MA Lesson 13 Notes Section 4.1 (calculus part of textbook, page 196) Techniques for Finding Derivatives

Transcription

1 Notation for the Derivative: MA Lesson 13 Notes Section 4.1 (calculus part of tetbook, page 196) Techniques for Finding Derivatives The derivative of a function y f ( ) may be written in any of these ways (these notations). 1) ' prime' notation: f ( ) or y (read " f prime of " or " y prime") dy ) (read "dee y to dee ", derivative of y with respect to ) d 3) f ( ) (derivative of function f ( ) with respect to ) 4) D [ f ( )] (derivative of function f ( ) with respect to ) Note: In the notations above, the independent variable is. Other letters could be used for the independent variable and other names could be given to the function (other than f ). A) Constant Rule: If f ( ) k, where k represents any real number (a constant), then f( ) 0. The derivative of a constant is zero. This rule is reasonable. Derivative represents a rate of change. If something is constant, it has no change. Also, the graph of f ( ) k is a horizontal line. At any point on this line, the tangent to f() at that point would be the line itself and the slope of a horizontal line is 0. (Remember, the derivative is the slope of a tangent line to a graph at a specified point.) Eamples: 1 a) If g( ) 9, find g( ). g( )? b) If y, find y. y? c D D ) Find t[ ]. t[ ]? 1

2 B) Power Rule: n n If f ( ), where n is a real number, then f ( ) n (The derivative of a power is found by multiplying the eponent by to one less power.) 1. The proof of this rule is found in the tetbook on page 199. It is tedious, so I will not prove this rule during class time. Eamples: 10 a) g( ), g( ) 1 dy b) y, 5 Hint for (b): Rewrite the equation as y = 5. c 3/ ) D[ ] C) Derivative of a Constant time a Function: If k is any real number and if the derivative of g eists, then the derivative of f ( ) k g( ) is f ( ) k g( ). (The derivative of a constant times a function is the constant times the derivative of the function.) Eamples 3: a y 4 ) 1, dy 3 b g g 4 4 ) ( ), ( ) c) D [ 5 t] t

3 D) Sum or Difference Rule: If f ( ) u( ) v( ), then f ( ) u( ) v( ) (as long as the derivatives of u and v eist. (The derivative of a sum or difference of functions if the sum or difference of the derivatives.) Eamples 4: 3 dy a) y 5 5 9, b) p( n) 6n 3 n, p( n) n 4 3 c) y (Hint: Rewrite equation without a denominator.) y ) ( ) 3 (Hint: Rewrite by finding the product.) d f f( ) Marginal cost, marginal revenue, or marginal profit: In business and economics the rates of change of variables such as cost, revenue, and profit are called marginal cost, marginal revenue, or marginal profit. Since the derivative of a function gives the instantaneous rate of change of the function; a marginal cost (or revenue or profit) function is found by taking the derivative. Roughly, the marginal cost at represents the cost of the net ( + 1) item and approimates the value C( 1) C( ). Similar statements can be made for marginal profit or marginal revenue 3

4 Eample 5: If the total cost (in hundreds of dollars) to produce thousand barrels of a beverage is given by the cost function C( ) , find and interpret C (4). Compare with the value of C(5) C(4). Marginal cost evaluated at is a good approimation of the actual cost to produce the ( + 1)st unit. Marginal revenue evaluated at is a good approimation of the actual revenue from the sale of the ( + 1)st unit. Marginal profit evaluated at is a good approimation of the actual profit from the sale of the ( + 1)st unit. The demand function relates the number of units of an item that consumers are willing to purchase at the price p. The revenue function can be found if the demand function is known and is R( ) p (number of items times the price/item). 4

5 Eample 6: 5000 The demand function for a certain product is given by p dollars (where is number 500 of products made and sold). Write a revenue function of the number of items sold. Find the marginal revenue when 1000 units are sold and interpret. Eample 7: Suppose the revenue function from the sale of items is given by R( ) and the cost of items is given by C( ) for 0 10,000. Write a profit function for this situation. Find the marginal profit (or loss) for 1500 items and interpret. 5

6 Eample 8 (eample 9 of tetbook): The number of Americans (in thousands) who are epected to be over 100 years old can be 3 approimated by the function f ( t) t 0.470t t where t is the year, with t = 0 corresponding to 000 and 0 t 50. Find the derivative of f. Evaluate f (5) and interpret. 6