Math 260 - Week 2 Lab Name: f ( x +h )f (x ) to find a derivative each time is a long process and h h0 often cumbersome. But, there are a few algorithms that work perfectly for finding derivatives. They are the power rule, the product rule, the quotient rule, and the chain rule. As we have seen, using lim ------------------------------------------------------------------------------------------------------------------------------------------------------------- Category 1: Power Rule Directions: Look at the examples below then answer questions 1 & 2. a.) f ( x )=x 33 x+ 4 f ' ( x )=3 x2 3 d.) f ( x )= ' f ( x )= b.) f ( x )=5 x 4 c.) f ' ( x )=20 x3 f ( x )=3 x f ' ( x )= 3 2 x 6 4 x 24 x5 1.) Describe in your own words how to find a derivative 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 f(x) 3x 4 2 x 2 x2 Category 2: Product Rule Directions: Look at the examples below then answer questions 3. a.) f ( x )=( x 2 +4 x 2 ) (5 x7) f ' ( x )=5 ( x 2+ 4 x2 ) + ( 2 x+ 4 )( 5 x7 )=5 x2 +20 x10+10 x 214 x +20 x28= 2 15 x +26 x38 b.) SIMPLIFIED ANSWER f ( x )=2 x x f ' ( x )= ( 2 x ) 3.) 1 2x +2 ( x ) = RULE x + 2 x= x+ 2 x=3 x x answer SIMPLIFIED i.) Describe, in your own words, how to find a derivative using the product rule. ii.) Find the derivative for f(x) = (2x + 3)(5x - 3x + 1) iii.) Explain what steps were taken to change the answer to the simplified form for b.) 2 Category 3: Quotient Rule Directions: Look at the examples below then answer the questions 4. x f ( x )= x+3 f ' ( x )= b.) ( x +3 ) ( 1 )( x ) (1) 3 = 2 ( x +3 ) ( x +3 )2 RULE SIMPLIFIED ANSWER ' f ( x )= 3 ( x )= x 24 f x +5 ( x2 +5 ) ( 3 x 2 )( x 34 ) ( 2 x ) x 4 +15 x 2+ 8 x = 2 2 ( x2 +5 ) ( x 2 +5 ) RULE SIMPLIFIED ANSWER 4.) Describe, in your own words, how to find the derivative using the Quotient rule, then find f '(x) for 3x 1 f(x) 2 2x 3 5.) Research and explain the Sum Rule for derivatives. Category 4: Chain Rule Directions: Look at the examples below then answer the question 6 & 7. a.) 3 b.) f ( x )=( 2 x+1 ) ' 2 f ( x )=( 3 ) ( 2 x +1 ) (2)=6 ( 2 x +1 ) c.) 2 f ( x )=( 3 x + 2 x ) 2 ' f ( x )= ( 3 ) f ( x )= 3 x1 () 5 2 4 3 x +1 ( 3 x +2 x ) 4 ' 2 f ( x )=( 5 ) ( 3 x +2 x ) ( 6 x+2 )=10 Rule 1 1 ( 3 x1 ) 2 = 2 Simplified Answer 3 2 ( 3 x1 ) 1 2 = 3 2 3 x1 6.) 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 graph of 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. 8.) Check the box below each valid form of notation for a higher order derivative. d3 y dx 3 y''' ''(x) dy 3 d3 x y'''' f y(6) 9.) Examine the higher derivatives a, b, c below, then answer questions i and ii. a.) f ( x )=x 32 x 2+5 x7 f ' ' ' ( x )=6 b.) c.) f ( x )=3 x f ' ' ' ( x )= f ( x )= 5 x4 3 4x 3 2 f ( 4 ) ( x )= 4200 x8 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 10.) 2 f(x) 3 ex 2 2 2 3x , show all work. 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) = -26t 2 + 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). 2x 3 x 2 1 , x 2 f(x) f '(x) f '(3) f '(2) f ''(4) f ''(10) 4 3x x2 6 x 3 7 , , , x2 2x8 x8 13.) Find the derivative of f(r) with respect to r given S and n are constant. f(r) 4S2r 2 2nr 2 3nr 4Sn 14.) Find the rate of change of f(x) at x = 0: f(x) 2x(x 1) 3x 4 \f