The function
𝑠 𝑥 = 60(𝑥 − 1)! + 60
represents the distance, s, in
miles, traveled by some object as
a function of time, t, in hours.
Use this representation to
respond to the following
questions.
MAT 145 – Calculus 1
140
Section
2.1: Position and Velocity
130
110
100
s:
distance
traveled
(miles)
90
80
70
60
50
40
30
1. Calculate the value of s(1.5) and
describe its meaning for this
context.
–0.4
s(x) = (60)·(x – 1)3 + 60
120
20
10
–0.2
0.2
–10
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x: time (hrs)
2. For the 2-hour time span
represented here, how far did the object travel? Describe at least two ways to determine this.
3. Calculate the average velocity of this object for the 2-hour time span shown here. Show your complete
calculation.
4. How far did the object travel for the time interval 1.2 ≤ t ≤ 1.8? Calculate the object’s average velocity
for this time interval. Show your complete calculation.
5. How far did the object travel for the time interval a ≤ t ≤ b, with a and b both within the interval 0 ≤ t ≤ 2
and a < b? Calculate the object’s average velocity for the time interval a ≤ t ≤ b. Describe your
calculation.
6. The fixed point P(7, 2) lies on the curve 𝑦 = 𝑥 − 3.
(a) If a moving point Q has coordinates 𝑥, 𝑥 − 3 , use your calculator to determine the slope of the
secant line 𝑃𝑄 (correct to six decimal places) for each of the following values of x.
(i)
x = 7.5
(ii) x = 7.1
(iii) x = 7.01
(iv) x = 7.001
(v) x = 7.0001
(vi) x = 6.9999
(vii) x = 6.999
(viii) x = 6.99
(ix) x = 6.9
(x) x = 6.5
(b) Study your results in (a). State a prediction for the slope of the line tangent to 𝑦 = 𝑥 − 3
at point P. Write a sentence or two to describe the basis for your prediction.
(c) Using the slope you determined in (b), write an equation for the line tangent to 𝑦 = 𝑥 − 3
at point P. Express your equation in the form y = mx + b.
I) The table here shows residential electrical costs for a home in
Bloomington, IL, for each month of 2016. Using these data,
calculate each of the following values for 2016 electricity costs:
a)
b)
c)
d)
e)
f)
Month (2016)
January
February
March
April
May
June
July
August
September
October
November
December
the total cost for electricity
the average monthly cost
the average daily cost
the average hourly cost
the average cost per minute
the average cost per second
g) Describe the process you used to determine averages (b) through (f).
Cost of Electricity
$172.19
$123.62
$144.64
$190.54
$205.78
$199.47
$134.21
$153.47
$176.75
$186.24
$197.21
$227.84
h) Describe how you would determine the 2016 average electricity cost for any instant in time.
II) The figure here shows the number of occupied beds in a regional hospital for each day of a recent
month. a) On what September
date(s) were the most
beds occupied? the
least?
c) Calculate the average
daily increase in
occupancy for the time
period Sept 8th to
Sept 14th.
Number of Beds Occupied b) By how many beds did
the occupancy decline
from Sept 3rd to
Sept 8th? Calculate the
average occupancy for
this time period.
Number of Hospital Beds Occupied: September 70 60 62 56 60 54 48 50 60 58 56 47 44 43 50 44 46 51 48 42 51 47 47 22 25 46 38 40 55 56 54 55 52 40 40 40 30 20 10 0 1 4 7 10 13 16 19 September Date (1 through 30) d) By how many beds per
day is the occupancy
changing on Sept 27th? Describe how you determined that value.
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