The average rate of change is the change in the price divided by the change in time.. The average rate of change of a function f on the interval [a, b] can be represented by the slope of
Trang 1(b) Suppose you tried
C(x) = 220 + 5x for 0 ≤ x ≤ 6
250 + 25x for x > 6
For x > 6 this model correctly counts the $5 commission for the first six items in with
the base rate, but then it gives an additional $25 for each of those first six items This
amounts to giving a $30 commission for each of the first six items sold For instance, when x = 7 this model gives
($220) + ($5/item)(6 items) + ($25/item)(7 items) instead of the correct
($220) + ($5/item)(6 items) + ($25/item)(1 item)
REMARK The function C(x) is defined piecewise: It is defined as one function on one interval and as another function on a second interval Because each piece is linear, the
function is called piecewise linear.
The slope of C(x) corresponds to the commission rate, the rate of change of salary with respect to the number of items sold The commission rate changes at x = 6 Below is
a sketch of the slope function, typically denoted by C
25
5
C ′ (dollars/item)
(items)
Figure 4.15
EXAMPLE 4.8 The resale value of a used TI-81 calculator is a function that decreases with time; as newer
and more advanced models come out there is less demand for the older TI-81 Let’s call
P (t )the price, in dollars, of a used TI-81 at time t, where t is measured in years, with t = 0 corresponding to January 1, 1992 Suppose that on January 1, 1992, the resale value was
$75 and was decreasing at a rate of $10 per year and that on January 1, 1995, the resale value was $51 Furthermore, let’s assume that although the value is always going down, it
is going down less and less steeply as time passes (The rationale behind this assumption might be that inflation tends to drive prices up over time, and that the calculator will always have some positive value.)
i Sketch a possible graph of P (t) incorporating all the information given
ii What is the average rate of change of the calculator’s value between t = 0 and t = 3? iii What can we say about the value of the calculator on January 1, 1994, at t = 2? Give good upper and lower bounds for the price of the calculator on that date
Trang 2i P (t) is decreasing and concave up
t (in yrs since 1/1/92)
51
75
P(t) dollars
Figure 4.16
ii The average rate of change is the change in the price divided by the change in time This is
P (3) − P (0)
51 − 75
year ,
or a decrease of $8 per year
Graphically, this is the slope of the line connecting the points (0, 75) and (3, 51)
t (in yrs since 1/1/92)
51
75
P(t)
dollars the slope of this line represents the
average rate of change the slope is
3 dollars/yr = –8 dollars /yr
24
–
Figure 4.17
iii To find upper and lower bounds for P (2) means that we must find a price that is greater than P (2) and a price that is less than P (2) For instance
51 < P (2) < 75,
so 51 is a lower bound and 75 is an upper bound We could do worse (for instance, using 0 as a lower bound and 75 as an upper bound), but we could do better! To “do better” means to find a larger lower bound and a smaller upper bound so we can pin down the price at t = 2 as much as possible
First, let’s find a good lower bound We know that the value was dropping at a rate
of $10 per year at t = 0 and that after this time the value dropped at a slower and slower rate So, we know that in the two years between t = 0 and t = 2, the value dropped by
some amount less than (2 years) · ($10/year) = $20 If the value dropped by less than
$20, then at t = 2 it must have been more than P (0) − $20 = $75 − $20 = $55.
How can we visualize this graphically? We know that at t = 0 the price is $75
If we assume that the price drops by $10 per year, this corresponds graphically to the line through (0, 75) with slope −10 The value of the calculator is going down less and
Trang 3less steeply as time passes, therefore this line lies below the graph of P Therefore, the
point on this line with the t-coordinate of 2 lies below the point (2, P (2))
51 55
75
lower bound actual value P(t)
t (in years since 1/1/92)
dollars
(2, 55)
Figure 4.18
Now, let’s find an upper bound for the price at t = 2 We know that the average rate of change of price between t = 0 and t = 3 is represented by the line we drew in Figure 4.17 We can see that the point on this line at t = 2 is an overestimate for the actual value of the calculator at that time since this secant line lies above the curve The slope of the secant line is −8, so the point on the secant line with a t-coordinate
of 2 corresponds to a price of P (0) − 2 · $8 = $75 − $16 = $59 Thus, the value of a
calculator at t = 2 must be under $59.
