A Long Way from Euclid by Constance Reid

By Constance Reid

This full of life consultant via a in demand historian makes a speciality of the position of Euclid's Elements in mathematical advancements of the final 2,000 years. No mathematical historical past past effortless algebra and aircraft geometry is important to understand the clear and easy reasons, that are augmented via greater than eighty drawings. 1963 version.

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F ͑x͒ ෇ 44. f ͑x͒ ෇ 0 2 4 6 8 10 12 14 T 68 65 63 63 65 76 85 91 (a) Use the readings to sketch a rough graph of T as a function of t. (b) Use your graph to estimate the temperature at 11:00 AM. 23 1 4 x 2 Ϫ 5x s 33. h͑x͒ ෇ from midnight to 2:00 PM in Baltimore on September 26, 2007. The time t was measured in hours from midnight. t FOUR WAYS TO REPRESENT A FUNCTION 45. f ͑x͒ ෇ 46. f ͑x͒ ෇ 23. If f ͑x͒ ෇ 3x 2 Ϫ x ϩ 2, find f ͑2͒, f ͑Ϫ2͒, f ͑a͒, f ͑Ϫa͒, Խ Խ Խ 3x ϩ x x ͭ ͭ ͭ 4 Ϫ t2 2Ϫt Խ Խ Խ 42. t͑x͒ ෇ x Ϫ x x ϩ 2 if x Ͻ 0 1 Ϫ x if x ജ 0 3 Ϫ 12 x 2x Ϫ 5 ͭ if x ഛ 2 if x Ͼ 2 x ϩ 2 if x ഛ Ϫ1 x2 if x Ͼ Ϫ1 xϩ9 Ϫ2x Ϫ6 if x Ͻ Ϫ3 if x ഛ 3 if x Ͼ 3 Խ Խ f ͑a ϩ 1͒, 2 f ͑a͒, f ͑2a͒, f ͑a 2 ͒, [ f ͑a͒] 2, and f ͑a ϩ h͒.

The ability to do this is a useful skill in solving calculus problems that ask for the maximum or minimum values of quantities. v EXAMPLE 5 Expressing a cost as a function A rectangular storage container with an open top has a volume of 10 m3. The length of its base is twice its width. Material for the base costs $10 per square meter; material for the sides costs $6 per square meter. Express the cost of materials as a function of the width of the base. SOLUTION We draw a diagram as in Figure 12 and introduce notation by letting w and 2w be the width and length of the base, respectively, and h be the height.

In general, a sequence ͕a n͖ is a set of numbers written in a definite order. For instance, the sequence {1, 12 , 13 , 14 , 15 , . } can be described by giving the following formula for the nth term: an ෇ a¢ a£ a™ 0 1 n We can visualize this sequence by plotting its terms on a number line as in Figure 10(a) or by drawing its graph as in Figure 10(b). Observe from either picture that the terms of the sequence a n ෇ 1͞n are becoming closer and closer to 0 as n increases. In fact, we can find terms as small as we please by making n large enough.

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