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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
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AI-Robot
HOW CAN MAN ORGAN FUNCTION
the diagram of the digestive system
allimentary cannel
Ogenrwot
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
Genetics is the study of heredity
Misack
how does twins formed?
Misack
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
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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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