3.7.1 Use the tangent line and secant line to estimate the following values. Make sure to identify the base point a you
used for your tangent line approximation and the second point you used
for the secant line approximation. Use a calculator to compare the
estimates with the exact answer. ...
Get solution
3.7.2 Use the tangent line and secant line to estimate the following values. Make sure to identify the base point a you
used for your tangent line approximation and the second point you used
for the secant line approximation. Use a calculator to compare the
estimates with the exact answer. ...
Get solution
3.7.3 Use the tangent line and secant line to estimate the following values. Make sure to identify the base point a you
used for your tangent line approximation and the second point you used
for the secant line approximation. Use a calculator to compare the
estimates with the exact answer. ...
Get solution
3.7.4 Use the tangent line and secant line to estimate the following values. Make sure to identify the base point a you
used for your tangent line approximation and the second point you used
for the secant line approximation. Use a calculator to compare the
estimates with the exact answer. ...
Get solution
3.7.5 Use the tangent line and secant line to estimate the following values. Make sure to identify the base point a you
used for your tangent line approximation and the second point you used
for the secant line approximation. Use a calculator to compare the
estimates with the exact answer. sin(0.02)
Get solution
3.7.6 Use the tangent line and secant line to estimate the following values. Make sure to identify the base point a you
used for your tangent line approximation and the second point you used
for the secant line approximation. Use a calculator to compare the
estimates with the exact answer. cos(−0.02)
Get solution
3.7.7
Use the quadratic approximation to estimate the following values.
Compare the estimates with the exact answer. ... (based on Exercise
1). Reference Exercise 1 ...
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3.7.8
Use the quadratic approximation to estimate the following values.
Compare the estimates with the exact answer. ... (based on Exercise
2). Reference Exercise 2 ...
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3.7.9
Use the quadratic approximation to estimate the following values.
Compare the estimates with the exact answer. ... (based on Exercise
3). Reference Exercise 3 ...
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3.7.10
Use the quadratic approximation to estimate the following values.
Compare the estimates with the exact answer. ... (based on Exercise
4). Reference Exercise 4 ...
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3.7.11 Use the quadratic approximation to estimate the following values. Compare the estimates with the exact answer. sin(0.02) (based on Exercise 5).
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3.7.12 Use the quadratic approximation to estimate the following values. Compare the estimates with the exact answer. cos(−0.02) (based on Exercise 6).
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3.7.13 Use
the tangent line approximation to evaluate the following in two ways.
First, find the tangent line to the whole function using the chain rule.
Second, break the calculation into two pieces by writing the function
as a composition, approximate the inner function with its tangent line,
and use this value to plug into the tangent line of the outer function.
Do your answers match? ...
Get solution
3.7.14 Use
the tangent line approximation to evaluate the following in two ways.
First, find the tangent line to the whole function using the chain rule.
Second, break the calculation into two pieces by writing the function
as a composition, approximate the inner function with its tangent line,
and use this value to plug into the tangent line of the outer function.
Do your answers match? ...
Get solution
3.7.15 Use
the tangent line approximation to evaluate the following in two ways.
First, find the tangent line to the whole function using the chain rule.
Second, break the calculation into two pieces by writing the function
as a composition, approximate the inner function with its tangent line,
and use this value to plug into the tangent line of the outer function.
Do your answers match? ...
Get solution
3.7.16 Use
the tangent line approximation to evaluate the following in two ways.
First, find the tangent line to the whole function using the chain rule.
Second, break the calculation into two pieces by writing the function
as a composition, approximate the inner function with its tangent line,
and use this value to plug into the tangent line of the outer function.
Do your answers match? ...
Get solution
3.7.17 For
the following functions, find the tangent line approximation of the two
values and compare with the true value. Indicate which approximations
are too high and which are too low. From graphs of the functions, try to
explain what it is about the graph that causes this. ...
Get solution
3.7.18 For
the following functions, find the tangent line approximation of the two
values and compare with the true value. Indicate which approximations
are too high and which are too low. From graphs of the functions, try to
explain what it is about the graph that causes this. ln(1.1) and ln(0.9).
