2.1.1 For each of the following functions, find the average rate of change between the given base point ... and times ...for the four following values of Δt: Δt = 1.0, Δt = 0.5, Δt = 0.1 and Δt = 0.01. f( t ) = 2 + 3t with base point ... = 1.0.
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2.1.2 For each of the following functions, find the average rate of change between the given base point ... and times ...for the four following values of Δt: Δt = 1.0, Δt = 0.5, Δt = 0.1 and Δt = 0.01. g ( t ) = 2 − 3t with base point... = 0.0.
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2.1.3 For each of the following functions, find the average rate of change between the given base point ... and times ...for the four following values of Δt: Δt = 1.0, Δt = 0.5, Δt = 0.1 and Δt = 0.01. ...
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2.1.4 For each of the following functions, find the average rate of change between the given base point ... and times ...for the four following values of Δt: Δt = 1.0, Δt = 0.5, Δt = 0.1 and Δt = 0.01. ...
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2.1.5 For each of the following functions, find the average rate of change between the given base point ... and times ...for the four following values of Δt: Δt = 1.0, Δt = 0.5, Δt = 0.1 and Δt = 0.01. ...
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2.1.6 For each of the following functions, find the average rate of change between the given base point ... and times ...for the four following values of Δt: Δt = 1.0, Δt = 0.5, Δt = 0.1 and Δt = 0.01. ...
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2.1.7 For each of the following functions, find the equation of the secant line connecting the given base point ... and times ... + Δt for Δt = 1.0, Δt = 0.5, Δt = 0.1, and Δt = 0.01. Sketch the function and each of the secant lines. f ( t ) = 2 + 3t with base point ... = 1.0 (based on Exercise 1). Reference Exercise 1 f ( t ) = 2 + 3t with base point ... = 1.0.
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2.1.8 For each of the following functions, find the equation of the secant line connecting the given base point ... and times ... + Δt for Δt = 1.0, Δt = 0.5, Δt = 0.1, and Δt = 0.01. Sketch the function and each of the secant lines. f ( t ) = 2 + 3t with base point ... = 1.0 (based on Exercise 1). Reference Exercise 2 g ( t ) = 2 − 3t with base point ... = 0.0 (based on Exercise 2). Reference Exercise 2 g ( t ) = 2 − 3t with base point ... = 0.0.
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2.1.9 For each of the following functions, find the equation of the secant line connecting the given base point ... and times ... + Δt for Δt = 1.0, Δt = 0.5, Δt = 0.1, and Δt = 0.01. Sketch the function and each of the secant lines. h ( t )= ... with base point ... = 1.0 (based on Exercise 3). Reference Exercise 3 g ( t ) = 2 − 3t with base point ... = 0.0.
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2.1.10 For each of the following functions, find the equation of the secant line connecting the given base point ... and times ... + Δt for Δt = 1.0, Δt = 0.5, Δt = 0.1, and Δt = 0.01. Sketch the function and each of the secant lines. h ( t )= ... = 0.0 (based on Exercise 4). Reference Exercise 4 ...
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2.1.11 For each of the following functions, find the equation of the secant line connecting the given base point ... and times ... + Δt for Δt = 1.0, Δt = 0.5, Δt = 0.1, and Δt = 0.01. Sketch the function and each of the secant lines. G ( t ) = ... = 0.0 (based on Exercise 5). Reference Exercise 5 ...
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2.1.12 For each of the following functions, find the equation of the secant line connecting the given base point ... and times ... + Δt for Δt = 1.0, Δt = 0.5, Δt = 0.1, and Δt = 0.01. Sketch the function and each of the secant lines. G ( t ) = ...with base point ... = 0.0 (based on Exercise 6). Reference Exercise 6 ...
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2.1.13 Using
the results in Exercises 1–6, take a guess at the limit of the slopes
of the secants, and find the slope and equation of the tangent line. ...
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2.1.14 Using
the results in Exercises 1–6, take a guess at the limit of the slopes
of the secants, and find the slope and equation of the tangent line. g ( t )= 2 − 3t with base point ...= 0.0. Call the tangent line function ...) (based on Exercise 2).
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2.1.15 Using
the results in Exercises 1–6, take a guess at the limit of the slopes
of the secants, and find the slope and equation of the tangent line. ...
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2.1.16 Using
the results in Exercises 1–6, take a guess at the limit of the slopes
of the secants, and find the slope and equation of the tangent line. ...
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2.1.17 Using
the results in Exercises 1–6, take a guess at the limit of the slopes
of the secants, and find the slope and equation of the tangent line. ...
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2.1.18 Using
the results in Exercises 1–6, take a guess at the limit of the slopes
of the secants, and find the slope and equation of the tangent line. ...
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2.1.19
Important concepts have many names and formulas. The following
problems ask you to recall them. Give two other names for the
instantaneous rate of change of the function g(t).
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2.1.20 For each equation for population size, find the following and illustrate on a graph.
a. The population at times 0, 1, and 2.
b. The average rate of change between times 0 and 1.
c. The average rate of change between times 1 and 2.
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2.1.21
For each equation for population size, find the following and
illustrate on a graph. A population of bacteria is described by the
formula b(t)= ...where the time t is measured in hours.
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2.1.22
For each equation for population size, find the following and
illustrate on a graph. A population of bacteria described by the
formula b(t) = ...where the time t is measured in hours.
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2.1.23 For each equation for population size, find the following. A population following b(t) = ...
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2.1.24 For each equation for population size, find the following. A population following b(t) = ...
