More from Lesson 6 The Limit Definition of the Derivative and Rules for Finding Derivatives.

Transcription

1 Math 1314 ONLINE More from Lesson 6 The Limit Definition of the Derivative an Rules for Fining Derivatives Eample 4: Use the Four-Step Process for fining the erivative of the function Then fin f (1) f( ) 4 =

2 We can use the average rate of change to approimate the slope of the tangent line The f( + h) f( ) average rate of change is given by the ifference quotient, We can also write h f( b) f( a) this formulas as where a an b are the enpoints of a close interval [a, b] b a Eample 5: Suppose f ( ) = Fin the average rate of change of f over the interval [, 5] Eample 6: Suppose the istance covere by a car can be measure by the function st ( ) = 4t + 3t, where st () is given in feet an t is measure in secons A Fin the average velocity of the car over the interval [0, 4] B Fin the instantaneous velocity of the car when t = 4

3 Rules for Fining Derivatives We can use the limit efinition of the erivative to fin the erivative of every function, but it isn t always convenient Fortunately, there are some rules for fining erivatives which will make this easier First, a bit of notation: [ f ()] is a notation that means the erivative of f with respect to, evaluate at Rule 1: The Derivative of a Constant [ c] = 0, where c is a constant Eample 1: If f ( ) = 1, fin f '( ) Rule : The Power Rule n n 1 [ ] = n for any real number n 4 Eample : If f ( ) =, fin f '( ) 3 Eample 3: If f ( ) =, fin f '( )

4 1 Eample 4: If f ( ) =, fin f '( ) 5 [ cf ( ) ] c [ f ( ) ] Rule 3: Derivative of a Constant Multiple of a Function = where c is any real number 3 Eample 5: If f ( ) = 6, fin f '( ) Eample 6: If f ( ) = 3, fin f '( ) [ f ( ) ± g( ) ] = [ f ( ) ] ± [ g( ) ] Rule 4: The Sum/Difference Rule

5 4 3 6 Eample 7: Fin the erivative: f ( ) = Eample 8: Fin the erivative: f 3 ( ) = Note, there are many other rules for fining erivatives by han We will not be using those in this course Instea, we will use GeoGebra for fining more complicate erivatives Lesson 7: Derivatives at a Point an Numerical Derivatives Many of our problems will ask for the rate at which something is changing at a specific number The number may epress time, or quantity prouce an sol, or many other quantities To fin this rate, fin the erivative an substitute the esire number into the erivative an evaluate State your answer using the correct units Eample 1: Fin the erivative when = 3: f 3 ( ) = Eample : The height of a rocket can be moele by the function ht ( ) = 16t + 48t+ 6 where ht () gives the height in feet at time t given in secons At what rate is the height changing when t = 1?

6 We can fin numerical erivatives using GeoGebra What is the comman? Eample 3: Fin the erivative when = 3: f = + 3 ( ) Eample 4: Fin the numerical erivative of (, 40) f( ) = at the point Eample 5: Fin the numerical erivative of 3 f( ) = when = 4

7 Eample 6: Fin the numerical erivative of f( ) = at = + 1 Eample 7: Fin the numerical erivative of f( ) = at = 4 Eample 8: Fin the numerical erivative of = C ( ) = when 3e Eample 9: Fin the numerical erivative of C ( ) = 4 8 when = 5 Some Applications of the Derivative

8 Now we ll start looking at the kins of problems we can work using erivatives Eample 10: Fin the slope of the tangent line when = -1 if f( ) = + ( + 3) Eample 11: Write an equation of the line that is tangent to 1 = + when = f( ) 7 Eample 1: Write an equation of the line that is tangent to when = 375 f = + 3 ( ) With this group of wor problems, the first step will be to etermine what kin of problem we have for each problem Does it ask for a function value (FV), a rate of change (ROC) or an average rate of change (AROC) From there, we ll apply the appropriate methos

9 Eample 13: A country s gross omestic prouct (in millions of ollars) is moele by the 3 function Gt ( ) = t + 45t + 0t where 0 t 11 an t = 0 correspons to the beginning of 1997 (A) At what rate was GDP changing at the beginning of 00? At the beginning of 004? At the beginning of 009? (B) What was the average rate of growth of the GDP over the perio ?

10 Eample 14: Data from a homebuilers association shows that the average size of a single family home built after 1970 can be moele by the function St ( ) = 01666t + 998t where t is the number of years since 1970 Was the average size of a new single family home growing faster at the en of 1989 or at the en of 1999? Eample 15: Accoring to the US Department of Energy, a typical car s fuel economy epens on the spee it is riven Fuel economy is approimate by the function 4 3 f( ) = where is measure in miles per hour an f( ) is measure in miles per gallon Fin the rate of change of f when = 0 an when = 50 an interpret the results

11 Eample 16: A secretarial school knows that the average stuent taking an avance typing class 60t will progress accoring to the rule Nt () =, t 0, where Nt () measures the number of t + 6 wors per minute that a stuent can type t weeks into the program (A) At what rate will the average stuent s spee in wors per minute be changing at the en of the first, thir, fourth an seventh weeks of a twelve week course? (B) What will be the average stuent s typing spee at the beginning of a 1 week course? (C) What will be the average stuent s typing spee at the en of a 1 week course? (D) What will be the average stuent s average rate of change uring a 1 week course? 015 Eample 17: The moel Nt ( ) = 344( t) gives the number of people in the US who are between the ages of 45 an 55 Note, Nt () is given in millions an t correspons to the beginning of 1995 (A) How large is this segment of the population projecte to be at the beginning of 011? (B) How fast will this segment of the population be growing at the beginning of 011?

12 Solving problems that involve velocity is a common application of the erivative Velocity can be epresse using one of these two formulas, epening on whether units are given in feet or meters: ht = t + vt+ h ( ) 16 o 0 (feet) ht = t + vt+ h ( ) 49 o 0 (meters) In each formula, vo is initial velocity an ho is initial height Eample 18: Suppose you are staning on the top of a builing that is 8 meters high You throw a ball up into the air, with initial velocity of 10 meters per secon Write the equation that gives the height of the ball at time t Then use the equation to fin the velocity of the ball when t = Eample 19: Suppose you are staning on the top of a builing that is 11 feet high You throw a ball up into the air, with initial velocity of 36 feet per secon Write the equation that gives the height of the ball at time t Then use the equation to fin the velocity of the ball when t = 3

13 We want to be able to fin all values of for which the tangent line to the graph of f is horizontal Since the slope of any horizontal line is 0, we ll want to fin the erivative, set it equal to zero an solve the resulting equation for Eample 0: Fin all values of for which f '( ) = 0 : 1 5 f 3 3 ( ) = Eample 1: Fin all points on the graph of ( ) horizontal f = 5 3+ where the tangent line is We can also etermine values of for which the erivative is equal to a specifie number Set the erivative equal to the given number an solve for either algebraically or by graphing Eample : Fin all values of for which f '( ) = 3 : f( ) =

14 Higher Orer Derivatives Sometimes we nee to fin the erivative of the erivative Since the erivative is a function, this is something we can reaily o The erivative of the erivative is calle the secon erivative, an is enote f ''( ) To fin the secon erivative, we will apply whatever rule is appropriate given the first erivative 5 4 Eample 3: Fin the secon erivative when = -3: f ( ) = Eample 4: Fin the value of the secon erivative when = : f = ( ) Eample 5: Fin the value of the secon erivative when = 1 if f( ) = Eample 6: Fin the value of the secon erivative when = 5 if f( ) = ln 1 3 ( + 3)