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Quiz 8 Solution - MA 113 Quiz 8 Name November 6th 2013 1...

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MA 113 Name Quiz 8 November 6th, 2013 1. Use L’Hˆ opital’s Rule to evaluate the limit, or state that L’Hˆopital’s Rule does not apply and explain why it does not. (2 points each) (a) lim x 0 sin(sin(sin( x ))) x 2 - 1 (b) lim x →- 1 9 x 7 + 2 x 2 + 7 x 2 - 1 Solutions: (a) lim x 0 sin(sin(sin( x ))) x 2 - 1 = 0 - 1 = 0 L’Hˆ opital’s Rule does not apply here, as we do not have an indeterminate form. This limit can simply be found using direct substitution. (b) 9( - 1) 7 + 2( - 1) 2 + 7 ( - 1) 2 - 1 = 0 0 Direct substitution of - 1 yields the indeterminate form 0 / 0, so we can apply L’Hˆopital’s Rule. Thus, lim x →- 1 9 x 7 + 2 x 2 + 7 x 2 + 1 = lim x →- 1 63 x 6 + 4 x 2 x = 63( - 1) 6 + 4( - 1) 2( - 1) = - 59 2
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2. A box with no top is to be constructed from a single rectangular piece of card- board, with side lengths A and B . The box will be constructed by cutting out squares of length x from each corner and folding up the sides. (a) Draw a picture to describe the situation. Label all pertinent information. (1 point) (b) Find the value of x that maximizes the volume of the box if A = 15 and B = 24. (4 points) (c) What are the dimensions of the box that maximizes volume? (1 point)
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