2.2.1 Using a computer or calculator, estimate the following limits. Sketch the function. ...
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2.2.2 Using a computer or calculator, estimate the following limits. Sketch the function. ...
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2.2.3 Using a computer or calculator, estimate the following limits. Sketch the function. ...
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2.2.4
Using a computer or calculator, estimate the following limits. Sketch
the function. ... (the function is defined only for positive
values of x).
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2.2.5 Using a computer or calculator, estimate the following limits. Sketch the function. ...
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2.2.6 Using a computer or calculator, estimate the following limits. Sketch the function. ...
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2.2.7 Using a computer or calculator, estimate the following limits. Sketch the function. ...
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2.2.8 Using a computer or calculator, estimate the following limits. Sketch the function. ...
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2.2.9
Using the results from the earlier problems, find the combined limits
using Theorem 2.2. Say how the new function was built. ...
Reference Exercise 1). ... Reference Theorem 2.2 ...
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2.2.10
Using the results from the earlier problems, find the combined limits
using Theorem 2.2. Say how the new function was built. ...
Reference Exercise 2. ... Reference Theorem 2.2 ...
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2.2.11
Using the results from the earlier problems, find the combined limits
using Theorem 2.2. Say how the new function was built. ...
Reference Exercise 3. ... Reference Theorem 2.2 ...
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2.2.12
Using the results from the earlier problems, find the combined limits
using Theorem 2.2. Say how the new function was built. ...
Reference Exercise 4. ... (the function is defined only for
positive values of x). Reference Theorem 2.2 ...
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2.2.13 The given functions all have limits of 0 as x →
0+. For each function, find how close the input must be to 0 for the
output to be a) within 0.1 of 0, and b) within 0.01 of the limit. Sketch
a graph of each function for x < 1, and say which functions approach 0 quickly and which approach 0 slowly. ...
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2.2.14 The given functions all have limits of 0 as x →
0+. For each function, find how close the input must be to 0 for the
output to be a) within 0.1 of 0, and b) within 0.01 of the limit. Sketch
a graph of each function for x < 1, and say which functions approach 0 quickly and which approach 0 slowly. ...
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2.2.15 The given functions all have limits of 0 as x →
0+. For each function, find how close the input must be to 0 for the
output to be a) within 0.1 of 0, and b) within 0.01 of the limit. Sketch
a graph of each function for x < 1, and say which functions approach 0 quickly and which approach 0 slowly. ...
Get solution
2.2.16 The given functions all have limits of 0 as x →
0+. For each function, find how close the input must be to 0 for the
output to be a) within 0.1 of 0, and b) within 0.01 of the limit. Sketch
a graph of each function for x < 1, and say which functions approach 0 quickly and which approach 0 slowly. ...
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2.2.17 The given functions all have limits of ∞ as x →
0+. For each function, find how close the input must be to 0 for the
output to be a) greater than 10, and b) greater than 100. Sketch a graph
of each function for x < 1, and say which functions approach infinity quickly and which approach infinity slowly. ...
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2.2.18 The given functions all have limits of ∞ as x →
0+. For each function, find how close the input must be to 0 for the
output to be a) greater than 10, and b) greater than 100. Sketch a graph
of each function for x < 1, and say which functions approach infinity quickly and which approach infinity slowly. ...
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2.2.19 The given functions all have limits of ∞ as x →
0+. For each function, find how close the input must be to 0 for the
output to be a) greater than 10, and b) greater than 100. Sketch a graph
of each function for x < 1, and say which functions approach infinity quickly and which approach infinity slowly. ...
Get solution
2.2.20 The given functions all have limits of ∞ as x →
0+. For each function, find how close the input must be to 0 for the
output to be a) greater than 10, and b) greater than 100. Sketch a graph
of each function for x < 1, and say which functions approach infinity quickly and which approach infinity slowly. ...
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2.2.21 From the following pictures, find the left-hand and righthand limits as x approaches 1. ...
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2.2.22 From the following pictures, find the left-hand and right hand limits as x approaches 1. ...
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2.2.23 From the following pictures, find the left-hand and right hand limits as x approaches 1. ...
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2.2.24 From the following pictures, find the left-hand and right hand limits as x approaches 1. ...
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2.2.25 Find the average rate of change of the following functions as a function of Δx, and find the limit as Δx→ 0. Graph the function and indicate the rate of change on your graph. f ( x )=5x + 7 near x =0.
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2.2.26 Find the average rate of change of the following functions as a function of Δx, and find the limit as Δx→ 0. Graph the function and indicate the rate of change on your graph. f ( x )=5x + 7 near x =1.
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2.2.27 Find the average rate of change of the following functions as a function of Δx, and find the limit as Δx→ 0. Graph the function and indicate the rate of change on your graph. f ( x )= ...near x =0.
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2.2.28 Find the average rate of change of the following functions as a function of Δx, and find the limit as Δx→ 0. Graph the function and indicate the rate of change on your graph. f ( x ) = ... near x =1.
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2.2.29 Find the average rate of change of the following functions as a function of Δx, and find the limit as Δx→ 0. Graph the function and indicate the rate of change on your graph. ...
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2.2.30 Find the average rate of change of the following functions as a function of Δx, and find the limit as Δx→ 0. Graph the function and indicate the rate of change on your graph. ...
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2.2.31 Suppose
we are interested in measuring the properties of a substance at
temperature of absolute zero (which is 0 degrees Kelvin). However, we
cannot measure these properties directly because it is impossible to
reach absolute 0. Instead, properties are measured for small values of
the temperature T , measured in kelvins. For each of the following, find
a. The limit as ...
b. How close would we be to the limit if we measured the property at 2K?
c. How close would we be to the limit if we measured the property at 1K?
d. About how cold would the temperature have to be for the property to be within 1% of its limit? ...
