2.7.1 On the figures, label
a. One critical point.
b. One point with a positive derivative.
c. One point with a negative derivative.
d. One point with a positive second derivative.
e. One point with a negative second derivative.
f. One point of inflection. ...
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2.7.2 On the figures, label
a. One critical point.
b. One point with a positive derivative.
c. One point with a negative derivative.
d. One point with a positive second derivative.
e. One point with a negative second derivative.
f. One point of inflection. ...
Get solution
2.7.3 On the figures, label
a. One critical point.
b. One point with a positive derivative.
c. One point with a negative derivative.
d. One point with a positive second derivative.
e. One point with a negative second derivative.
f. One point of inflection. ...
Get solution
2.7.4 On the figures, label
a. One critical point.
b. One point with a positive derivative.
c. One point with a negative derivative.
d. One point with a positive second derivative.
e. One point with a negative second derivative.
f. One point of inflection. ...
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2.7.5 Draw graphs of functions with the following properties. A function with a positive, increasing derivative.
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2.7.6 Draw graphs of functions with the following properties. A function with a positive, decreasing derivative.
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2.7.7
Draw graphs of functions with the following properties. A function
with a negative, increasing (becoming less negative) derivative.
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2.7.8
Draw graphs of functions with the following properties. A function
with a negative, decreasing (becoming more negative) derivative.
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2.7.9
Although there is no easy way to recognize all points with positive
or negative third derivative, it is possible for some points (usually
points of inflection). On the figure for Exercise 1, find one point
with negative third derivative. figure for Exercise 1 ...
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2.7.10
Although there is no easy way to recognize all points with positive
or negative third derivative, it is possible for some points (usually
points of inflection). On the figure for Exercise 2, find one point
with positive third derivative. figure for Exercise 2 ...
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2.7.11 Find the first and second derivatives of the following functions. ...
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2.8.12 Compute the first and second derivatives of the following functions. ...
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2.7.13 Find the first and second derivatives of the following functions. ...
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2.7.14 Find the first and second derivatives of the following functions. ...
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2.7.15 Find the first and second derivatives of the following functions. F ( z )= z(1 + z)(2 + z).
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2.7.16 Find the first and second derivatives of the following functions. ...
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2.7.17 Find the first and second derivatives of the following functions. ...
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2.7.18 Find the first and second derivatives of the following functions. ...
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2.7.19 Find the first and second derivatives of the following functions and use them to sketch a graph. ...
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2.7.20 Find the first and second derivatives of the following functions and use them to sketch a graph. ...
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2.7.21 Find the first and second derivatives of the following functions and use them to sketch a graph. h ( x )=(1 − x)(2 − x)(3 − x).
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2.7.22 Find the first and second derivatives of the following functions and use them to sketch a graph. ...
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2.7.22 Find the first and second derivatives of the following functions and use them to sketch a graph. ...
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2.7.24 Find the first and second derivatives of the following functions and use them to sketch a graph. ...
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2.7.25 Find the first and second derivatives of the following functions and use them to sketch a graph. ...
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2.7.26 Find the first and second derivatives of the following functions and use them to sketch a graph. ...
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2.7.27 Some higher derivatives can be found without a lot of calculation. Find the 10th derivative of ...
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2.7.28 Some higher derivatives can be found without a lot of calculation. Describe the graph of the 5th derivative of ...
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2.7.29 Some higher derivatives can be found without a lot of calculation. Is the eighth derivative of ... ... positive or negative?
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2.7.30 Some higher derivatives can be found without a lot of calculation. Find the fifth derivative of x(1 + x)(2 + x)(3 + x)(4 + x).
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2.7.31 The
following equations give the positions as functions of time of objects
tossed from towers in various exotic solar system locations. For each,
a. Find the velocity and the acceleration of this object.
b. Sketch a graph of the position for 0 ≤ t ≤ 3.
c.
How high was the tower? Which way was the object thrown? How does the
acceleration compare with that on earth ...? An object on Saturn that
follows p(t)=−5.2 ... − 2.0t + 50.0.