t (in yrs since 1/1/92)
51
75 P(t)
dollars
(2, 59)
upper bound
Figure 4.19
In summary, our lower bound is $55 and our upper bound is $59 Notice that the difference between these two estimates for P (2) is $4 To obtain our two estimates, we used two different linear approximations, both using the point (0, 75) corresponding
to a price of $75 in 1992 The difference in the slopes of the lines we used was $2 per year We were looking at a period of 2 years; therefore it makes sense that the difference between the estimates is (2 years) · ($2/year) = $4 If we were to make estimates closer
to t = 0, this difference would be smaller; this reflects the fact that estimates based on the idea of local linearity are more accurate the nearer we are to the point at which we have definite information
Figure 4.20 shows both linear approximations Notice that the line through (0, 75) with slope −10 is a much better linear approximation of the curve near (0, 75) than is the line through (0, 75) and (3, 51) In fact, we will soon find that the former line is the best linear approximation of the curve at (0, 75)
Trang 41 2 3 51
55 75
t (years since 1/1/92)
y (dollars)
approximation by secant gives an overestimate of $59 at t=2 actual value somewhere between $55 and $59
approximation gives an underestimate of $55 at t=2
Figure 4.20
REMARKThe key ideas in Example 4.8 are the geometric ones
The average rate of change of a function f on the interval [a, b] can be represented by the slope of the secant line through (a, f (a)) and (b, f (b))
Where a curve is concave up, its secant lines lie above the curve
Where a curve is concave down, its secant lines lie below the curve
concave up Secant lines lie above the curve.
concave down Secant lines lie below the curve.
Figure 4.21
The particular information we were given in this problem determined the approach we took to solving it Our problem-solving strategy involved drawing pictures to represent the information available in a visual way
P R O B L E M S F O R S E C T I O N 4 4
1 A social worker gets paid $D per hour up to 40 hours per week If he puts in more than
40 hours, the hours over 40 count as overtime, which pays an additional 50% per hour Express his weekly wages as a function of x, where x is the number of hours he has worked that week (You’ll have to write a function in two pieces since the pay equation
is described in two different ways depending upon the value of x.)
2 Inflation in Turkey has caused prices of small everyday items to be measured in tens
of thousands of lira One day I went to a market and purchased one container of yogurt and two packets of honey for 180,000 Turkish lira Two days later I returned to the
Trang 5same market and purchased two containers of yogurt and three packets of honey for 310,000 Turkish lira
(a) Assuming that the price remained constant over this two-day period, what is the price of a yogurt? What is the price of a packet of honey?
(b) The figures given in this problem are accurate for the summer of 1998 in the town of Iznik, a beautiful, tiny lakeside town founded nearly 3000 years ago The exchange rate at the time was 258,000 Turkish lira per dollar Convert the prices of yogurt and honey into dollars
3 (a) Determine the equation of the supply and demand curves shown in the figure below (b) What are the equilibrium price and quantity? Assume price is measured in dollars and quantity in thousands of units (The equilibrium occurs when supply and demand are equal.)
(9, 12) 16
12
p
q
supply
demand
4 A moving company charges a minimum of $250 for a move An additional $100 per hour is charged for time in excess of two hours Write a function C(t) that gives the cost of a move that takes t hours to complete
5 The graph that follows indicates the salary scheme at Company A for a certain job The pay scheme at Company B for the analogous job is as follows: Workers get paid $80 per week plus an additional $10 for each item sold How many items must a worker sell over the course of a week in order to have the job at Company B to be to her advantage? Please give all possible answers We would like you to answer this question in three different ways
(a) First approach it numerically Make a salary table for each job
(b) Now approach the problem graphically Use numerical methods only for fine-tuning
(c) Finally, approach the problem algebraically: Let SA be the salary scheme at Company A and SBbe the salary scheme at Company B Write SAand SBeach as functions of x, the number of items sold per week Solve the required inequality
by solving the corresponding system of equations
Trang 65 10 15 20 100
220 280
items
dollars
(15, 220) (20, 280)
6 You’ve been presented with two different pay plans for the same job Plan A offers $12 per hour with overtime (hours above 40 per week) paying time and a half Plan B offers
$14 per hour with no overtime Let x denote the number of hours you work each week Let PA(x)give the weekly pay under plan A and PB(x)give the weekly pay under plan B
(a) What is the algebraic formula for PB(x)?
