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Definition of a Derivative

Calc Proofs

1) Using the following two functions X and X^2 develop their derivatives using the Definition of a Derivative
for three values of "h"

h = .1
h = .01
h= .001

and then repeat the calculation

in the limit as h->0

2) Using the two functions above, show that the Finite Sum approximation of the area between the two functions approaches the value of the the actual value in the interval 0.5 to 2.5.

3) Using Proper Mathish explain the Definition of a Derivative.

4) Using Proper Mathish provide a Definition of Integral-- both Indefinite (anti derivative) and Definite Integral.

5) Using Proper Mathish, Describe a Finite Sum approximation and how it used

6) Describe what a Well-Behaved Function is? and is NOT?

7) Describe in Proper Mathish what a Limit is? and is NOT?

8) Describe what the Fundamental Theorem of Calculus means to you for well-behaved functions?

9) Finally, starting with a Finite Sum approximation develop the the relationship between the Lower and Upper Limits of a Definite Integral? Include the effect of changing the width of the rectangles starting with an initial approximation and then "in the limit" case where the width approaches zero.

Solution Preview

Please see the attachment.

It is in Q7. "I can always find some term" , "I" means "we". Limit means that no matter how small the given number is, there is always some term ...

Solution Summary

Definition of a derivative is applied in the solution.