Functions and their Applications Indefinite Integration ...€¦ · Indefinite Integration and its Applications Chapter 8 Definite Integration Chapter 9 Applications of Definite Integration
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. Find the equation of the curve if it passes through (1, 2)
and (4, 4).
66. The growth rate of the population of a city is given by )0(12)(' 015.0 ≥= tetP t , where t is the time measured
in years from the beginning of 2000, )(tP (in thousands) is the population at time t. It is known that the
population of the city was 900 thousand at the beginning of 2004.
(a) Find )(tP .
(b) Find the population of the city at the beginning of 2020. (Give your answer correct to 4 significant figures.)
67. The rate of change of the number of flats sold in a private housing estate can be modelled by
)0()(
90023.03.0
≥+
= − teedt
dNtt ,
where t is the number of days elapsed since the start of the selling of the flats, N is the number of flats sold at time t. It is known that 100=N when 0=t .
(a) (i) Prove that 26.0
6.0
)1(
900
+= t
t
ee
dtdN
.
(ii) Using the substitution 16.0 += teu , or otherwise, express N in terms of t.
(b) Can the number of flats sold be 900? Explain briefly.
68. The rate of change of the daily number of people infected with common cold in a town can be modelled by
)70(3
)25(32 3
12
<<−= tttdtdN
,
where t is the time measured in days with 1=t corresponds to last Monday, N is the daily number of
infected people.
(a) When did the daily number of people infected with common cold become the greatest?
(b) If the daily number of infected people was 150 on last Monday, find the daily number of infected people on the day obtained in (a). (Give your answer correct to the nearest integer.)
69. The slope at any point (x, y) of the curve C is given by xdxdy 48 −= . The line 32 += xy is a tangent to the
curve at the point P.
x
y
y = 2x + 3
O
C
P
(a) Find the coordinates of P.
(b) Find the equation of C.
70. The slope at any point (x, y) of the curve C is given by )5)(2(6 +−−= xxdxdy . The y-intercept of C is 10.
(a) Find the equation of C.
(b) (i) Prove that the slope of C cannot exceed
2147 .
(ii) Find the point of C with the greatest slope.
71. The rate of change of the temperature of a city yesterday can be modelled by
)100(1 ≤≤+−=θ − thedtd kt ,
where t is the time in hours measured from 9:00 a.m., θ (in °C) is the temperature at time t. At 9:00 a.m., the temperature was 7.4°C, h and k are positive constants.
(a) (i) Express )1ln( +θdtd
as a linear function of t.
(ii) If the slope and the intercept on the vertical axis of the graph of the linear function in (a)(i) are −0.5 and 2 respectively, find the values of h and k. (Give your answers correct to 4 significant figures if necessary.)
Take 4.7=h and 5.0=k .
(b) Express θ in terms of t.
(c) Find the greatest temperature. (Give your answer correct to 1 decimal place.)
72. The rate of change of the number of visitors in a library during a day can be modelled by
)120(10016
)8(1602
≤≤+−−= t
ttt
dtdN
,
where t is the time elapsed in hours since 8:00 a.m., N is the number of visitors in the library at time t. When the library is just open (i.e. t = 0), there are 88 visitors.
(a) (i) Let 100162 +−= ttu . Find
dtdu
.
(ii) Hence express N in terms of t.
(b) There is a period of time where the number of visitors in the library exceeds 160. How long does the period last for? (Give your answer correct to the nearest 0.1 hour.)
(c) Can the number of visitors in the library reach 180? Explain briefly.
7.10
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