Explanation Given information :. f ( x ) = x x + 1 Formula Used: Product rule: [ f ⋅ g ] ' = f g ' + g f ' Chain rule: [ f ( g ( x ) ) ] ' = f ' ( g ( x ) ) ⋅ g ' ( x ) Concept Involved: Step 1: The roots of the function are the values of x that makes the function equal to zero. Step 2: Write our intervals where the function will be positive or negative using the roots of the function. Step 3: Pick a test point from each interval, substitute in the function If we get f ( x ) > 0 , then function is positive in that interval & If we get f ( x ) < 0 , the function is negative in that interval. Step 4: Find the first derivative of the function. Step 5: Critical numbers are values of x that makes the first derivative a zero or undefined. Set up equation to find the critical numbers ‘c’. Step 6: Write all the intervals where the function will be increasing or decreasing using the critical number. Step 7: Pick a test point ‘ x 0 ’ from each interval, substitute it in the first derivative; If f ' ( x 0 ) < 0 , then function is decreasing in that interval & If f ' ( x 0 ) > 0 , then function is increasing in that interval. Step 8: Based on the below cases, identify whether the critical number that we found is local maximum or local minimum. If to the left of critical number function is decreasing and to the right function is increasing, then it means at c we have local minimum given by ( c , f ( c ) ) If to the left of critical number function is increasing and to the right function decreasing, then it means at c we have local maximum. given by ( c , f ( c ) ) Step 9: Find the second derivative of the function. Step 10: Find the values of x that makes the f ' ' ( x ) = 0 . Step 11: Write all the intervals where the function will be concave up or concave down using the values of x that makes the equation f ' ' ( x ) = 0 TRUE. Step 12: Pick a test point ‘ x 1 ’ from each interval, substitute it in the first derivative; If f ' ' ( x 1 ) < 0 , then function is concave down in that interval & If f ' ' ( x 1 ) > 0 , then function is concave up in that interval. Step 13: Inflection points are the values of x where the function changes concavity. Calculation: Description Steps Rewrite the function using the exponent rule f ( x ) = x x + 1 = x ( x + 1 ) 1 / 2 Find the zeros of the function that makes f ( x ) = 0 x ( x + 1 ) 1 / 2 = 0 By using the zero factor property which states that if a ⋅ b = 0 Then a = 0 ( o r ) b = 0 set each factor of function equal to zero ( x + 1 ) 1 / 2 = 0 be the 1 s t equation x = 0 be the 2 n d equation Solving the equation ( x + 1 ) 1 / 2 = 0 by ➢Square on both sides, then ➢ Subtract 1 on both sides ( ( x + 1 ) 1 / 2 ) 2 = ( 0 ) 2 x + 1 = 0 x + 1 − 1 = 0 − 1 x = − 1 Calculation (Continued): Description Steps List the values of x that makes the function as zero f ( x ) = 0 x = 0 ; x = − 1 List the intervals using the zeros of the function to check where the function is positive or negative ( − ∞ , − 1 ) ; ( − 1 , 0 ) ; ( 0 , ∞ ) Pick a test point from each interval and check the sign of f ( x ) = x x + 1 Substituting the test point x = − 3 ∈ ( − ∞ , − 1 ) , we get… f ( − 3 ) = ( − 3 ) − 3 + 1 = U n d e f i n e d Substituting the test point x = − 0.5 ∈ ( − 1 , 0 ) , we get… f ( − 0.5 ) = ( − 0.5 ) − 0.5 + 1 ≈ − 0.35 < 0 Substituting the test point x = 3 ∈ ( 0 , ∞ ) , we get… f ( 3 ) = ( 3 ) 3 + 1 = 6 > 0 Sign chart of the function Analyze the sign chart and identify the intervals where the function is positive and where the function is negative Function is negative in the intervals ( − 1 , 0 ) Function is positive in the interval ( 0 , ∞ ) Find the derivative of the function f ( x ) = x x + 1 f ' ( x ) = x ⋅ d ( ( x + 1 ) 1 / 2 ) / d x + ( x + 1 ) 1 / 2 ⋅ d ( x ) / d x = x ⋅ 1 2 ( x + 1 ) − 1 / 2 + ( x + 1 ) 1 / 2 ⋅ 1 = x 2 ( x + 1 ) 1 / 2 + 2 ( x + 1 ) 2 ( x + 1 ) 1 / 2 = x + 2 x + 2 2 ( x + 1 ) 1 / 2 = 3 x + 2 2 ( x + 1 ) 1 / 2 = 0.5 ( 3 x + 2 ) ( x + 1 ) − 1 / 2 Calculation (Continued): Description Steps The critical number is the value of x that makes the first derivative f ' ( x ) = 3 x + 2 2 ( x + 1 ) 1 / 2 equal to zero or undefined. 3 x + 2 = 0 be the 1st equation x + 1 = 0 be the 2nd equation Solving the equation 3 x + 2 = 0 by ➢Subtracting 2 on both sides ➢Dividing 3 on both sides 3 x + 2 − 2 = 0 − 2 3 x = − 2 3 x / 3 = − 2 / 3 x = − 2 / 3 Solving the equation x + 1 = 0 by ➢ Subtract 1 on both sides x + 1 = 0 x + 1 − 1 = 0 − 1 x = − 1 List the critical number that makes first derivative as zero x = − 2 / 3 ; x = − 1 List the intervals using the critical number where the function will be increasing or decreasing ( − ∞ , − 1 ) ; ( − 1 , − 2 / 3 ) ; ( − 2 / 3 , ∞ ) Pick a test point from each interval and check the sign of f ' ( x ) = 3 x + 2 2 ( x + 1 ) 1 / 2 Substituting the test point x = − 2 ∈ ( − ∞ , − 1 ) , f ' ( − 2 ) = D o e s n o t e x i s t Substituting the test point x = − 5 / 6 ∈ ( − 1 , − 2 / 3 ) , f ' ( − 5 / 6 ) = 3 ( − 5 / 6 ) + 2 2 ( ( − 5 / 6 ) + 1 ) 1 / 2 ≈ − 0

5th Edition

ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 3, Problem 19SC

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To construct : sign charts for f, f', & f'' , identify the end behavior and graph the function.

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Chapter 3, Problem 18SC

Chapter 3, Problem 20SC

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Chapter 3 Solutions

Calculus

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To construct : sign charts for f , f ' , & f ' ' , identify the end behavior and graph the function.

Solution Summary: The author constructs a graph of the function based on the root values of x and the critical numbers.

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To construct : sign charts for f, f', & f'' , identify the end behavior and graph the function.

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Chapter 3 Solutions

To construct : sign charts for f , f ' , &amp; f ' ' , identify the end behavior and graph the function.

Solution Summary: The author constructs a graph of the function based on the root values of x and the critical numbers.

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To construct : sign charts for f, f', & f'' , identify the end behavior and graph the function.

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Chapter 3 Solutions

To construct : sign charts for f , f ' , & f ' ' , identify the end behavior and graph the function.