MHF4U
NAME:
AROC&IROC
1. The function
models the Harrington Paper Company’s revenue, where x is the
time measured in weeks and f(x) is the revenue measured in thousands of dollars. At what rate was the
company’s revenue growing from week 50 to week 75?
2. Find the average rate of change of
on
3. Estimate the instantaneous rate of change of the function
.
at
.
4. Explain how can you use the zeros of a quadratic function to help you determine where the function has a
positive rate of change and a negative rate of change.
5. A bicyclist is competing in a tournament. He is currently in a very hilly part of the tournament’s course. For
his current hill, his height in metres over time in minutes is modelled by the function
. Estimate the bicyclist’s instantaneous rate of change and give his
. Round your answer to two decimal places.
direction at
7. Estimate the slope of the tangent line to the graph of
at the point where
8. For what value of x will there not be a tangent line to the graph of
10. For
, find the average rate of change from
11. For
?
.
, at what value(s) of x can you not determine an instantaneous rate of change?
12. Given the function
, determine the coordinates of a point on
for
the tangent line is equal to the slope of the secant line passing through A(3, 9) and B
where the slope of
.
13. A biologist finds the average production in which cancerous cells are produced in rats is given by
where is measured in hours and
is measured in 1000 cells per hour. Estimate the rate
at which the average production is changing at 4 hours.
14. The concentration of mosquito eggs in a stagnant pond is modelled by
, where
is the
number of eggs per million and x is the time in days. Estimate the rate at which the concentration is changing
after one week.
15. Find the equation of the tangent line to
at (1, 1).
16. A fish swimming is modelled by the equation
. On the interval
, when is the
instantaneous rate of change 0?
17. What is the average rate of change of the interval
20. For the function
for the function
?
, state an interval that gives an average rate of change of 0?
21. A Ferris wheel has a maximum height of 56 m and a minimum height of 14 m. It can make one rotation in 45
seconds. During the first rotation, when is the instantaneous rate of change of the height of a chair on the
Ferris wheel 0, if it starts at the height of the Ferris wheel’s axle?
23. A hamster wheel has a point on the wheel with height modelled by the equation
,
where h is in cm and t is in seconds. What is the average rate of change for the interval
?
26. A certain radioactive substance decays exponentially. The percent, P, of the substance left after t years is
given by the formula
has reached its half-life.
. Determine the instantaneous rate of decay of the substance when it
27. Consider the function
. As
, what happens to the instantaneous rate of change?
28. After college you invest $8500 at 7.2%/a compounded quarterly. Estimate the instantaneous rate of change at
10 years.
30. The population in Toronto was 2.4 million in 2001. One growth estimate predicts the population will be 3.0
million in 2030. If the estimate is correct, and the population grew exponentially, what will the instantaneous
rate of change of the population be in the year 2030?
31. A hearing test gradually raises the noise a listener is subjected to. If the test starts with an intensity of
, and the intensity is increased by 15% every 20 seconds, what will the average rate of change be, in
dB per minute, when the dB level reaches the level of a whisper, 30dB?
AROC&IROC
1. The revenue is growing at a rate of $636.72 dollars a week.
2.
192
3. 27
4. The maximum or minimum of a quadratic function falls in the exact middle between the zeros of the function.
If the function has a maximum, the function will be increasing before the maximum and decreasing afterward.
If the function has a minimum, it will be decreasing before the minimum and increasing afterward.
5. The bicyclist is moving downward at a rate of 0.73 metres per minute.
6. 0
7. 2
12. (4, 6) 13. 0.94
10.
11.
14. 1.06
15.
16. t = 7
20. Answers may vary. For example: t = 0 to t = 32
23. 0 cm/s
28. $1238.18 per year
, x = 4
8. .
26. 1.96%/year
21.
17. -3.3079
t = 11.25 and t = 33.75
27. The instantaneous rate of change
30. 23 000 people/year
31. 0.182 dB/min
.