Question

After the second recitation video of the first session, am I understanding the graph of the derivative of a function is the graph of the behaviour of that function's slope? Or did I completely screw that up?

Also, how is any derivative line equation used? It seems like it could be used to track trends in any data plotted on a graph... but then again I'm pretty new at this...

Answer 1

I think you are right it's a behaviour of the function slope but any derivative line equation can't be used just when the slope = 0 is possible determine that behavior

Answer 2

Explain the "derivative equation can't be used just when the slope = 0". I'm not sure what you mean. Do you mean as the it approaches 0 as a limit?

Answer 3

*** am I understanding the graph of the derivative of a function is the graph of the behaviour of that function's slope?***

That sounds about right, but I would say the graph of the derivative of a function is the graph of the function's slope.... where "slope" is the slope of the line that is tangent to the function's curve at each point.

**how is any derivative line equation used?***

I assume you mean, how can we use the information about the tangent line (either the slope of the tangent line, or the equation of the tangent line)?

The course will give applications. Off-hand, there are at least two uses:
1) linear approximation... which means use the tangent line as an estimate of what the curve is doing... which works reasonably well for "short distances"
2) minimization (or maximizing) some function. Notice that if you plot y= x^2 it will have a well-defined minimum... and the slope of the tangent line is 0 (a horizontal line) at this minimum. You can solve a lot of problems using that idea.