Explanation Given: The linear function is y = 1 + 1 − x 2 between the points ( 1 , 1 ) and ( 3 2 , 3 2 ) . The above linear function is rewritten in the form of y is x = 1 − ( 1 − y ) 2 . Let x = f ( y ) then f ( y ) = 1 − ( y − 1 ) 2 . Formula used: The surface area of the revolution is S = ∫ a b 2 π f ( y ) 1 + f ′ ( y ) 2 d y , where f ( y ) is mentioned above. Calculation: The linear function y = 1 + 1 − x 2 is rewritten in terms of y as follows, y − 1 = 1 − x 2 ( 1 − y ) 2 = 1 − x 2 x 2 = 1 + ( 1 − y ) 2 x = 1 + ( 1 − y ) 2 From the above coordinate points the inequality in terms of y is 1 ≤ y ≤ 3 2 . Let l = 1 + f ′ ( y ) 2 . Calculate the derivative of f ( y ) . That is, f ′ ( y ) = d ( f ( y ) ) d y = d ( 1 − ( 1 − y ) 2 ) d y = 2 ( 1 − y ) 2 1 − ( 1 − y ) 2 = ( 1 − y ) 1 − ( 1 − y ) 2 To obtain the surface area of revolution, first calculate the

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 6.6, Problem 20E

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The surface area generated through revolving the given curve about the y-axis.

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Chapter 6.6, Problem 19E

Chapter 6.6, Problem 21E

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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.

The graph below is the function f(z) 4 3 -2 -1 -1 1 2 3 -3 Consider the function f whose graph is given above. (A) Find the following. If a function value is undefined, enter "undefined". If a limit does not exist, enter "DNE". If a limit can be represented by -∞o or ∞o, then do so. lim f(z) +3 lim f(z) 1-1 lim f(z) f(1) = 2 = -4 = undefined lim f(z) 1 2-1 lim f(z): 2-1+ lim f(x) 2+1 -00 = -2 = DNE f(-1) = -2 lim f(z) = -2 1-4 lim f(z) 2-4° 00 f'(0) f'(2) = = (B) List the value(s) of x for which f(x) is discontinuous. Then list the value(s) of x for which f(x) is left- continuous or right-continuous. Enter your answer as a comma-separated list, if needed (eg. -2, 3, 5). If there are none, enter "none". Discontinuous at z = Left-continuous at x = Invalid use of a comma.syntax incomplete. Right-continuous at z = Invalid use of a comma.syntax incomplete. (C) List the value(s) of x for which f(x) is non-differentiable. Enter your answer as a comma-separated list, if needed (eg. -2, 3, 5).…

A graph of the function f is given below: Study the graph of f at the value given below. Select each of the following that applies for the value a = -4. f is defined at = a. f is not defined at 2 = a. If is continuous at x = a. Of is discontinuous at x = a. Of is smooth at x = a. f is not smooth at x = a. If has a horizontal tangent line at x = a. f has a vertical tangent line at x = a. Of has a oblique/slanted tangent line at x = a. Of has no tangent line at x = a. f(a + h) − f(a) h lim is finite. h→0 f(a + h) - f(a) lim is infinite. h→0 h f(a + h) - f(a) lim does not exist. h→0 h f'(a) is defined. f'(a) is undefined. If is differentiable at x = a. If is not differentiable at x = a.

Chapter 6 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 6.1 - A police officer leaves his station on a...Ch. 6.1 - Describe a possible motion of an object along a...Ch. 6.1 - Is the position s(t) a number or a function? For...Ch. 6.1 - Without doing further calculations, what are the...Ch. 6.1 - Prob. 5QC Ch. 6.1 - Prob. 6QC Ch. 6.1 - Explain the meaning of position, displacement, and...Ch. 6.1 - Suppose the velocity of an object moving along a...Ch. 6.1 - Given the velocity function v of an object moving...Ch. 6.1 - Prob. 4E

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The surface area generated through revolving the given curve about the y -axis.

Solution Summary: The linear function y=1+sqrt1-x2 is rewritten in the form of x.

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