2.3.1 Describe
how the following functions are built out of the basic continuous
functions. Identify points where they might not be continuous. l ( t )=5t + 6.
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2.3.2 Describe
how the following functions are built out of the basic continuous
functions. Identify points where they might not be continuous. ...
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2.3.3 Describe
how the following functions are built out of the basic continuous
functions. Identify points where they might not be continuous. ...
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2.3.4 Describe
how the following functions are built out of the basic continuous
functions. Identify points where they might not be continuous. ...
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2.3.5 Describe
how the following functions are built out of the basic continuous
functions. Identify points where they might not be continuous. ...
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2.3.6 Describe
how the following functions are built out of the basic continuous
functions. Identify points where they might not be continuous. ...
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2.3.7 Describe
how the following functions are built out of the basic continuous
functions. Identify points where they might not be continuous. ...
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2.3.8 Describe
how the following functions are built out of the basic continuous
functions. Identify points where they might not be continuous. ...
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2.3.9 Describe
how the following functions are built out of the basic continuous
functions. Identify points where they might not be continuous. ...
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2.3.10 Describe
how the following functions are built out of the basic continuous
functions. Identify points where they might not be continuous. ...
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2.3.11 Using
the given functions, find the limits by plugging in (if possible). Say
whether the limit is infinity or negative infinity. Compute the value of
the function 0.1 and 0.01 above and below the limiting argument to see
if your answer is correct. ... reference Exercise 1 l ( t )=5t + 6.
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2.3.12 Using
the given functions, find the limits by plugging in (if possible). Say
whether the limit is infinity or negative infinity. Compute the value of
the function 0.1 and 0.01 above and below the limiting argument to see
if your answer is correct. ... reference Exercise 2 ...
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2.3.13 Using
the given functions, find the limits by plugging in (if possible). Say
whether the limit is infinity or negative infinity. Compute the value of
the function 0.1 and 0.01 above and below the limiting argument to see
if your answer is correct. ... reference Exercise 3 ...
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2.3.14 Using
the given functions, find the limits by plugging in (if possible). Say
whether the limit is infinity or negative infinity. Compute the value of
the function 0.1 and 0.01 above and below the limiting argument to see
if your answer is correct. ... reference Exercise 4 ...
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2.3.15 Using
the given functions, find the limits by plugging in (if possible). Say
whether the limit is infinity or negative infinity. Compute the value of
the function 0.1 and 0.01 above and below the limiting argument to see
if your answer is correct. ... reference Exercise 5 ...
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2.3.16 Using
the given functions, find the limits by plugging in (if possible). Say
whether the limit is infinity or negative infinity. Compute the value of
the function 0.1 and 0.01 above and below the limiting argument to see
if your answer is correct. ... reference Exercise 4 ...
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2.3.17 Using
the given functions, find the limits by plugging in (if possible). Say
whether the limit is infinity or negative infinity. Compute the value of
the function 0.1 and 0.01 above and below the limiting argument to see
if your answer is correct. ... reference Exercise 5 ...
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2.3.18 Using
the given functions, find the limits by plugging in (if possible). Say
whether the limit is infinity or negative infinity. Compute the value of
the function 0.1 and 0.01 above and below the limiting argument to see
if your answer is correct. ... reference Exercise 6 ...
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2.3.19 Using
the given functions, find the limits by plugging in (if possible). Say
whether the limit is infinity or negative infinity. Compute the value of
the function 0.1 and 0.01 above and below the limiting argument to see
if your answer is correct. ... reference Exercise 9 ...
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2.3.20 Using
the given functions, find the limits by plugging in (if possible). Say
whether the limit is infinity or negative infinity. Compute the value of
the function 0.1 and 0.01 above and below the limiting argument to see
if your answer is correct. ... reference Exercise 8 ...
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2.3.21
For the following functions, find the input tolerance necessary to
achieve the given output tolerance. How close must the input be to x = 0 for f (x) = x + 2 to be within 0.1 of 2?
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2.3.21
For the following functions, find the input tolerance necessary to
achieve the given output tolerance. How close must the input be to x = 0 for f (x) = x + 2 to be within 0.1 of 2?
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2.3.23
For the following functions, find the input tolerance necessary to
achieve the given output tolerance. How close must the input be to x = 1 for f (x)= ... to be within 0.1 of 1?
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2.3.24
For the following functions, find the input tolerance necessary to
achieve the given output tolerance. How close must the input be to x = 2 for f (x)= 5 ... to be within 0.1 of 20?
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2.3.23
For the following functions, find the input tolerance necessary to
achieve the given output tolerance. How close must the input be to x = 1 for f (x)= ... to be within 0.1 of 1?
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2.3.26
For the following functions, find the input tolerance necessary to
achieve the given output tolerance. How close must the input be to x = 0 for H(x) to be within 0.1 of 0?
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2.3.27
We can build different continuous approximations of signum (the
function giving the sign of a number) as follows. For each, case
a. Graph the continuous function.
b. Find the formula.
c. How close would the input have to be to 0 for the output to be within 0.1 of 0? A continuous function that is -1 for x ≤−0.1, 1 for x ≥ 0.1, and is linear for −0.1 < x < 0.1.
