Unformatted text preview: 136 CHAPTER 2 LIMITS AND DERIVATIVES 48. (a) T (C) (IT/d1 -> t
0 t (b) The initial temperature of the water is close to room temperature because of the water that was in the pipes.
When the water from the hot water tank starts coming out. dT/dt is large and positive as T increases to the
temperature of the water in the tank. In the next phase. dT/dt : 0 as the water comes out at a constant, high
temperature. After some time. dT/dt becomes small and negative as the contents of the hot water tank are
exhausted. Finally. when the hot water has run out. dT/dt is once again 0 as the water maintains its (cold) temperature. 49. In the right triangle in the diagram. let Ay be the side opposite angle ¢ and A11: the side adjacent angle (1). Then the slope of the tangent line E is m : Ay/Am : tan (1). Note that 0 < (b < %. We know (see Exercise 19)
that the derivative of f(1‘.) = m2 is f'($) : 23:. So the slope of the tangent
to the curve at the point (1. 1) is 2. Thus, (15 is the angle between 0 and 325 whose tangent is 2; that is. d) : tan‘1 2 3 63°. 2 Review
CONCEPT CHECK ____——————————— 1. (a) lim f(:1:) : L: See Deﬁnition 2.2.1 and Figures 1 and 2 in Section 2.2. I—‘Va (b) lim f (at) : L: See the paragraph after Deﬁnition 2.2.2 and Figure 9(b) in Section 2.2. I—ra‘l' (c) lim ﬂan) : L: See Deﬁnition 2.2.2 and Figure 9(a) in Section 2.2. z—HI’ (d) lim ﬂan) : 00: See Deﬁnition 2.2.4 and Figure 12 in Section 2.2. Iaa (6) lim ﬂan) : L: See Deﬁnition 2.6.1 and Figure 2 in Section 2.6. (Ii—’00
2. In general. the limit of a function fails to exist when the function does not approach a ﬁxed number. For each of the following functions. the limit fails to exist at an : 2. The left— and right—hand There is an There are an inﬁnite limits are not equal. inﬁnite discontinuity. number of oscillations. 3. (a)— (g) See the statements of Limit Laws 1—6 and l l in Section 2.3. ...
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