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Calculus volume 1
Derivatives
Defining the derivative
For the following functions,
use
[link] to find the slope of the tangent line
m
tan
=
f
′
(
a
)
, and
find the equation of the tangent line to
f at
x
=
a
.
For the following functions
y
=
f
(
x
)
, find
f
′
(
a
) using
[link] .
For the following exercises, given the function
y
=
f
(
x
)
,
find the slope of the secant line
P
Q for each point
Q
(
x
,
f
(
x
)
) with
x value given in the table.
Use the answers from a. to estimate the value of the slope of the tangent line at
P
.
Use the answer from b. to find the equation of the tangent line to
f at point
P
.
[T]
f
(
x
)
=
x
2
+
3
x
+
4
,
P
(
1
,
8
) (Round to
6 decimal places.)
x
Slope
m
P
Q
x
Slope
m
P
Q
1.1
(i)
0.9
(vii)
1.01
(ii)
0.99
(viii)
1.001
(iii)
0.999
(ix)
1.0001
(iv)
0.9999
(x)
1.00001
(v)
0.99999
(xi)
1.000001
(vi)
0.999999
(xii)
a.
(i)
5.100000
,
(ii)
5.010000
,
(iii)
5.001000
,
(iv)
5.000100
,
(v)
5.000010
,
(vi)
5.000001
,
(vii)
4.900000
,
(viii)
4.990000
,
(ix)
4.999000
,
(x)
4.999900
,
(xi)
4.999990
,
(x)
4.999999 b.
m
tan
=
5 c.
y
=
5
x
+
3
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[T]
f
(
x
)
=
x
+
1
x
2
−
1
,
P
(
0
,
−1
)
x
Slope
m
P
Q
x
Slope
m
P
Q
0.1
(i)
−0.1
(vii)
0.01
(ii)
−0.01
(viii)
0.001
(iii)
−0.001
(ix)
0.0001
(iv)
−0.0001
(x)
0.00001
(v)
−0.00001
(xi)
0.000001
(vi)
−0.000001
(xii)
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[T]
f
(
x
)
=
10
e
0.5
x
,
P
(
0
,
10
) (Round to
4 decimal places.)
x
Slope
m
P
Q
−0.1
(i)
−0.01
(ii)
−0.001
(iii)
−0.0001
(iv)
−0.00001
(v)
−0.000001
(vi)
a.
(i)
4.8771
,
(ii)
4.9875
(iii)
4.9988
,
(iv)
4.9999
,
(v)
4.9999
,
(vi)
4.9999 b.
m
tan
=
5 c.
y
=
5
x
+
10
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[T] For the following position functions
y
=
s
(
t
)
, an object is moving along a straight line, where
t is in seconds and
s is in meters. Find
the simplified expression for the average velocity from
t
=
2 to
t
=
2
+
h
;
the average velocity between
t
=
2 and
t
=
2
+
h
, where
(i)
h
=
0.1
,
(ii)
h
=
0.01
,
(iii)
h
=
0.001
, and
(iv)
h
=
0.0001
; and
use the answer from a. to estimate the instantaneous velocity at
t
=
2 second.
For the following exercises, use the limit definition of derivative to show that the derivative does not exist at
x
=
a for each of the given functions.
[T] The position in feet of a race car along a straight track after
t seconds is modeled by the function
s
(
t
)
=
8
t
2
−
1
16
t
3
.
Find the average velocity of the vehicle over the following time intervals to four decimal places:
[4, 4.1]
[4, 4.01]
[4, 4.001]
[4, 4.0001]
Use a. to draw a conclusion about the instantaneous velocity of the vehicle at
t
=
4 seconds.
a.
(i)
61.7244 ft/s,
(ii)
61.0725 ft/s
(iii)
61.0072 ft/s
(iv)
61.0007 ft/s b. At
4 seconds the race car is traveling at a rate/velocity of
61 ft/s.
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[T] The distance in feet that a ball rolls down an incline is modeled by the function
s
(
t
)
=
14
t
2
, where
t is seconds after the ball begins rolling.
