MATH 260 Week 2 LAB A++++++++++

Math 260 - Week 2 Lab As we have seen, using lim┬(h→0)⁡〖(f(x+h)-f(x))/h〗   to find a derivative each time is a long process and often cumbersome.  But, there are a fewalgorithms that work perfectly for finding derivatives.  They are the power rule, the product rule, the quotient rule, and the chain rule.Category 1: Power Rule

Directions:  Look at the examples below then answer questions 1&2.  
	f(x)=x^3-3x+4            b.)   f(x)=5x^4                    c.)f(x)=3√x

f^' (x)=3x^2-3f^' (x)=20x^3 f^' (x)=3/(2√x)

 f(x)=6/x^4 

f^' (x)=-24/x^5 
1.)	Describe in your own words how to find aderivative using the Power Rule.  What must be done to c.) and d.) above before using the power rule to differentiate ?

2.)  Find the derivative of  

Category 2: Product Rule

Directions:  Look at the examples below then answer questions 3.  
	f(x)=(x^2+4x-2)(5x-7)

f^' (x)=5(x^2+4x-2)+(2x+4)(5x-7)=5x^2+20x-10+10x^2-14x+20x-28=

= 15x^2+26x-38  ←   SIMPLIFIED ANSWER

	f(x)=2x√x

f^' (x)=(2x)  1/(2√x)+2(√x)=x/√x+2√x=√x+2√x=3√x

3.)i.)  Describe, in your own words, how to find a derivative using the product rule.
ii.)Find the derivative for f(x) = (2x + 3)(5x2 - 3x + 1)
iii.) Explain what steps were taken to change the answer to the simplified form for b.)

Category 3: Quotient Rule

Directions:  Look at the examples below then answer the questions 4.  
f(x)=x/(x+3)b.)  f(x)=(x^3-4)/(x^2+5)

f^' (x)=((x+3)(1)-(x)(1))/(x+3)^2 =3/(x+3)^2  f^' (x)=((x^2+5)(3x^2 )-(x^3-4)(2x))/(x^2+5)^2 =(x^4+15x^2+ 8x)/(x^2+5)^2 

RULE               SIMPLIFIED ANSWERRULE                            SIMPLIFIED ANSWER

4.)  Describe, in your own words, how to find the derivative using the Quotient rule, then find f ’(x) for 
 
5.) Research and explain the Sum Rule for derivatives

Category 4: Chain Rule 

Directions:  Look at the examples below then answer the question6 &7
	f(x)=(2x+1)^3			b.)  f(x)=√(3x-1)

f^' (x)=(3) (2x+1)^2 (2)=6(2x+1)^2 f^' (x)=(3)(1/2) (3x-1)^(-1/2)=3/(2(3x-1)^(1/2) )=3/(2√(3x-1))


c.) f(x)=(3x^2+2x)^5

f^' (x)=(5) (3x^2+2x)^4 (6x+2)=10(3x+1〖)(3x^2+2x)〗^4
Rule            Simplified Answer

Describe, in your own words, how to find the derivative using the Chain Rule, then find the 
      derivative for  
7.)  For example b.) above, explain why the exponent is  -1/2  in the first step ?  Explain how the 
      answer to c.) has a 10 in front.
.  
Part 2: Higher Order Derivatives

Higher order derivatives are derivatives of derivatives.  The significance of a derivative depends on what x and f(x) represent.  

For example, for the graphof f(x), and a point x on the graph f’(x) is the slope of the curve, and f’’(x) is the rate at which the slope is changing from point to point, called the curvature.

If s(t) is a displacement or position function, then s’(t) is the velocity, s’’(t) is the acceleration, s’’’(t), s(4)(t), s(5)(t), s(6)(t), are the jerk, jounce/snap, crackle and pop.
Examine the higher derivatives a, b, c below, then answer questions i and ii.


	f(x)=x^3-2x^2+5x-7	b.)   f(x)=3√x		c.)  f(x)=5/x^4 

f'''(x)=6			f^''' (x)=-3/(4x^â–¡(3/2) )	f^((4) ) (x)=4200/x^8 

i. For example a.) what value does every higher derivative have starting with the fourth derivative ?  

ii.  Does the same value occur for example b.) or c.)  ?  Explain why or why not ?


9.)  Find the 3rd derivative for , show all work.


10.)	As a dare devil is shot from a cannon.  His wallet drops from his pocket when he is 70 feet from the ground.  The position function of the wallet is given by s(t) = -26t2 + 73t + 70,  t in seconds.Use your calculator, round all answers to the 100ths

a.)	How long will it take the wallet to hit the ground ? 
b.)  What is the instantaneous velocity of the wallet when t = 2.2 seconds ?


  c.) What is the velocity of the wallet as it hits the ground ?


   d.)  What is the velocity of the wallet after it lands in the mud on the ground ?



11.)  To find the derivative of a piecewise function, you must differentiate each piece.  If 
asked to find the derivative at a particular value of x, the correct derivative must be 
chosen according to the domain of each piece of f(x).

         Find f ’(x) and f ’(-3), f’(2), and f ’(3), f ’(10). 




12.)  The energy output E of an electric device is a function of time given by the formula 
         E = t(1 + 3t)2.Power, P is the first derivative of E.  Find the Power (in Watts) generated by the
device at 3 seconds.
13.)  Find the derivative of f(r) with respect to r given S and n are constant. 

14.)  Find the rate of change of f(x) at x = 0: