Question

Question about recitation section 4 (Limits & cont.). When asked to choose values of a and b to make the function continuous, what would you do when x= 1 ( a condition met by the linear function whose a and b function you get to choose). How are you supposed to choose a and b values if the slope and y-int of a vertical line is undefined?

Answer 1

I'm a bit confused as to what you're asking. The recitation is about a piecewise function, but the only function you've given is x = 1. You can't make that into a piecewise function.

Can you please clarify?

Answer 2

Yes, I did not explain that well upon review. The question asks to make the piecewise continuous, which is accomplished only when the left-hand function (y=ax+b) passes through the point (1,2). However, this left-hand function account for values less-than and EQUAL to 1. So at x=1, the left-hand function would need to be vertical in order for it to pass through (1,2). So how would you choose values of a and b to make y=ax+b vertical at x=1? Hope this clarifies things, although I am not as literate in mathematical language as I would like so this may be difficult to explain. Thanks

Answer 3

When the left-hand function is at x = 1 the slope does not need to be vertical. What it needs is an equivalent slope as the function to the right. That is how to make it differentiable.

To find the slope for the function on the right we need to know it's derivative. The derivative of f(x) = x^2 + 1 is f'(x) = 2x. Then all you do is plug in the x which is 1 so you get the slope is 2. So the slope of the right-hand function at x=1 is 2.

So what you need to do is find a left-hand function that's slope is 2 at x=1.

Obviously the hangup here was the idea that the slope at x=1 had to be vertical. I'm not sure where you got that.