## Monday, May 26, 2008

### Introduction to Anti-Derivatives (sounds like some sort of medicine)

So, on the previous class, we were introduced to anti-derivatives.. At the beginning of the class, we were asked to find the derivatives of the above functions. Needless to say, we found it difficult... NOT. It's 2x for all functions. And then we were given the derivative, f ' (x) = x and were asked to find the parent function. We were quite stumped at first, but then managed to figure it out. We had to go back to a previous lesson since we were asked whether if given a constant and a function and if we had to find the derivative of the function, does the constant play a role? No, it doesn't. Anyway, we were asked to find the parent function of the derivative f ' (x) = x. If x is the derivative, then obviously the parent function is x2. But the derivative of x2 is 2x and the derivative we want is x. So we figured, we have to multiply 2x by 1/2 to get x. From that, we got the parent function, which is 1/2x2. Mr. K then asked us, what about if I give you the derivative x2? What would be its parent function? We then thought about it. If x2 is the derivative, obviously the parent function is x3. But the derivative of that function will give us 3x2, which we don't want since we're given the derivative x2. So we thought multiplying 3x2 by 1/3 would give us x2. We reckon that the parent function is 1/3x3. From that we came up with a rule for all power functions, which is:

f ' (x) = xn -------> f (x) = xn+1/n+1 + C, where C is the constant. I'll explain later. We then made anti-derivative rules for almost all derivative rules we remember. Mr. K said that the anti-derivative rules for the product and quotient rule are much more complicated and we won't be taking it until First Year University Calculus, so yeah. Now this is where I explain the C part in the rule:

f ' (x) = xn -------> f (x) = xn+1/n+1 + C, where C is the constant

We can't really find C not unless we're given a point in the graph. C is basically the y-intercept, but if we're just given the derivative and asked to solve for the parent function, all we can do is find the parent function and then + C, not unless we're given a point in the graph.

Then we were just given a practice problem, which was pretty straightforward.  The next scribe is etimz, or Ethan.