Question

The difference Quotient- A first look at the derivative

Answer 1

not sure how to answer the first question
what doe the expression [f(x+h)-f(x)]/[(x+h)-x] represent? What does this expression simplify to?

Answer 2

average rate of change

Answer 3

so it represent average rate of change?

Answer 4

average rate of change between "x" and "x+h"

Answer 5

yep it is also called the difference quotient

Answer 6

what does it simplify to? or does it

Answer 7

denominator - we can cancel x's

Answer 8

that is the only thing that simplifies

Answer 9

Recall the limit definition of derivative :
\[\large f'(x) = \lim \limits_{h\to 0} \dfrac{f(x+h)-f(x)}{h}\]

Answer 10

doesn't the secant line turn into the tangent line as h approaches zero

Answer 11

I guess I am trying to answer the second part of the question

Answer 12

yes, `average rate of change` becomes `instantaneous rate of change` when u take the limit \(h \to 0\)

Answer 13

average rate of change :: slope of secant line
instantaneous rage of change :: slope of tangent line

Answer 14

probably best to show an animation of this, a drawing does not do it justice since it is so stationary..... Thanks

Answer 15

http://math.bu.edu/people/tkohl/teaching/spring2013/secant.html

Answer 16

thanks :)

Answer 17

nice :)