Lily

7. An influenza virus is spreading through a school according to the function N(t) = 3(2)^t, where N is the number of people infected and t is the time, in days. 

a) How many people have the virus initially, when t = 0? 

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b) Determine the average rate of change between 

i) day 1 and day 2 ii) day 2 and day 3 

6. a) Given f(x) = log(x) and g(x) = 1/(𝑥+3), identify the steps you would take to determine the domain of (f ∘ g)(x). What is the domain of (f ∘ g)(x)? 

 b) Would the domain of (g ∘ f)(x) be the same? Explain. 

4. Given f(x) = 1/𝑥 and g(x) = sinx, provide a characteristic that the two functions have in common and a characteristic that distinguishes them. 

4. Given f(x) = 1/x

and g(x) = sinx, provide a characteristic that the two functions have in common and a characteristic that distinguishes them.

5. Describe the difference between finding the average rate of change and the instantaneous rate of change. How are they related to secants and tangents? Use a diagram to help you explain. 

6. a) Given f(x) = log(x) and g(x) = 1 /(𝑥+3), identify the steps you would take to determine the domain of (f ∘ g)(x). What is the domain of (f ∘ g)(x)? 

b) Would the domain of (g ∘ f)(x) be the same? Explain. 

Prove the following identity:

(1-sin^(2)x-2cosx)/(cos^(2)x-cosx-2)=1/(1+secx)

𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥: 2 sin(𝑥) tan(𝑥) − tan(𝑥) = 1 − 2 sin(𝑥) in the interval [0, 2𝜋]

Write cos(4𝑥) cos(3𝑥) − sin(4𝑥) sin(3𝑥) as a single trig function.

Solve the following trig equation: sin(𝑥) cos(3𝑥) + cos(𝑥) sin(3𝑥) =√3/2 

in the interval [0, 2𝜋]

A large wheel is attached to a boat and spins as the boat moves. A rock becomes nudged in the wheel as it spins in the water. It is noticed that at t = 2 s, the rock is at the highest point 3 m above the water. At time t = 6 seconds, the rock is submerged in the water 5 m below the water(the lowest point).

a. Graph 5 points to represent one cycle of the above problem. Label these points clearly on your graph.

b. Determine a cosine model to represent the motion of the wheel.

c. Determine whether the rock was above the water’s surface at t = 0 or below the water’s surface and how far is the rock above or below the water at this time?

3. Suppose you compressed the function 𝑦 = sec(𝑥) horizontally by a factor of 2. Will it still have the same vertical asymptotes? Explain. If not, list all vertical asymptotes in 

the interval [0, 2𝜋]

Junala substituted x =π/2 into the expression. 4sinx + sin^(2) x + cos^(2) x = 5 and saw the following: 4 sin (π/2) + cos^2 (π/2) + sin^2 (π/2) = 5 and concluded this was a trig identity. Is Junala correct? Provide a brief explanation of Junala’s thought process.

1. Suppose you wanted to model a Ferris wheel using a sine function that took 60 seconds to complete one revolution. The Ferris wheel must start 0.5 m above ground. 

Provide an equation of such a sine function that will ensure that the ferris wheel’s minimum height of the ground is 0.5 m. 

Explain why your equation works.

A ferris wheel completes 2 revolutions in 30 seconds. Determine how far it has travelled in 15 seconds. The radius of the ferris wheel is 10 m. 

Solve the following trig equation: sin(𝑥) cos(3𝑥) + cos(𝑥) sin(3𝑥) =√3/2 in the interval [0, 2𝜋]

Solve for x algebraically: (3^𝑥) + (3^𝑥+1) = (11^𝑥) + (11^𝑥+1) 

Leave your answer as an exact value (do not round).

What is the inverse of 𝑦 = 4^𝑥?

Is the following statement true or false? Explain. 

log(120) − log(−10) = log(−12)

Students participating in a psychology experiment and took MHF4U were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model: f(t)=75 - (6log(t+1))/(log e) where e is the number in your calculator and t is time in months 0< t < 12.

a) What was the average score on the original exam (t=0)

b) When was the average score equal to 61.2%

Between 2007 and 2017, the annual average pH of precipitation in a northern Ontario town dropped from 5.6 to 4.3. How many times more acidic was the precipitation in 2017 than the precipitation in 2007?

The growth rate of a population in a small town is about 3%. If the town has a population of 1250 people in 2006, in which year does the population reach 2000 people?