2.5.1 Find the derivatives of the following power functions. ...
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2.5.2 Find the derivatives of the following power functions. ...
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2.5.3 Find the derivatives of the following power functions. ...
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2.5.4 Find the derivatives of the following power functions. ...
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2.5.5 Find the derivatives of the following power functions. ...
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2.5.6 Find the derivatives of the following power functions. ...
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2.5.7 Find the derivatives of the following power functions. ...
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2.5.8 Find the derivatives of the following power functions. ...
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2.5.9
Find the derivatives of the following polynomial functions. Say where
you used the sum, constant product, and power rules. ...
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2.5.10
Find the derivatives of the following polynomial functions. Say where
you used the sum, constant product, and power rules. ...
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2.5.11
Find the derivatives of the following polynomial functions. Say where
you used the sum, constant product, and power rules. ...
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2.5.12
Find the derivatives of the following polynomial functions. Say where
you used the sum, constant product, and power rules. ...
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2.5.13 Use the binomial theorem to compute the following. ...
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2.5.14 Use the binomial theorem to compute the following. ...
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2.5.15 Use the binomial theorem to compute the following. ...
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2.5.16 Use the binomial theorem to compute the following. ...
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2.5.17 Use the derivative to sketch a graph of each of the following functions. ...
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2.5.18 Use the derivative to sketch a graph of each of the following functions. ...
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2.5.19 Use the derivative to sketch a graph of each of the following functions. ...
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2.5.20 Use the derivative to sketch a graph of each of the following functions. ...
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2.5.20 Use the derivative to sketch a graph of each of the following functions. ...
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2.5.22 Try to guess functions that have the following as their derivatives. 2x
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2.5.23 Try to guess functions that have the following as their derivatives. ...
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2.5.24 Try to guess functions that have the following as their derivatives. ...
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2.5.25 Try to guess functions that have the following as their derivatives. ...
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2.5.26 Try to guess functions that have the following as their derivatives. ...
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2.5.27
Find the derivatives of the following functions. In its early phase,
the number of AIDS cases in the United States grew approximately
according to a cubic equation, ...where t is measured in years since the beginning of the epidemic in 1972. Find and interpret the derivative.
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2.5.28 Find the derivatives of the following functions. ... where F represents the flow in cubic centimeters per second through a pipe of radius r . If r = 1, how much will a small increase in radius change the flow (try it with Δr = 0.1)? If r = 2, how will a small increase in radius change the flow?
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2.5.29 Find the derivatives of the following functions. The area of a circle as a function of radius is ...
with area measured in cm2 and radius measured in centimeters. Find the
derivative of area with respect to radius. On a geometric diagram,
illustrate the area corresponding to Δ A = A(r + Δr ) − A(r ). What is a geometric interpretation of the derivative? Do the units make sense?
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2.5.30 Find the derivatives of the following functions. The volume of a sphere as a function of radius is V (r )= .... Find the derivative of volume with respect to radius. On a geometric diagram, illustrate the volume corresponding to ΔV = V (r + Δr ) − V (r ). What is a geometric interpretation of this derivative? Do the units make sense?
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2.5.31 One
car is towing another using a rigid 50-ft pole. Sketch the positions
and speeds of the two cars as functions of time in the following
circumstances. The car starts from a stop, slowly speeds up, cruises for a while, and then abruptly stops.
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2.5.32
One
car is towing another using a rigid 50-ft pole. Sketch the positions
and speeds of the two cars as functions of time in the following
circumstances. The car starts from a stop, goes slowly in reverse for a
short time, stops, goes forward slowly, and then more quickly.
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2.5.33
A passenger is traveling on a luxury train that is moving west at 80
mph (miles per hour). The passenger starts running east at 10 mph. What
is her velocity relative to the ground?
