Rates of Change Homework: p236 #3, 5bd, 10, 11, 12 p370 #2, 4bc, 6, 8, 1113 p303 #46, 812 p507 #46, 8 Average and Instantaneous Rates of Change for All Functions ☺ The average rate of change is represented by And the instantaneous rate of change is represented by the difference quotient: !!() EXAMPLE 1: Determine the average rate of change from = −4 and = −2 on the function () = 2(+ 3) ! + 1 EXAMPLE 2: Estimate the instantaneous rate of change at = 2 for the function = ! − + 1 Average rate of change 5 f ( x 2 ) 2 f ( x 1 ) x 2 2 x 1 5 Dy Dx 5 change in y change in x
Average and Instantaneous Rates of Change for All Functions ☺ The average rate of change is represented by And the instantaneous rate of change is represented
by the difference quotient: 𝒇 𝒂!𝒉 !𝒇(𝒂)𝒉
EXAMPLE 1: Determine the average rate of change from 𝑥 = −4 and 𝑥 = −2 on the function 𝑓(𝑥) = 2(𝑥 + 3)! + 1 EXAMPLE 2: Estimate the instantaneous rate of change at 𝑥 = 2 for the function 𝑓 𝑥 = 𝑥! − 𝑥 + 1
NEL 75Chapter 2
2.1
In Summary
Key Ideas• The average rate of change is the change in the quantity represented by the dependent variable divided by the
corresponding change in the quantity represented by the independent variable over an interval. Algebraically, the average rate of change for any function over the interval can be determined by
Average rate of change
• Graphically, the average rate of change for any function over theinterval is equivalent to the slope of the secant line passing through two points and
Average rate of
Need to Know• Average rate of change is expressed using the units of the two quantities that are related to each other.• A positive average rate of change indicates that the quantity represented by the dependent variable is increasing on
the specified interval, compared with the quantity represented by the independent variable. Graphically, this is indicated by a secant line that has a positive slope (the secant line rises from left to right).
• A negative average rate of change indicates that the quantity represented by the dependent variable is decreasing onthe specified interval, compared with the quantity represented by the independent variable. Graphically, this is indicated by a secant line that has a negative slope (the secant line falls from left to right).
• All linear relationships have a constant rate of change. Average rate of change calculations over different intervals of the independent variable give the same result.
• Nonlinear relationships do not have a constant rate of change. Average rate of change calculations over differentintervals of the independent variable give different results.
5y2 2 y1
x2 2 x1
5DyDx
change 5 msecant 5change in ychange in x
(x2, y2).(x1, y1)x1 # x # x2
y 5 f(x)
5f(x2) 2 f(x1)
x2 2 x1
5DyDx
5change in ychange in x
x1 # x # x2y 5 f(x)(Dx)
(Dy)
x
(x1, y1)y = f (x)
(x2, y2)
y
EXAMPLE 3: For the rational function 𝑓 𝑥 = !!!!
a) Find the average rate of change for 0 ≤ 𝑥 ≤ 4 b) Estimate the slope of the tangent at the point 𝑥 = 4 c) Why can there not be a tangent line where 𝑥 = −4
EXAMPLE 4:
NEL304 5.6 Rates of Change in Rational Functions
5. Select a strategy to estimate the instantaneous rate of change of eachfunction at the given point.
a)
b)
c)
d)
6. Determine the slope of the line that is tangent to the graph of eachfunction at the given point. Then determine the value of x at whichthere is no tangent line.
a)
b)
c)
d)
7. When polluted water begins to flow into an unpolluted pond, theconcentration of pollutant, c, in the pond at t minutes is modelled by
where c is measured in kilograms per cubic metre.
Determine the rate at which the concentration is changing aftera) 1 h b) one week
8. The demand function for snack cakes at a large bakery is given by the function The x-units are given in thousands of cakes, and the price per snack cake, , is in dollars.a) Find the revenue function for the cakes.b) Estimate the marginal revenue for What is the marginal
revenue for ?
9. At a small clothing company, the estimated average cost function for
producing a new line of T-shirts is where x is the number of T-shirts produced, in thousands. is measured indollars.a) Calculate the average cost of a T-shirt at a production level of
3000 pairs.b) Estimate the rate at which the average cost is changing at a
production level of 3000 T-shirts.
C(x)C(x) 5 x 2 2 4x 1 20
x ,
x 5 2.00x 5 0.75.
p(x)p(x) 5 15
2x 2 1 11x 1 5.
c(t) 5 27t10 000 1 3t ,
f (x) 55
x 2 6, where x 5 4
f (x) 52x 2 2 6x3x 1 5
, where x 5 22
f (x) 5x 2 6x 1 5
, where x 5 27
f (x) 525x
2x 1 3, where x 5 2
y 523x 2 1 5x 1 6
x 1 6, where x 5 23
y 5x 1 3x 2 2
, where x 5 4
y 517x 1 3x 2 1 6
, where x 5 25
y 51
25 2 x, where x 5 13
A
K
EXAMPLE 5: For this graph of a function, a) State two points where the function has an instantaneous rate of change of
i. Zero ii. A negative value iii. A positive value
b) Calculate the average rate of change on the interval 2 ≤ 𝑥 ≤ 5
EXAMPLE 6:
NEL370 NEL370
2. For this graph of a function, state two points where the function hasan instantaneous rate of change in that isa) zerob) a negative valuec) a positive value
3. Use the graph to calculate the average rate of change in on theinterval
4. Determine the average rate of change of the function
for each interval.