(b) What is the algebraic formula for PA(x)?
Note that you must define this function differently for x ≤ 40 and for x > 40 Check your answer and make sure that the pay for a 50-hour work week is $660 (c) i For what value(s) of x are the two plans equivalent?
ii For what values of x is plan B better?
(Hint: A good problem-solving strategy is to draw a graph so you can really
see what is going on.) (d) True or False:
i PB(x + y) = PB(x) + PB(y)
ii PA(x + y) = PA(x) + PA(y)
(Hint: If you are not sure how to approach a problem, a good strategy—
frequently used by mathematicians everywhere!—is to try a concrete case
If the statement is false for this special case, then you know the statement is definitely false If the statement holds for this special case, then the process
of working through the special case may help you determine whether the
statement holds in general.) Caution: Since the rule for PA(x)changes for
x >40, you need to check several cases If any case doesn’t hold, then the
statement is false
7 You’ve written a book and have two publishers interested in putting it out Both publishers anticipate selling the book for $20 The first publisher guarantees you a flat sum of $8000 for up to the first 10,000 copies sold and will pay 12% royalty for any copies sold in excess of 10,000 For instance, if 10,001 copies were sold, you would receive $8002.40 The second publisher offers a royalty of 10%
Let x be the number of books sold
Let A(x) give the income under plan A and B(x) give the income under plan B (a) What is the algebraic formula for A(x)?
(b) What is the algebraic formula for B(x)?
(c) i For what value(s) of x are the two plans equivalent?
ii For what values of x is plan B better?
iii For what values of x is plan A better?
Trang 78 An investment fund has two different investment options Option C, the more conser-vative option, puts 70% of the investor’s money into slow-growing reliable stocks and 30% of the money into high-risk stocks with high growth potential Option R, the riskier option, puts 60% of the money into high-risk stocks and 40% into low-growth stocks
If a client has $2 million invested in high-risk stocks and $3 million in low-risk stocks, how much of the client’s money is in option C and how much in option R?
9 Below is a graph of temperature, T , plotted as a function of time, t The temperature function is increasing on [0, 21] It is concave down on [0, 14] and concave up on [14, 21]
2 4 6 8 10 12 14 16 18 20 55
60 65 70
(4, 60)
(10, 63)
(18, 65) (20,68)
time (in hrs)
Temperature (in degrees F)
(a) On average, between hours 4 and 10, what is the rate of increase of temperature with respect to time? In other words, what is the average rate of change of temperature between hour 4 and hour 10?
(b) What is the average rate at which temperature is increasing between hour 18 and hour 20?
(c) Draw the secant line through the points on the graph where t = 4 and t = 10 Find the equation of this line
(d) Draw the secant line through the points on the graph where t = 18 and t = 20 Find the equation of this line
(e) Using your answer to part (d) and approximating the graph by the secant line, estimate the temperature at hour 21 Would you guess that the T -coordinate of the point on the secant line is slightly higher than the temperature at hour 21 or slightly lower?
10 After 3 miles of difficult climbing in the morning, a group of hikers has reached a plateau and they are confident they can maintain a steady pace for the next 10 miles After covering a total of 13 miles, they’ll set camp Twenty minutes after reaching the plateau, they’ve covered 113miles Express the total daily mileage as a function of t, where t is the number of hours spent hiking since they reached the plateau What is the domain of the function?
11 At 8:00 a.m., a long-distance runner has run 10 miles and is tiring She runs until 9:00 a.m.but runs more and more slowly throughout the hour By 9:00 she has run 16 miles (a) Sketch a possible graph of distance traveled versus time on the interval from 8:00
to 9:00 What are the key characteristics of this graph?
