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StudySheetTest3_10

# StudySheetTest3_10 - Study Sheet for Test 3 Honors Calculus...

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Study Sheet for Test 3 Honors Calculus Keesling Test on 12/3/10 1. Define a function ln x = 1 t dt 1 x ! . Show that ln( x ) with this definition has the property that ln( a ! b ) = ln a + ln b . 2. Show that ln( x ) as defined above has the following properties. (a) ln( a ! b ) = ln a + ln b (b) ln( x ) is an increasing function on (0, ! ) (c) lim x ! + " ln x = + " (d) lim x ! 0 + ln x = "# 3. Define exp( x ) to be the inverse function for ln( x ) . Show that d exp( x ) dx = exp( x ) . 4. Define a b = exp( b ! ln a ) . Show the following properties of this function. (a) dx a dx = a ! x a " 1 for a a constant and x > 0 (b) x n = x ! x ! ! ! x n " times " # \$ % \$ for n a positive integer and x > 0 5. Determine the area under the curve of f ( x ) = 1 x over the interval [1, ! ) . Determine the volume obtained by rotating this area around the x –axis. 6. Determine the volume of the cone over the given area. Determine the centroid. h A x

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7. The following figure is rotated around the y –axis. Determine the volume and surface area of the result. 8. Determine the arc length of the graph of the function f ( x ) = x 2 over the interval [0, a ] .
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StudySheetTest3_10 - Study Sheet for Test 3 Honors Calculus...

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