Explanation Given: The derivative d d x ∫ 1 x ln t d t . Concept Used: Second Fundamental Theorem of Calculus stats that if f is continuous on [ a , b ] and we define F ( x ) = ∫ a x f ( t ) d t for all x ∈ [ a , b ] , then F is continuous on [ a , b ] and differentiable on ( a , b ) . Fis antiderivative of f , that is, F ' ( x ) = f ( x ) . If f is continuous on [ a , b ] and u ( x ) is differentiable on [ a , b ] , then for all x ∈ [ a , b ] , d d x ∫ a u ( x ) f (

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.

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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 4.7, Problem 42S

To determine

The derivative ddx∫1xlnt dt using the Second Fundamental Theorem of Calculus.

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Chapter 4.7, Problem 41S

Chapter 4.7, Problem 43S

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Evaluate the limit along the stated paths, or type "DNE" if the limit Does Not Exist: lim xy+y³ (x,y)(0,0) x²+ y² Along the path = = 0: Along the path y = = 0: Along the path y = 2x:

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A graph of the function f is given below: Study the graph of ƒ at the value given below. Select each of the following that applies for the value a = 1 Of is defined at a. If is not defined at x = a. Of is continuous at x = a. If is discontinuous at x = a. Of is smooth at x = a. Of is not smooth at = a. If has a horizontal tangent line at = a. f has a vertical tangent line at x = a. Of has a oblique/slanted tangent line at x = a. If has no tangent line at x = a. f(a + h) - f(a) lim is finite. h→0 h f(a + h) - f(a) lim h->0+ and lim h h->0- f(a + h) - f(a) h are infinite. lim does not exist. h→0 f(a+h) - f(a) h f'(a) is defined. f'(a) is undefined. If is differentiable at x = a. If is not differentiable at x = a.

Chapter 4 Solutions

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The derivative d d x ∫ 1 x ln t d t using the Second Fundamental Theorem of Calculus.

Solution Summary: The author explains the second fundamental theorem of calculus, which states that if f is continuous on left[a,bright], then it is antiderivative of

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The derivative ddx∫1xlnt dt using the Second Fundamental Theorem of Calculus.

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