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Unformatted text preview: A. y = 0 y = f (t) = et 3 t
2 −− − − − −→
−−−−−− 4 6 8 10 12 14 16 18 20 t y = T (t) 3. From the graph, we see that the horizontal asymptote is y = 70. It is worth a moment or two
of our time to see how this happens analytically and to review some of the ‘number sense’
developed in Chapter 4. As t → ∞, We get T (t) = 70 + 90e−0.1t ≈ 70 + 90every big (−) . Since
e > 1, every big (−) = every 1 (+) ≈ very big (+) ≈ very small (+). The larger t becomes, the smaller
e−0.1t becomes, so the term 90e−0.1t ≈ very small (+). Hence, T (t) ≈ 70 + very small (+)
which means the graph is approaching the horizontal line y = 70 from above. This means
that as time goes by, the temperature of the coﬀee is cooling to 70◦ F, presumably room
temperature. As we have already remarked, the graphs of f (x) = bx all pass the Horizontal Line Test. Thus the
exponential functions are invertible. We now turn our attention to these inverses, the logarithmic
functions, which ar...
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