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2.3.28
We can build different continuous approximations of signum (the
function giving the sign of a number) as follows. For each, case
a. Graph the continuous function.
b. Find the formula.
c. How close would the input have to be to 0 for the output to be within 0.1 of 0? A continuous function that is -1 for x ≤−0.01, 1 for x ≥ 0.01, and is linear for −0.01 < x < 0.01.
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2.3.29
Find the accuracy of input necessary to achieve the desired output
accuracy. Suppose the mass of an object as a function of volume is
given by M = ρ V . If ρ = 2.0 g/ ..., how close must V be to 2.5 ... for M to be within 0.2 g of 5.0 g?
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2.3.30
Find the accuracy of input necessary to achieve the desired output
accuracy. The area of a disk as a function of radius is given by ... of 4π ?
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2.3.31 Find the accuracy of input necessary to achieve the desired output accuracy. The flow rate F through a vessel is proportional to the fourth power of the radius, or... Suppose a =1.0/cm s. How close must r be to 1.0 cm to guarantee a flow within 5% of 1 mL/s?
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2.3.32 Find the accuracy of input necessary to achieve the desired output accuracy. Consider an organism growing according to .... Suppose α = 0.001/s, and S(0)= 1.0 mm. At time 1000 s, S(t) = 2.71828 mm. How close must t be to 1000 s to guarantee a size within 0.1 mm of 2.71828 mm?
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2.4.33
A
bear sets off in pursuit of a hiker. Graph the position of a bear and
hiker as functions of time from the following descriptions. The bear
increases speed and the hiker steadily slows down until the bear catches
the hiker.
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2.3.34
Suppose a population of bacteria follows the discrete-time dynamical
system ... and we wish to have a population within ...at t =10.
What values of ... produce a result within the desired tolerance?
What is the input tolerance? Why is it harder to hit the target from
here?
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2.3.35
Suppose a population of bacteria follows the discrete-time dynamical
system ... and we wish to have a population within ...at t =10. What values of ... produce a result within the desired tolerance? What is the input tolerance?
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2.3.36
Suppose a population of bacteria follows the discrete-time dynamical
system ... and we wish to have a population within ...at t =10. How would your answers differ if the discrete-time dynamical system were ... ?Would the tolerances be larger or smaller? Why?
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2.3.37 Suppose the amount of toxin in a culture declines according to ...and we wish to have a concentration within 0.02 of 0.5 g/L at t = 10. What values of ... produce a result within the desired tolerance? What is the input tolerance?
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2.3.38 Suppose the amount of toxin in a culture declines according to ...and we wish to have a concentration within 0.02 of 0.5 g/L at t = 10. What values of ... produce a result within the desired tolerance? What is the input tolerance?
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2.3.39 Suppose the amount of toxin in a culture declines according to ...and we wish to have a concentration within 0.02 of 0.5 g/L at t = 10. What values of ...produce a result within the desired tolerance? What is the input tolerance?
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2.3.40 Suppose the amount of toxin in a culture declines according to ...and we wish to have a concentration within 0.02 of 0.5 g/L at t = 10. How would your answers differ if the discrete-time dynamical system were ...Would the tolerances be larger or smaller? Why?
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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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2.3.42 Suppose a neuron has the following response to inputs. If it receives a voltage input V greater than or equal to a threshold of ..., it outputs a voltage of kV for some constant k. If it receives an input less than the threshold value of ..., it outputs a fixed voltage V ∗. Suppose that k =1.5, ...=60, and V ∗ =100. Write and graph the function giving output in terms of input as a function defined in pieces.
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2.3.43 Suppose a neuron has the following response to inputs. If it receives a voltage input V greater than or equal to a threshold of ..., it outputs a voltage of kV for some constant k. If it receives an input less than the threshold value of ..., it outputs a fixed voltage V ∗. If k =2.0 and ... =50, what would V ∗ have to be for the function to be continuous? Graph the resulting function.
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2.3.44 Suppose a neuron has the following response to inputs. If it receives a voltage input V greater than or equal to a threshold of ..., it outputs a voltage of kV for some constant k. If it receives an input less than the threshold value of ..., it outputs a fixed voltage V ∗. If ...= 50 and V ∗ = 80, what would k have to be to make the function continuous? Graph the resulting function.
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2.3.45 The following questions are based on examples of hysteresis involving children. A
child outside is swinging on a swing that makes a horrible screeching
noise. Starting from when the swing is furthest back, the pitch of the
screeching noise increases as it swings forward and then decreases as it
swings back.
a. Draw a graph of the pitch as a function of position without hysteresis.
b. Draw a graph with hysteresis. Which graph seems more likely?
c. Imagine what each sounds like. Which is more irritating?
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2.3.46 The following questions are based on examples of hysteresis involving children.
Little Billy walks due east to school, but must cross from the south
side to the north side of the street. Because he is a very careful
child, he crosses quickly at the first possible opportunity.
a. Graph little Billy’s latitude as a function of distance from home on the way to school.
b. Graph little Billy’s latitude as a function of distance from home on the way home.
c. Is this an example of hysteresis?
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2.3.47 Graph the function ... What happens near x = 0?
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