a) c)
b) d)
PRACTISING5. State two intervals where the function has an
average rate of change that isa) zerob) a negative valuec) a positive value
y 5 3 cos (4x) 2 4
p
2# x #
5p
4p
6# x #
p
2
p
3# x #
p
20 # x #
p
2
y 5 2 cos Qx 2p3R 1 1
0 2 31–1
6
2
4
–2
–4
x
y
4 5
y = f (x)
2 # x # 5.f (x)
0
3
1
2
4
5
6
–1
–2
x
y f (x) = 4 cos (x– ) + 2p4
p2
p 2p3p2
5p2
f (x)
K
6.7 Rates of Change in Trigonometric Functions
NEL 377NEL 377
12. The current, I, in amperes, of an electric circuit isgiven by the function where t is the time in seconds.a) Draw a graph that shows one cycle.b) What is the singular period?c) At what value of t is the current a maximum
in the first cycle?d) When is the current a minimum in
the first cycle?
Lesson 6.5
13. State the period of the graph of each function,in radians.a) c)b)
Lesson 6.6
14. A bumblebee is flying in a circular motionwithin a vertical plane, at a constant speed. The height of the bumblebee above the ground,as a function of time, can be modelled by asinusoidal function. At , the bumblebee isat its lowest point above the ground.a) What does the amplitude of the sinusoidal
function represent in this situation?b) What does the period of the sinusoidal
function represent in this situation?c) What does the equation of the axis of the
sinusoidal function represent in this situation?d) If a reflection in the horizontal axis was
applied to the sinusoidal function, was thesine function or the cosine function used?
15. The population of a ski-resort town, as afunction of the number of months into the year,can be described by a cosine function. Themaximum population of the town is about 15 000 people, and the minimum population isabout 500 people. At the beginning of the year,the population is at its greatest. After six months,the population reaches its lowest number ofpeople. What is the equation of the cosinefunction that describes the population of this town?
16. A weight is bobbing up and down on a springattached to a ceiling. The data in the followingtable give the height of the weight above thefloor as it bobs. Determine the sine functionthat models this situation.
Lesson 6.7
17. State two intervals in which the function
has an average rate of
change that isa) zerob) a negative valuec) a positive value
18. State two points where the function has an instantaneous
rate of change that isa) zerob) a negative valuec) a positive value
19. A person’s blood pressure, in millimetres of mercury (mm Hg), is modelled by the
function
where t is the time in seconds.a) What is the period of the function? b) What does the value of the period mean in
this situation?c) Calculate the average rate of change in a
person’s blood pressure on the interval
d) Estimate the instantaneous rate of change ina person’s blood pressure at t 5 0.5.
tP 30.2, 0.34.P(t) 5 100 2 20 cos Q8p
3 tR,P(t),
y 514 cos (4px) 2 3
y 5 7 sin Q15 xR 1 2
t 5 0
y 5 sec xy 5 cot xy 5 csc x
I(t) 5 4.5 sin (120pt),
t (s) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
h(t)(cm)
120 136 165 180 166 133 120 135 164 179 165 133
Chapter Review
Chapter 6NEL 377
EXAMPLE 7:
EXAMPLE 8:
NEL 505Chapter 8
8.8
Solution B: Calculating graphically
The equation of the tangent is
The slope of the tangent is
At year 8, the instantaneous rate of change in students per computer decreased by about 4 students per computer.
23.8.
y 5 23.8x 1 46.9.
Plot the data, and then fit anexponential curve to the data.
The slope of the tangent at year 8gives the instantaneous rate ofdecline.
The calculator provides theequation of the tangent.
Position the cursor at year 8, anddraw the tangent.
EXAMPLE 3 Comparing instantaneous rates of change in exponential and logarithmic functions
The graphs of and are shown below. Discuss how the instantaneous rate of change in the y-values for each function changes as x grows larger.
y
8
10
6
0
4
2
–2
–4
–2 42 6 8 10x
y = 10x
y = log x
y 5 log xy 5 10x
NEL 507Chapter 8
8.8
CHECK Your UnderstandingUse the data from Example 1 for questions 1 to 3.
1. Calculate the average rate of change in number of students percomputer during the following time periods.a) years 2 to 10 b) years 1 to 5 c) years 10 to 13
2. Predict when the instantaneous rate of change in number of studentsper computer was the greatest. Give a reason for your answer.
3. Estimate the instantaneous rate of change in number of students percomputer for the following years.a) year 2 b) year 7 c) year 12
PRACTISING4. Jerry invests $6000 at 7.5% a, compounded annually.
a) Determine the equation of the amount, A, after t years. b) Estimate the instantaneous rate of change in the value at 10 years.c) Suppose that the interest rate was compounded semi-annually
instead of annually. What would the instantaneous rate of changebe at 10 years?
5. You invest $1000 in a savings account that pays 6% a, compoundedannually.a) Calculate the rate at which the amount is growing over the first
i) 2 years ii) 5 years iii) 10 yearsb) Why is the rate of change not constant?
6. For 500 g of a radioactive substance with a half-life of 5.2 h, theamount remaining is given by the formula where M is the mass remaining and t is the time in hours.a) Calculate the amount remaining after 1 day.b) Estimate the instantaneous rate of change in mass at 1 day.
7. The table shows how the mass of a chicken embryo inside an eggchanges over the first 20 days after the egg is laid.a) Calculate the average rate of change in the mass of the embryo
from day 1 to day 20.b) Determine an exponential equation that models the data.c) Estimate the instantaneous rate of change in mass for the
following days.i) day 4 ii) day 12 iii) day 20
d) According to your model, when will the mass be 6.0000 g?
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