Question

1. Demonstrate your understanding of limits (including right hand and left hand), continuity, and differentiability with labeled sketches and explanations.

2. Demonstrate that you understand the difference between average rate of change and instantaneous rate of change with a graph sketching in tangent line, secant line, and including necessary math

Answer 1

For #2, you could draw a line defined at two points on the curve nearing one of the two. As for the first, simply draw a function like \(f(x)=\frac{1}{x}\), which encompasses all of these, and explain it.

Answer 2

For #1, the only thing that I don't get it is differentiability. I tried to search for some explanation but I just don't get it.
For #2, I just have no idea how to do exactly what Wolf said

Answer 3

A function is differentiable at a point \(x\) if its slope is a non-vertical line at that point. Essentially, it must be continuous at each point which it is being differentiated over, otherwise, it has an undefined derivative at such point.

Answer 4

http://www.math.hmc.edu/calculus/tutorials/tangent_line/ <- Try to manipulate each point, and draw them nearer.

Answer 5

The following attached file is where these questions come from. It's an exam that the teacher give us to do. None of us have any idea how to them. It's like she tell us to write out our own questions and solve it.