What is a derivative? What information can it provide us? What is the limit defi

Question

What is a derivative? What information can it provide us?

What is the limit definition of the derivative? From what basic formula relating to lines is it developed? How is that related to part a?

Use the limit definition of the derivative to find the derivative of the following function.

f(x)=?2x?^3

d) Given that the generic form of an objects path through time in relationship to earth is

s(t)=-16t^2+V_0 t+s_0 Do the following:

Find the model for the Velocity of the object through time where the initial velocity is 50ft/sec and the initial height is 600ft.

Find the average velocity of the object between t = 1 and t = 4 seconds

Find the instantaneous velocity of the object at t = 2 seconds

e) Remembering, that the derivative is equal to the slope of the tangent line to the curve at any point on the curve, do the following:

Plot f(x)=x^2 and f^' (x)=2x in your graphing calculator

How does f

Explanation / Answer

1)

derivative is rate of change in variable

it provides change in the variable with respect to some factor(time,...etc)

2)let f(x) is some function

f '(x)=lim h->0 [f(x+h)-f(x)]/(x+h-x)

f '(x)=lim h->0 [f(x+h)-f(x)] /h

3)

f(x)=2x^3

f '(x)=lim h->0 [f(x+h)-f(x)]/(x+h-x)

f '(x)=lim h->0 [2(x+h)^3-2x^3]/(x+h-x)

=lim h->0 [2(x^3+h^3+3xh(x+h)-2x^3]/h

=lim h->0 [2x^3+2h^3+6xh(x+h-2x^3]/h

=lim h->0 [2h^3+6xh(x+h]/h

=lim h->0 [2h^2+6x(x+h]

=6x^2

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