Calculus Calculating the Derivative Chapter 4 Section 1 Techniques for Finding Derivatives

Transcription

1 Calculus Calculating the Derivative Chapter 4 Section 1 Techniques for Fining Derivatives Essential Question: How is the erivative etermine of a single term? Stuent Objectives: The stuent will etermine the erivative of a constant function. The stuent will unerstan the ifferent notations for a erivative. The stuent will etermine the erivative of a sum. The stuent will etermine the erivative of a ifference. The stuent will etermine the use erivatives to perform a marginal analysis on business an economic wor problems. Terms: Notations for the erivative Constant Rule Power Rule Constant Times a Function Sum or Difference Rule Marginal Revenue Marginal Profit Marginal Cost

2 Key Concepts: Notations for a Derivative The erivative of y = f x f '( x), y x, ( ) may be written in any of the following ways: x f ( x) ( ), or D x f x Constant Rule If f ( x) = k,where k is any real number, then f '( x) = 0 ( The erivative of a constant is 0. ) Power Rule If f x ( ) = x n for any real number n, then f '( x) = nx n 1 ( ) = x n is foun by multiplying by The erivative of f x the exponent n an ecreasing the exponent on x by 1. Constant Times a Function Let k be a real number. If g' x ( ) exists, then the erivative of f ( x) = k g( x) is ( ) = k g' ( x) f ' x The erivative of a constant timesa function is the constant times the erivative of the function. Sum or Difference Rule If f ( x) = u( x) ± v( x), an if u' x ( ) an v' ( x) exists, then ( ) = u' ( x) ± v' ( x) f ' x The erivative of a sum or ifference of functions is the sum or ifference of the erivatives.

3 Graphing Calculator Skills: Determine the erivative of a function at a point, x = a. ( ) = 3x 3 5x 2 10x + 3 Determine f '( 3) f x GRAPH APPROACH for the erivative at a point 1. Enter f ( x)into Y 1 on the calculator. 2. Graph the function using an appropriate winow. 3. Make sure the value of f 3 ( ) is in the graphing winow. 4. While in the graphing winow, press 2 n an then Calc. Select option 6 6 :y / x. The calculator will go back to the graphing winow. At this point, type in the numeric value that you wish to fin the erivative of: in this case type in 3 an then press Enter. 5. You shoul see y = at the bottom of the graphing winow. x 6. The actual erivative value is 41, but the metho that the calculator is using involves several numeric calculations to etermine the erivative. It is very common to see the answer slightly off from the actual answer ue to rouning errors. The GRAPH of the erivative 1. Press Y =, pick a function (Y 1, Y 2, or... Y 0 ) an then type in the erivative accoring to your operating system (see the next two pages). 2. Replace the number where you woul calculate the value of the erivative with the variable x. You shoul see either Y 1 = ( 3x 3 5x 2 10x + 3 ) x or Y 1 = nderiv( 3x 3 5x 2 10x + 3, x, x). 3. Graph the erivative function using an appropriate winow. x = x

4 HOME WINDOW APPROACH for the erivative at a point (NEWER OPERATING SYSTEM) 1. Press Math an select option 8 8 : nderiv. You shoul see ( ) = on the home winow. 2. Type the variable x into the 1 st box (the enominator of the fraction). You shoul see x ( ) = on the home winow..3. Enter the function between the parentheses OR enter Y 1. (If the function has been entere in the calculator in Y 1, press VARS for YVARS, select option 1 1: Function, then select the appropriate location for your function, ex: 1. You shoul see ( 3x 3 5x 2 10x + 3 ) x = on the home winow. 3. Next, press the variable x key an then enter the number where you wish to calculate the erivative of the function. You shoul see ( 3x 3 5x 2 10x + 3 ) x on the home winow. 4. You shoul see in the home winow. 5. The actual erivative value is 41, but the metho that the calculator is using involves several numeric calculations to etermine the erivative. It is very common to see the answer slightly off from the actual answer ue to rouning errors. x = 3

5 HOME WINDOW APPROACH for the erivative at a point (OLDER OPERATING SYSTEM) 1. Press Math an select option 8 8 : nderiv. You shoul see nderiv( on the home winow. 2. Enter the function ening with a comma OR enter Y, 1 (If the function has been entere in the calculator in Y 1, press VARS for YVARS, select option 1 1: Function, then select the appropriate location for your function, ex: 1. En this process by pressing,. You shoul now see nderiv(3x 3 5x 2 10x + 3,. 3. Next, press the variable x key an another comma: press x an then,. You shoul now see nderiv(3x 3 5x 2 10x + 3, x,. 4. Lastly, type in the number where you wish to take the erivative, then type in a closing paranthesis, an press the enter key: press 3, ), an then Enter. You shoul now see nderiv(3x 3 5x 2 10x + 3, x, 3). 5. You shoul see in the home winow. 6. The actual erivative value is 41, but the metho that the calculator is using involves several numeric calculations to etermine the erivative. It is very common to see the answer slightly off from the actual answer ue to rouning errors.

6 Sample Exercises: 1. Determine the erivative of the following function. f ( x) = 9x 5 4x 4 + 8x 3 5x 2 + 6x Determine the erivative of the following function. Rewrite the function into exponential form. Calculate the erivative of the function in exponential form. Finally, write the erivative in raical form. f x ( ) = 6 x x x + 1 6x 4 x x x 7

7 3. Determine the profit an marginal profit when 300 items are prouce. C x ( ) = 2.50x R x ( ) = 8x 500 x Homework: Pages Exercises: 7, 13, 17, 19, 21, 29, 31, 37, 41, 45, 53, 65, an 73 Exercises: 4, 12, 14, 20, 22, 30, 34, 38, 42, 44, 54, 62, an 72