1) Graph the following inequalities on the number line
a) b)
2) Solve
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a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
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3) The position in kilometres of a particle at t hours is given by , where .
a) What is the initial position of the particle?
b) What is the particle’s average velocity from 3 hours to 5 hours?
c) What is the particle’s instantaneous velocity at 7 hours?
d) Is there a sign change between the velocities in parts b) and c)? Explain.
4) Find the domain, intercepts, and asymptotes for each of the following. Sketch the graph.
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a)
b)
c)
d)
e)
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5) is divided by to give a remainder of 2. Determine k.
6) The volume of a rectangular solid is
a) Determine the expression for each of its dimensions.
b) State the restrictions in the context of the question.
c) Calculate the dimensions of a solid with a volume of 30 cm3.
7) For each of the following functions state:
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i) the domain
ii) the range
iii) the transformation(s) it has undergone
iv) asymptote(s)
v) intercepts
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then use the information to sketch the graph
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a) , compare to
b) , compare to
c) , compare to
d) , compare to . Also state the period and amplitude.
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8) A polynomial of degree 5 has a negative leading coefficient.
a) How many turning points could the polynomial have?
b) How many zeros could the function have?
c) Describe the end behaviour.
d) Sketch two possible graphs, each passing through the point .
9) Find the slope of the secant of that passes through the points where and
10) Determine when and and state the domain and range of h(x).
11) Calculopia had a population of 16 000 in 1960 and a population of 20 000 in 1970. Assume exponential growth.
a) Write a function that models the population growth. Round the base to 4 decimal places.
b) What does the model predict as the population in 2010?
c) Predict when the population of Calculopia will reach 1 000 000.
12) For the function defined by
a) Determine the value of k, if (1, -24) is a point on the graph of the function
b) solve for p if (3, p) is a point on the graph of the function
c) considering the end behaviours and the zeros, state where
13) The concentration of medicine in a patient’s bloodstream is given by , where C is measured in milligrams per cubic centimetre and t is the time in hours after the medicine was taken. Determine:
a) the concentration in the bloodstream 3 hours after the medicine was taken.
b) the average rate at which the concentration is decreasing from 4 hours after taking the medicine to 7 hours after taking the medicine.
c) the instantaneous rate of change for the concentration 2 hours after the medicine was taken. Interpret the meaning of your answer.
14) The population of Epoville has been increasing exponentially by 15% per year. How long will it take for the population to double?
15) Carbon-14 is a radioactive substance with a half-life of 5730 years. It is used to determine the age of artefacts. An archaeologist discovers that the burial cloth on an Egyptian mummy has 45% of the carbon-14 that it contained originally. How old is the mummy?
16) The table represents the number of bacteria in a researcher’s sample at various times.
Time (min.) / Number of bacteria10 / 3297
20 / 5437
30 / 8963
40 / 14778
50 / 24365
a) Determine an algebraic model for the number of bacteria, in the form . Round the base to 4 decimal places.
b) Use your model to estimate the number of cells after one hour.
c) Use your model to estimate the doubling time of these cells.
d) Determine the rate of growth after 30 minutes.
17) Prove the following identities.
a) / b)c) / d)
18) Given the graph of shown on the right,
state the intervals of x for which
(a) the function is decreasing
(b) the function is concave up
19) Given the curve defined by ,
a) state the equations of the vertical and horizontal asymptotes.
b) state the intercepts.
c) sketch the graph of .
20) The point P(-5, 4) is on the terminal arm of an angle in standard position.
a) Sketch the principal angle.
b) Determine the measure of the related acute angle to the nearest degree.
c) Determine the measure of the principal angle.
21) Convert the following radians to degrees. Round your answer to one decimal place, if necessary.
a) b) c) 2.678
22) Determine the smallest positive co-terminal angle to 420. Determine a co-terminal angle that is larger than 420. Determine a negative co-terminal angle.
23) a) Find the inverse of
b) What is the domain of the inverse?
c) Is the inverse a function?
24) Sketch the inverse of the following functions on the same axes.
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a)
b)
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25) Sketch and its inverse. What would the domain have to be so that the inverse is a function?
26) Determine the exact value for each of the following:
a) b) c) d)
27) Distance in kilometres above sea level is given by the formula , where P is the atmospheric pressure measured in kiloPascals, kPa.
a) At the top of the highest mountain in Shelbyville, the atmospheric pressure was recorded as being 220 kPa. Calculate the height of the mountain above sea level.
b) The town of Springfield has a mountain with a peak 4.5 km above sea level. Calculate the atmospheric pressure at the top of the mountain.
c) In the year 1980, both towns had an earthquake. Springfield’s earthquake measured 7.5 on the Richter Scale. The magnitude of the earthquake was 107.5. The earthquake in Shelbyville measured 6.4. How many times more intense was Springfield’s earthquake when compared to Shelbyville’s earthquake.
earthquake was than that in Shelbyville.following:
point
d) The Earthquake uncovered an archaeological find in Shelbyville and a fossil was uncovered. The formula for the amount of carbon-14 remaining in a fossil is , where M(t) is the amount of carbon-14, in grams, in the fossil at time t, in years, and M0 is the original amount of carbon-14. Calculate the age of the fossil if 20% of the original amount of carbon is remaining.
28) In 1960, the city of Springfield set up a disaster relief fund based on donations. The amount of money, A, that Springfield must invest compounded annually at 8%/a in order to have B dollars in 20 years is represented by the equation 20(log1.08) + logA = logB.
a) How much money should Springfield have invested in 1960 in order to have its investment grow to $1 000 000 in 1980 when the earthquake hit?
b) How would this information be represented in an exponential equation?
29) Point P(– 4,–5) is on the terminal arm of an angle of measure in standard position.
a) Determine the exact value of .
b) Determine the exact value of
c) Determine the value of , to four decimal places, where .
30) Determine whether each of the following functions is even, odd, or neither. Justify your answer.
a)/ b)
/ c)
d) f(x) = 3x2 + 4 / e) f(x) = -3x3 + x / f)
g) / h) / i)
31) The graph of the function is shown on the right. Use the graph to estimate the answer to the following questions then verify your answer(s) using the equation.
a) evaluate
b) solve for a if
c) the interval for which
32) Express each of the following as a co-function
a) b)
33) Express as a single logarithm.
34) Express as a single logarithm with base 2.
35) Given and determine
a) b)
36) Given,
a) determine b) at most, what is the range of the function?
37) Given,
a) determine b) at most, what is the range of the function?
38) What is the maximum number of zeros possible for ? Do you think there will actually be that many zeros? Justify your answers.
39) Given and , determine (Keep your answers within .)
a) b)
40) The volume of air in the lungs during normal breathing can be modeled by a sinusoidal function of time. Suppose a person’s lungs contain from 2200 mL to 2800 mL of air during normal breathing. Suppose a normal breath takes 4 seconds, and that t = 0 s corresponds to a minimum volume.
a) Let V represent the volume of air in a person’s lungs. Draw a graph of Volume versus time for 20 seconds.
b) State the period, amplitude, phase shift and vertical translation for the function.
c) Write a possible equation for the volume of air as a function of time.
d) Describe how the graph would change if the person breaths more rapidly.
e) Describe how the graph would change if the person took bigger breaths.
f) Determine the amount of air in the lungs after 8 seconds.
g) Determine when, within the first 8 seconds, the volume is 2400 mL.
41. You should also do some review from the textbook (chapter reviews, cumulative reviews).
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