Trang 8(b) Suppose that at 8:00 a.m she is running at a speed of 9 miles per hour Find good upper and lower bounds for the total distance she has run by 8:30 a.m Explain your reasoning with both words and a graph
12 This problem focuses on the difference between being piecewise linear (made up of straight lines) and being locally linear (being approximately linear when magnified enough) Consider the functions f , g, and h below
f (x) = |x + 2| − 3 g(x) =
x for x ≤ 0
x2 for x > 0 h(x) = (x − 1)10+ 1 (a) Graph f , g, and h
(b) Specify all intervals for which the given function is linear (a straight line.)
i f
ii g iii h
(c) Specify the point(s) at which the given function is not locally linear (that is, where
it does not look like a straight line, no matter how much you zoom in)
i f
ii g iii h
13 As part of a conservation effort we want to buy a monogrammed mug for every student, staff, and faculty member in the mathematics department We check with several companies and get the following price quotes
Great Mugs will charge $20 just to place the order and then they charge an additional
$6 for each mug that we order
Name It will only charge $10 to process the order and has a varying scale depending
upon the number of mugs ordered For the first 20 mugs we order, the cost is $7 per mug; for the next 50 mugs, the cost is $6 per mug; and for all mugs after that, the cost
is $5 per mug
Let G(x) be the cost of ordering x mugs from Great Mugs.
Let N (x) be the cost of ordering x mugs from Name It.
(a) Graph G(x) and N (x)
(b) Write functions for G(x) and N (x)
(c) For which values of x is it cheaper to order from Great Mugs as compared to ordering from Name It?
(d) How much can the difference in prices between the two companies ever be if we place an order for the same number of mugs from each company?
Trang 9The Derivative Function
5.1 CALCULATING THE SLOPE OF A CURVE
AND INSTANTANEOUS RATE OF CHANGE
In Chapter 4 we looked at linear functions, functions characterized by a constant rate of change This characteristic is unique to linear functions; the rate of change of the output of any nonlinear function varies with the value of the independent variable
Consider, for example, the bucket calibration problem for a bucket as drawn in Figure 5.1 The change in height produced by adding one gallon of water to an empty bucket is greater than the change in height produced by adding the same amount of water to a partially filled bucket The more water in the bucket the less impact an additional gallon of water will have on the height For the bucket, the change in height produced by the addition of water is a function of the volume of water already in the bucket Similarly, for the conical flask the change in height
change in volumeratio depends upon volume.
bucket conical flask
volume height
height versus volume for the bucket
height
volume height versus volume for the conical flask
Figure 5.1
169
Trang 10We have looked at several examples in which a rate of change is a function of the
independent variable While we have not yet defined the instantaneous rate of change of a
function, we already have some notion of it For instance:
Velocity is the instantaneous rate of change of position over change in time When riding in a car, we look at the speedometer to determine our velocity at an instant
In Chapter 2 we analyzed a graph of a cyclist’s velocity plotted as a function of time Implicit in this graph is the idea that at each instant the cyclist’s velocity can be determined
In Chapter 3 we looked at graphs of the rates of flow of water in and out of a reservoir plotted as a function of time Again, implicit in these graphs is the idea that at each instant such a rate of change can be determined
For a given function we know how to calculate the average rate of change over an interval We’ll now tackle the problem of calculating an instantaneous rate of change To find an average rate of change we need two data points; an instant provides us with only one data point This is the fundamental challenge of differential calculus We need a strategy for approaching this problem
Problem-Solving Strategies
Look at a concrete problem and determine what methods of attack can be applied to the more abstract problem
Use the method of successive approximations Approximate what you’re looking for, determining upper and lower bounds if possible Then improve on the approximation Repeat this process until the approximation is good enough for your purposes or until you arrive at an exact answer
Let’s consider a rock dropped from a height of 256 feet From the moment the rock is dropped the fundamental forces acting on it are the force of gravity and the opposing force
of air resistance (We will consider only the force of gravity because air resistance in this situation is negligible.) The force of gravity results in a downward acceleration of the rock The rock’s speed will increase as it falls Its position, s, is a function of time Let’s set t = 0
to be the moment the rock is dropped and measure time in seconds Suppose we are given the following data:
t(time in seconds) s(position in feet)