QUESTION
Biologists have noticed that the chirping rate of crickets of a certain species
is related to temperature, and the relationship appears to be very nearly
linear. A cricket produces 113 chirps per minute at 21
◦
C and 173 chirps
per minute at 27
◦
C.
(a) Find a linear equation that models the temperature T as a function
of the number of chirps per minute N.
(b) What is the slope of the graph and what does it represent?
(c) What is the T-intercept of the graph and what does it represent?
(d) If the crickets are chirping at 150 chirps per minute, estimate the
temperature.
2. Consider the piecewise function f(x)=
(x − 1)
2
if x>1
tan x if −
<x≤ 0
(a) Sketch a graph of f.
π
2
(b) State the domain and range of f.
(c) Does f have an inverse? If not, explain why, otherwise if so then
sketch the inverse function f
.
3. Find the domain of the function g(t)=
−1
3
√
.
4. Is the function given by f(x)=(x − 5)(x +5)x even or odd or neither?
Please justify your reasoning.
5. Factorize the quintic polynomial P(x)=x
t
2
5
−7t
−x
4
+7x
3
−9x
−18x into a
product of irreducible linear and quadratic factors. What are the roots of
P?
6. Consider the rational function g(t)=
.
(a) What is the domain of g?
t+3
2t−4
.
(c) What is the range of g?
(b) Given that g is one-to-one, find the inverse g
.
8. Solve the equation 2 sin
7. Use partial fractions to simplify the rational function
2
x − sin x =1for0≤ x ≤ 2π.
SOLUTION
(1)
(a) Let chirp rate is represented by c and temperature by T.
So, according to the question
C is linearly proportional to T.
C = kT + b
Where k = proportional constant
b = T intercept
Given that, c = 113 when T = 21 degree c
And c = 173 when T = 27 degree c.
113 = 21k + b ——-equation (i)
And 173 = 27k + b ——-equation (ii)
Substracting (i) from (ii), we get
6k = 60 => k = 10
And when k = 10, b = -97
Equation become c = 10T
(b)
Slope of Graph = k = 10
And this represent ,The graph is making constant angle arctan(10) with counterclock direction of x-axis.
(c)
T intercept of graph represent y-axis. At T = -97 degre c ,chirpying is ceases.
(d) And when c = 150
150 = 10T – 97 10 T = 150 + 97 = 247
T = 24.7 degree C
II95
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