Find the average velocity of the ball over the following time intervals:
[5, 5.1]
[5, 5.01]
[5, 5.001]
[5, 5.0001]
Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at
t
=
5 seconds. Got questions? Get instant answers now!
Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by
s
=
f
(
t
) and
s
=
g
(
t
)
, where
s is measured in feet and
t is measured in seconds.
Which vehicle has traveled farther at
t
=
2 seconds?
What is the approximate velocity of each vehicle at
t
=
3 seconds?
Which vehicle is traveling faster at
t
=
4 seconds?
What is true about the positions of the vehicles at
t
=
4 seconds?
a. The vehicle represented by
f
(
t
)
, because it has traveled
2 feet, whereas
g
(
t
) has traveled
1 foot. b. The velocity of
f
(
t
) is constant at
1 ft/s, while the velocity of
g
(
t
) is approximately
2 ft/s. c. The vehicle represented by
g
(
t
)
, with a velocity of approximately
4 ft/s. d. Both have traveled
4 feet in
4 seconds.
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[T] The total cost
C
(
x
)
, in hundreds of dollars, to produce
x jars of mayonnaise is given by
C
(
x
)
=
0.000003
x
3
+
4
x
+
300
.
Calculate the average cost per jar over the following intervals:
[100, 100.1]
[100, 100.01]
[100, 100.001]
[100, 100.0001]
Use the answers from a. to estimate the average cost to produce
100 jars of mayonnaise. Got questions? Get instant answers now!
[T] For the function
f
(
x
)
=
x
3
−
2
x
2
−
11
x
+
12
, do the following.
Use a graphing calculator to graph
f in an appropriate viewing window.
Use the ZOOM feature on the calculator to approximate the two values of
x
=
a for which
m
tan
=
f
′
(
a
)
=
0
.
a.
b.
a
≈
−
1.361
,
2.694
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[T] For the function
f
(
x
)
=
x
1
+
x
2
, do the following.
Use a graphing calculator to graph
f in an appropriate viewing window.
Use the ZOOM feature on the calculator to approximate the values of
x
=
a for which
m
tan
=
f
′
(
a
)
=
0
. Got questions? Get instant answers now!
Suppose that
N
(
x
) computes the number of gallons of gas used by a vehicle traveling
x miles. Suppose the vehicle gets
30 mpg.
Find a mathematical expression for
N
(
x
)
.
What is
N
(
100
)? Explain the physical meaning.
What is
N
′
(
100
)
? Explain the physical meaning.
a.
N
(
x
)
=
x
30 b.
∼
3.3 gallons. When the vehicle travels
100 miles, it has used
3.3 gallons of gas. c.
1
30
. The rate of gas consumption in gallons per mile that the vehicle is achieving after having traveled
100 miles.
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[T] For the function
f
(
x
)
=
x
4
−
5
x
2
+
4
, do the following.
Use a graphing calculator to graph
f in an appropriate viewing window.
Use the
nDeriv function, which numerically finds the derivative, on a graphing calculator to estimate
f
′
(
−2
)
,
f
′
(
−0.5
)
,
f
′
(
1.7
)
, and
f
′
(
2.718
)
. Got questions? Get instant answers now!
[T] For the function
f
(
x
)
=
x
2
x
2
+
1
, do the following.
Use a graphing calculator to graph
f in an appropriate viewing window.
Use the
nDeriv function on a graphing calculator to find
f
′
(
−4
)
,
f
′
(
−2
)
,
f
′
(
2
)
, and
f
′
(
4
)
.
a.
b.
−0.028
,
−0.16
,
0.16
,
0.028
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Questions & Answers
how does Neisseria cause meningitis
is the branch of biology that deals with the study of microorganisms.
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
How bacteria create energy to survive?
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
which site have a normal flora
Many sites of the body have it
Skin
Nasal cavity
Oral cavity
Gastro intestinal tract
Safaa
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Binomial nomenclature
adeolu
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Source:
OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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