and the derivative of a constant is 0.

gathering all the derivatives since the derivative of a sum is the sum of the derivatives:

∂E = 0  + 2bc +  2c
∂b

Simarly for
∂E = 2ac + b^2 + 2b
∂c

Question #3:
Integrate:   a) ∫ 3t^2 + 4t + 8  dt
           b)  2∫^5  t^3 + 9   dt (the 2 in front of the ∫ is at the bottom of the ∫, I didn't know how to express it in here)

I think there is still a problem with part b because you have a carot in front of the 5. If you mean that the 2 is small there needs to be another number at the top that is small and this would indicate what range of numbers you are integrating over. I can't do the problem as it is written, sorry.

part a is done like so: Integration is the opposite of derivatives. You integrate term by term since the integral of a sum is the sum of the integrals. ∫ 3t^2 + 4t + 8  dt = ∫ 3t^2 dt + ∫ 4t dt + ∫ 8 dt
= t^3 + 2t^2 + 8t

When you integrate a polynomial you raise the power by 1 and divide by the new power so 3t^2 has a power of two so make it a power of three and then divide the 3 by 3 to get 1*t^3. Likewise 4t, t has a power of 1 so make it a 2 but divide 4t^2 by 2 to get 2t^2 and 8 has a 'hidden' variable power of 0 so make that power 1 and divide by 1 so ∫ 8 dt = 8t/1.

There is a theorem in calculus that says when you take the derivative you can find the minimum or the maximum by setting the derivative = 0 and finding the 'critical values'. So the derivative of
x^3 + 3x^2 + 3x is 3x^2 + 6x + 3 and  3x^2 + 6x + 3 = 0 -> 3(x^2 + 2x +1) = 0 which factors as
(x+1)^2 so the only 'critical' point is x = -1. So you examine what happens to the original function before and after x = -1 on a number line. f(-2)=(-2)^3 +3(-2)^2+3(-2) = -8 + 12 -6 = -2
and at a number to the right of -1, use 0, f(0) = 0^3 +3*0 +3 = 3. This tells you that at x = -1 the graph has negative slope and at x = -1 the slope changes to positive slope which means that you have a local minimum at x = -1. To find the y coordinate of the point calculate f(-1)= -1 so the minimum is at (-1,-1)

Math Prof

I am busy, I teach at a college and I am a grad student so I have limited time but I will answer what I can when I can.