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2.1.25 For each equation for population size, find the following. A population following h(t) = ...
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2.1.26 For each equation for population size, find the following. A bacterial population following b(t) = ...
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2.1.27 For
the following bacterial populations, find the average rate of change
during the first hour, and during the first and second half hours. Graph
the data and the secant lines associated with the average rates of
change. Which populations change faster during the first half hour? ...
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2.1.28 For
the following bacterial populations, find the average rate of change
during the first hour, and during the first and second half hours. Graph
the data and the secant lines associated with the average rates of
change. Which populations change faster during the first half hour? ...
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2.1.29 For
the following bacterial populations, find the average rate of change
during the first hour, and during the first and second half hours. Graph
the data and the secant lines associated with the average rates of
change. Which populations change faster during the first half hour? ...
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2.1.30 For
the following bacterial populations, find the average rate of change
during the first hour, and during the first and second half hours. Graph
the data and the secant lines associated with the average rates of
change. Which populations change faster during the first half hour? ...
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2.1.31
Follow the steps in the text used to derive the approximate
differential Equation 2.1.1 ... with the following values of Δt. This requires computing the value of the function b(t)= ...at times separated by Δt, and finding the average rate of change between those times. Δ t = 1.0.
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2.1.32
Follow the steps in the text used to derive the approximate
differential Equation 2.1.1 ... with the following values of Δt. This requires computing the value of the function b(t)= ...at times separated by Δt, and finding the average rate of change between those times. Δ t = 0.5.
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2.1.33
Follow the steps in the text used to derive the approximate
differential Equation 2.1.1 ... with the following values of Δt. This requires computing the value of the function b(t)= ...at times separated by Δt, and finding the average rate of change between those times. Δ t = 0.01.
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2.1.34
Follow the steps in the text used to derive the approximate
differential Equation 2.1.1 ... with the following values of Δt. This requires computing the value of the function b(t)= ...at times separated by Δt, and finding the average rate of change between those times. Δ t = 0.001.
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2.1.35 Consider the following data on a tree. ... For each of the measurements,
a. Estimate the rate of change at each age.
b. Graph the rate of change as a function of age.
c. Find and graph the rate of change divided by the value as a function of age.
d. Use these results to describe the growth of this tree with a differential equation. The height
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2.1.36 Consider the following data on a tree. ... For each of the measurements,
a. Estimate the rate of change at each age.
b. Graph the rate of change as a function of age.
c. Find and graph the rate of change divided by the value as a function of age.
d. Use these results to describe the growth of this tree with a differential equation. The mass
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2.1.37 The
procedure banks use to compute continuously compounded interest is
similar to the process we used to derive a differential equation.
Suppose several banks claim to be giving 5% annual interest and that you
have $1000 to deposit. How much would you have after a year from a bank that has no compounding?
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2.1.38 The
procedure banks use to compute continuously compounded interest is
similar to the process we used to derive a differential equation.
Suppose several banks claim to be giving 5% annual interest and that you
have $1000 to deposit. A bank that compounds twice yearly
really gives 2.5% interest twice. How much would you have after a year
from this bank? How much better is this than a bank with no compounding?
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2.1.39
The
procedure banks use to compute continuously compounded interest is
similar to the process we used to derive a differential equation.
Suppose several banks claim to be giving 5% annual interest and that you
have $1000 to deposit. A bank that compounds monthly really gives
5/12% interest each month. How much would you have after a year from
this bank?
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2.1.40 The
procedure banks use to compute continuously compounded interest is
similar to the process we used to derive a differential equation.
Suppose several banks claim to be giving 5% annual interest and that you
have $1000 to deposit. How much would you have after a year from a bank that compounded daily?
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2.1.41 The
procedure banks use to compute continuously compounded interest is
similar to the process we used to derive a differential equation.
Suppose several banks claim to be giving 5% annual interest and that you
have $1000 to deposit. Write down a limit that expresses the
amount of money youwould get from a bank that compounded continuously,
and try to guess the answer.
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2.1.42 The
procedure banks use to compute continuously compounded interest is
similar to the process we used to derive a differential equation.
Suppose several banks claim to be giving 5% annual interest and that you
have $1000 to deposit. Follow the same steps to compare yearly, monthly, and daily compounding for a bank giving 20% interest.
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2.1.43 The
procedure banks use to compute continuously compounded interest is
similar to the process we used to derive a differential equation.
Suppose several banks claim to be giving 5% annual interest and that you
have $1000 to deposit. Follow the same steps to compare yearly,
monthly, and daily compounding for a bank giving 100% interest (in a
time of severe inflation). Why do you think compound interest makes a
bigger difference when the interest rate is higher?
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2.1.44 Consider the function ... defined for −1 ≤ x ≤ 1. This is the equation for a semi circle. The tangent line at the base point ...) has
slope −1. Graph this tangent line. Now zoom in on the base point. Does
the circle look more and more like the tangent line? How far do you need
to go before the circle looks flat? Would a tiny insect be able to tell
that his world was curved?
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2.1.45 Suppose a bacterial population oscillates with the formula b ( t ) = 2.0 + cos(t).
a. Graph this function
b. Find and graph the function that gives the rate of change between times t and t + 1 as a function of t for 0≤t ≤10.
c. Find and graph the function that gives the rate of change between times t and t + 0.1 as a function of t for 0≤t ≤10.
d. Try the same with smaller values of Δt. Do you have any idea what the limit might be?
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2.1.46 Follow the steps in Exercise 45 for a bacterial population that follows the formula ...
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