108Fall12Ex1VersionASolutions

A 2 x

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Unformatted text preview: 2 x x 6. 1 (3 pts.) The natural logarithm of x is defined by ln x = ∫ dt for x > 0 . Using this t 1 definition, which of the following theorems allows us to conclude that d ln x 1 = ? dx x b)***Fundamental Theorem of a) Mean Value Theorem Calculus c) Chain Rule d) Intermediate Value Theorem () 2 MthSc 108 Test 1-Version A Fall 2012 7. Find the derivative of y with respect to x for y = x ln x . (3 pts.) ⎛ 2⎞ ln x ln x−1 a) y′ = x ln x ⎜ ⎟ x b) y ′ = ⎝ x⎠ x ⎛ 2 ln x ⎞ x ln x−1 c) *** y′ = x ln x ⎜ d) y ′ = ⎝ x ⎟ ⎠ x () 8. Let R be the region bounded by the graphs of y = sin x 2 and the x-axis on the (3 pts.) interval ⎡0, π ⎤ . Which integral correctly computes the volume of the solid ⎣ ⎦ formed when R is rotated around the x − axis? a) 2π π ∫ x sin 2 ( x 2 ) dx 1 () b) π ∫ sin 2 y 2 dy c) *** π 0 0 π 2 2 ∫ sin ( x ) dx d) 2π 0 π ∫ sin ( x ) dx 2 2 0 9. x2 (3 pts.) Evaluate ∫ ( 2 x ) 5 dx . 2 a) 5 x + C 2 1 c) 5 x + C 2 ln 5 x b) ( ln 5 )...
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