Algebra vs. Calculus

1. What is the DIFFERENCE between finding the limit of a function at x=c and actually plugging in the number x=c? When are the two cases the SAME?

Difference: when function is discontinuous

(eg.removable discontinuity at x=c)

Finding the limit of f(x) at x=c is finding the y value as the graph approaches c. Where the function may have a hole at x=c, the limit still exists. But when plugging in the number x=c, the output will ibe the exact y value of the function.

for example.

here, the limit f(x) as x approaches the point of discontinuity (lets call this c) does exist. the limit will be the y value of the open dot. but when plugged in x=c into the equation... the output will become the y value of the closed(black) dot.

Same: they are only the same when the graph of the function is continuous.

2.What are the SIMILARITIES between finding the derivative and finding the slope of a line? What are the DIFFERENCES between the two?

Similarities: find the slope and the derivative both are the change of y/ change of x. The process of finding both the slope and derivative require that you find the secant line first.

Differences: In the finding the derivative, we find the tangent line of the point by finding the secant line (picking two points close by) then bring them closer to x=c (or wherever you are finding the derivative for) this is done with limits.

so basically, finding the derivative is like finding the tangent line at that one exact point; whereas finding the slope, you pick two points .(well thats how i would put it)

Excited about monday. Its going to be funny!!