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2.5.34 A passenger is traveling on a luxury train that is moving west at 80 mph (miles per hour). While
running, the passenger flips a dinner roll over her shoulder (west) at
25 mph. What is the velocity of the roll relative to the train? What is
the velocity of the roll relative to the ground?
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2.5.35 A passenger is traveling on a luxury train that is moving west at 80 mph (miles per hour). A
roll weevil jumps east off the roll at 5 mph. What is the velocity of
the roll relative to the passenger? What is he velocity of the roll
relative to the train? What is the velocity of the roll relative to the
ground?
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2.5.36 A passenger is traveling on a luxury train that is moving west at 80 mph (miles per hour). A
roll weevil flea jumps west off the roll weevil at 15 mph. What is the
velocity of the roll weevil flea relative to the roll? What is the
velocity of the roll weevil flea relative to the passenger? What is the
velocity of the roll weevil flea relative to the train? What is the
velocity of the roll weevil flea relative to the ground?
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2.5.37 Each of the following measurements is the sum of two components.
a. Find the formula for the sum.
b. Find the derivative of each component. What are the units?
c. Find the derivative of the sum and check that the sum rule worked.
d. Describe in words what is happening.
e. Sketch a graph of each component and the total as functions of time. A population of bacteria consists of two types a and b. The first follows a(t)=1 + ..., and the second follows b(t)= 1 − 2t + ...where populations are measured in millions and time is measured in hours. The total population is P(t) = a(t) + b(t).
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2.5.38 Each of the following measurements is the sum of two components.
a. Find the formula for the sum.
b. Find the derivative of each component. What are the units?
c. Find the derivative of the sum and check that the sum rule worked.
d. Describe in words what is happening.
e. Sketch a graph of each component and the total as functions of time. ...
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2.5.39 An object tossed upward at 10 m/s from a height of 100 m has distance above the ground of ... where a is
the acceleration of gravity. For each of the following planets with the
given acceleration, find the time when the object reaches a critical
point, how high it gets, the time when it hits the ground, and the speed
of the object at that time. Sketch the position of the object as a
function of time. On Earth, where a = 9.78 ...
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2.5.40 An object tossed upward at 10 m/s from a height of 100 m has distance above the ground of ... where a is
the acceleration of gravity. For each of the following planets with the
given acceleration, find the time when the object reaches a critical
point, how high it gets, the time when it hits the ground, and the speed
of the object at that time. Sketch the position of the object as a
function of time. On the moon, where a = 1.62 ....
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2.5.41 An object tossed upward at 10 m/s from a height of 100 m has distance above the ground of ... where a is
the acceleration of gravity. For each of the following planets with the
given acceleration, find the time when the object reaches a critical
point, how high it gets, the time when it hits the ground, and the speed
of the object at that time. Sketch the position of the object as a
function of time. On Jupiter, where a = 22.88 ....
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2.5.42 An object tossed upward at 10 m/s from a height of 100 m has distance above the ground of ... where a is
the acceleration of gravity. For each of the following planets with the
given acceleration, find the time when the object reaches a critical
point, how high it gets, the time when it hits the ground, and the speed
of the object at that time. Sketch the position of the object as a
function of time. On Mars’ moon Deimos, where ...
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2.5.43 Try different power functions of the form ... to guess solutions of the following differential equations describing the size of an organism. Find the size at t = 1 and t = 2. Can you explain why some grow so much faster than others? ...
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2.5.44 Try different power functions of the form ... to guess solutions of the following differential equations describing the size of an organism. Find the size at t = 1 and t = 2. Can you explain why some grow so much faster than others? ...
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2.5.45 Try different power functions of the form ... to guess solutions of the following differential equations describing the size of an organism. Find the size at t = 1 and t = 2. Can you explain why some grow so much faster than others? ...
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2.5.46 Try different power functions of the form ... to guess solutions of the following differential equations describing the size of an organism. Find the size at t = 1 and t = 2. Can you explain why some grow so much faster than others? ...
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2.5.47 Consider the polynomials ...
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