Biologists have noticed that the chirping rate of crickets of a

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1. 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 −π

2 < x ≤ 0

(a) Sketch a graph of f.

(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−1.

 

 

3. Find the domain of the function g(t) = 3

√t2

−7t .

 

 

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) = x5 −x4 +7×3 −9×2 −18x into a

product of irreducible linear and quadratic factors. What are the roots of

P?

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6. Consider the rational function g(t) = t+3

2t−4 .

(a) What is the domain of g?

(b) Given that g is one-to-one, find the inverse g−1.

(c) What is the range of g?

 

 

7. Use partial fractions to simplify the rational function 5x−1

x3

−3x−2 .

 

 

8. Solve the equation 2 sin2 x − sin x = 1 for 0 ≤ x ≤ 2π.

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