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2.2.32 Suppose
we are interested in measuring the properties of a substance at
temperature of absolute zero (which is 0 degrees Kelvin). However, we
cannot measure these properties directly because it is impossible to
reach absolute 0. Instead, properties are measured for small values of
the temperature T , measured in kelvins. For each of the following, find
a. The limit as ...
b. How close would we be to the limit if we measured the property at 2K?
c. How close would we be to the limit if we measured the property at 1K?
d. About how cold would the temperature have to be for the property to be within 1% of its limit? The hardness H(T ) follows H(T ) = ...
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2.2.33 For each of the following populations, the instantaneous rate of change of the population size at t =0 is exactly 1.0million bacteria per hour. If you computed the average rate of change between t = 0 and t =Δt, how small would Δt have to be before your value was within 1% of the instantaneous rate of change? ...
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2.2.34 For each of the following populations, the instantaneous rate of change of the population size at t =0 is exactly 1.0million bacteria per hour. If you computed the average rate of change between t = 0 and t =Δt, how small would Δt have to be before your value was within 1% of the instantaneous rate of change? ...
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2.2.35 For each of the following populations, the instantaneous rate of change of the population size at t =0 is exactly 1.0million bacteria per hour. If you computed the average rate of change between t = 0 and t =Δt, how small would Δt have to be before your value was within 1% of the instantaneous rate of change? b ( t ) = ... (this cannot be solved algebraically).
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2.2.36 For each of the following populations, the instantaneous rate of change of the population size at t =0 is exactly 1.0million bacteria per hour. If you computed the average rate of change between t = 0 and t =Δt, how small would Δt have to be before your value was within 1% of the instantaneous rate of change? b ( t ) = sin(t) (this cannot be solved algebraically).
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2.2.37 Scientifically,
two quantities are close if it requires an accurate measuring device to
detect the difference. In real life, accuracy costs money. How much
would it cost to measure the differences in the following circumstances? A
piano tuner is trying to get the note A on a piano to have a frequency
of exactly 440 hertz (H), or cycles per second. An electronic tuner
capable of detecting a difference of x cycles per second costs ...dollars.
a. How much would it cost to make sure the note was within 1.0 H of 440?
b. How much would it cost to make sure the note was within 0.1 H of 440?
c. How much would it cost to make sure the note was within 0.01 H of 440?
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2.2.38
Scientifically,
two quantities are close if it requires an accurate measuring device to
detect the difference. In real life, accuracy costs money. How much
would it cost to measure the differences in the following circumstances?
The army is developing satellite-based targeting systems. A system
that can send a missile within y meters of its target costs ...million dollars.
a. How much would it cost to hit within 10 m?
b. How much would it cost to hit within 1 m?
c. How much would it cost to hit within 1 cm?
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2.2.39 Scientifically,
two quantities are close if it requires an accurate measuring device to
detect the difference. In real life, accuracy costs money. How much
would it cost to measure the differences in the following circumstances? Suppose a body has temperature B and is cooling toward room temperature of 20?C according to the function B(t) = 20 + ...where t is
measured in hours. A $10 thermocouple can detect a difference of 0.1?C,
a $100 thermocouple can detect a difference of 0.01?C, and so forth.
a. How much would it cost to detect the difference after 1 h?
b. How much would it cost to detect the difference after 5 h?
c. How much would it cost to detect the difference after 10 h?
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2.2.40 Scientifically,
two quantities are close if it requires an accurate measuring device to
detect the difference. In real life, accuracy costs money. How much
would it cost to measure the differences in the following circumstances? Some
dangerously radioactive and toxic radium was dumped in the desert in
1950. It has a half-life of 50 years, and the initial level of
radioactivity was r = 10.0 rads. Nobody remembers where it
was. How much will it cost to find it in the following years if
detecting radioactivity costs ...thousand dollars?
a. How much would it cost to find in the year 2000?
b. How much would it cost to find in the year 2050?
c. How much would it cost to find in the year 2130?
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2.2.41 Scientifically,
a quantity is large if it requires a tough measuring device to assess.
In real life, toughness costs money. How much would it cost to measure
the value in the following circumstances? We are interested in
measuring the pressure at different depths below the surface of the
ocean. Pressure increases by approximately 1 atmosphere for every 10 m
of depth below the surface (for example, at a depth of 20 m, there are
approximately 3 atmospheres of pressure, 2 due the ocean and 1 to the
atmosphere itself). Measuring a pressure of x atmospheres without crushing the device costs ...dollars.
a. How much would it cost to measure the pressure 100 m down?
b. How much would it cost to measure the pressure 1000 m down?
c. How much would it cost to measure the pressure 5000 m down?
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2.2.42 Scientifically,
a quantity is large if it requires a tough measuring device to assess.
In real life, toughness costs money. How much would it cost to measure
the value in the following circumstances? Solar scientists want
to measure the temperature inside the sun by sending in probes. Imagine
that temperature increases by 1 million?C for every 10,000 km below the
surface. A probe that can handle a temperature of x million degrees costs ...million dollars.
a. How much would it cost to measure the temperature 10,000 km down
b. How much would it cost to measure the temperature 100,000 km down?
c. How much would it cost to measure the temperature 200,000 km down?
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2.2.43 There are various ways to find the smallest and largest numbers your calculator or computer can handle.
a. Try doubling some number until you get an overflow.
b. Try halving some number until you get an underflow.
c. Compute 1.0 + ε for smaller and smaller values of ε. At what point does your calculator return 1.0 as the answer?
d. Compare the answers of b and c.
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