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2.7.32 The
following equations give the positions as functions of time of objects
tossed from towers in various exotic solar system locations. For each,
a. Find the velocity and the acceleration of this object.
b. Sketch a graph of the position for 0 ≤ t ≤ 3.
c.
How high was the tower? Which way was the object thrown? How does the
acceleration compare with that on earth ...? An object on the Sun that
follows p(t)=−137... + 20.0t + 500.0.
Get solution
2.7.33 The
following equations give the positions as functions of time of objects
tossed from towers in various exotic solar system locations. For each,
a. Find the velocity and the acceleration of this object.
b. Sketch a graph of the position for 0 ≤ t ≤ 3.
c.
How high was the tower? Which way was the object thrown? How does the
acceleration compare with that on earth ...? An object on Pluto that
follows p(t)=−0.325... − 20.0t + 500.0.
Get solution
2.7.34 The
following equations give the positions as functions of time of objects
tossed from towers in various exotic solar system locations. For each,
a. Find the velocity and the acceleration of this object.
b. Sketch a graph of the position for 0 ≤ t ≤ 3.
c.
How high was the tower? Which way was the object thrown? How does the
acceleration compare with that on earth ...? An object on Mercury that
follows ...
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2.7.35 The
total mass is the product of the following functions for mass and
number as functions of time in years (based on Section 2.6, Exercises
23–26). Find the second derivative of each and check your graph. The population P is ...and the weight per person W(t) is W(t)=80 − 0.5t (based on Section 2.6, Exercise 23).
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2.7.36 The
total mass is the product of the following functions for mass and
number as functions of time in years (based on Section 2.6, Exercises
23–26). Find the second derivative of each and check your graph. The population P is ...and the weight per person W(t) is W(t)=80 + 0.5t (based on Section 2.6, Exercise 24).
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2.7.37 The
total mass is the product of the following functions for mass and
number as functions of time in years (based on Section 2.6, Exercises
23–26). Find the second derivative of each and check your graph. The population P is ...and the weight per person W(t) is W(t)=80 − 0.5t (based on Section 2.6, Exercise 25).
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2.7.38 The
total mass is the product of the following functions for mass and
number as functions of time in years (based on Section 2.6, Exercises
23–26). Find the second derivative of each and check your graph. The population P is ...and the weight per person W(t) is ... (based on Section 2.6, Exercise 26).
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2.7.39 The
following graphs show the position of a roller coaster as a function of
time. When is the roller coaster going most quickly? When is it
accelerating most quickly? When is it decelerating most quickly? ...
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2.7.40 The
following graphs show the position of a roller coaster as a function of
time. When is the roller coaster going most quickly? When is it
accelerating most quickly? When is it decelerating most quickly? ...
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2.7.41 In
a model of a growing population, we find the new population by
multiplying the old population by the per capita production. For each
case, find the second derivative of the new population as a function of
the old population and sketch a graph. Per capita production is 2.0 ... Consider values of ... less than 1000.
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2.7.42 In
a model of a growing population, we find the new population by
multiplying the old population by the per capita production. For each
case, find the second derivative of the new population as a function of
the old population and sketch a graph. Per capita production is ... Consider values of ...less than 1000.
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2.7.43 We can use the second derivative to study Hill functions ... Find the second derivative of the Hill function with n = 1 and describe the curvature of the graph.
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2.7.44 We can use the second derivative to study Hill functions ... Find the second derivative of the Hill function with n = 2 and describe the curvature of the graph.
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2.7.45
In the current universe, acceleration due to gravity is constant, so
that position follows a differential equation rather like ...
when g points
in the downward direction. One can imagine a universe where gravity
changed over time, making objects accelerate according to ... for some power n. Set g = 10 m/s.
a. Find a solution of the normal differential equation.
b. Find solutions of the modified differential equation for different values of n. Would objects fall faster or slower?
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2.7.46
Have your computer find all critical points and points of inflection
of the function ... Show that these match what you see on a
graph.
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