Notes 2.1 - Limits

Velocity and Other Rates of ChangeNotes 3.4 I. Instantaneous Rate of ChangeA.) Def: The instantaneous rate of change of f with respect to x at a is the derivative

Provided the limit exists.

B.) Example 1.) Write the area A of an equilateral triangle as a function of the side length s. 2.) Find the instantaneous rate of change of the area A with respect to s.

3.) Evaluate the rate of change of A at s = 6.

4.) If s is measured in inches and A is measured in square inches, what would be appropriate units for

Note - Take notice of the NOTATION!!!refers to the change in the area of the triangle with respect to the change in the side of the triangle.

II. Rectilinear MotionA.) Def: The motion of a particle back and forth (or up and down), along an axis s over a time t.

B.) The displacement of the object over the time interval from is

i.e., the change in position.

C.) The average velocity of the object over the same time interval is

D.) The instantaneous velocity of the object at any time t is

E.) The speed of the object at any time t is

F.) The acceleration of the object at any time t is

G.) A particle in rectilinear motion is speeding up if the signs of the velocity and the acceleration are the same. H.) A particle in rectilinear motion is slowing down if the signs of the velocity and the acceleration are the opposite.

III. Derivatives in EconomicsA.) Let C(x) represent the cost of producing x-tons of steel in a week.

The average rate of change in cost on =

The instantaneous rate of change of the cost of production of x is Economists call this marginal cost. This is the prediction of extra cost of producing 1 additional ton of steel per week.

SPSE production changes from x to x+1 tons, i.e. h = 1, then the average rate of change in is

We can use C (x) to ESTIMATE C

i.e., MARGINAL COST ESTIMATES the ACTUAL COSTB.) Example: SPSE the cost of producing x number of stoves in a day when 8 to10 are produced per day is given by the following equation:

About how much more would it cost to produce 1 more stove in a day?

Calculate C(10) to predict the additional cost of producing 11 stoves per day.

C.) Find the MARGINAL REVENUE if the company produces 11 stoves per day if the revenue equation is given by