Unformatted text preview: e horizontal asymptote of the graph.
Solution.
1. To ﬁnd T (0), we replace every occurrence of the independent variable t with 0 to obtain
T (0) = 70 + 90e−0.1(0) = 160. This means that the coﬀee was served at 160◦ F.
2. To graph y = T (t) using transformations, we start with the basic function, f (t) = et . As we
have already remarked, e ≈ 2.718 > 1 so the graph of f is an increasing exponential with
y intercept (0, 1) and horizontal asymptote y = 0. The points −1, e−1 ≈ (−1, 0.37) and
(1, e) ≈ (1, 2.72) are also on the graph. Since the formula T (t) looks rather complicated, we
rewrite T (t) in the form presented in Theorem 1.7 and use that result to track the changes to
our three points and the horizontal asymptote. We have T (t) = 90e−0.1t + 70 = 90f (−0.1t) +
70. Multiplication of the input to f , t, by −0.1 results in a horizontal expansion by a factor
of 10 as well as a reﬂection about the y axis. We divide each of the x values of our points by
−0.1 (which amounts to multiplying them by −10) to obtain...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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