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3.7.19 For
the following functions, find the tangent line approximation of the two
values and compare with the true value. Indicate which approximations
are too high and which are too low. From graphs of the functions, try to
explain what it is about the graph that causes this. ...
Get solution
3.7.20 For
the following functions, find the tangent line approximation of the two
values and compare with the true value. Indicate which approximations
are too high and which are too low. From graphs of the functions, try to
explain what it is about the graph that causes this. ...
Get solution
3.7.21 Find the third order Taylor polynomials for the following functions. ...
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3.7.22 Find the third order Taylor polynomials for the following functions. ...
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3.7.23 Find the third order Taylor polynomials for the following functions. ...
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3.7.24 Find the third order Taylor polynomials for the following functions. ...
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3.7.25 Find the third order Taylor polynomials for the following functions. h ( x ) = ln(x) for x near 1.
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3.7.26 Find the third order Taylor polynomials for the following functions. h ( x ) = sin(x) for x near 0.
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3.7.27 Use the Taylor series from the text to evaluate the following sums. Check by adding up through the n = 5 term. How close do they get to their limit? ...
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3.7.28 Use the Taylor series from the text to evaluate the following sums. Check by adding up through the n = 5 term. How close do they get to their limit? ...
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3.7.29 Use the Taylor series from the text to evaluate the following sums. Check by adding up through the n = 5 term. How close do they get to their limit? ...
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3.7.30 Use the Taylor series from the text to evaluate the following sums. Check by adding up through the n = 5 term. How close do they get to their limit? ...
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3.7.31 Find the Taylor polynomial of degree n and
the Taylor series for the following functions. Add up the given series
by assuming that the sum of the Taylor series is equal to the function. Find the Taylor polynomial for h(x)= cos(x) with the base point a = 0. Use your result to find...……….
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3.7.32 Find the Taylor polynomial of degree n and
the Taylor series for the following functions. Add up the given series
by assuming that the sum of the Taylor series is equal to the function. f ( x ) = ln(1 − x) with the base point a = 0. Use your result to find ...
Get solution
3.7.33 Write
the tangent line approximation for the numerators and denominators of
the following functions and show that the result of applying
L’Hˆopital’s rule matches that of comparing the linear approximations. ...
Get solution
3.7.34 Write
the tangent line approximation for the numerators and denominators of
the following functions and show that the result of applying
L’Hˆopital’s rule matches that of comparing the linear approximations. ...
Get solution
3.7.35 Write
the tangent line approximation for the numerators and denominators of
the following functions and show that the result of applying
L’Hˆopital’s rule matches that of comparing the linear approximations. ...
Get solution
3.7.36 Write
the tangent line approximation for the numerators and denominators of
the following functions and show that the result of applying
L’Hˆopital’s rule matches that of comparing the linear approximations. ...
Get solution
3.7.37 Compare the tangent line approximation of the following absorption functions with the leading behavior at c = 0. If they do not match, can you explain why? ... (as in Section 3.6, Exercise 23).
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3.7.38 Compare the tangent line approximation of the following absorption functions with the leading behavior at c = 0. If they do not match, can you explain why? ... (as in Section 3.6, Exercise 24).
Get solution
3.7.39 Compare the tangent line approximation of the following absorption functions with the leading behavior at c = 0. If they do not match, can you explain why? ... (as in Section 3.6, Exercise 25).
Get solution
3.7.40 Compare the tangent line approximation of the following absorption functions with the leading behavior at c = 0. If they do not match, can you explain why? ... (as in Section 3.6, Exercise 26).
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3.7.41 Compare the tangent line approximation of the following absorption functions with the leading behavior at c = 0. If they do not match, can you explain why? ... (as in Section 3.6, Exercise 27).
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3.7.42 Compare the tangent line approximation of the following absorption functions with the leading behavior at c = 0. If they do not match, can you explain why? α( c ) = 5c(1 + c) (as in Section 3.6, Exercise 28).
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3.7.43
Consider a declining population following the formula ...
(measured in millions). Approximate the population at each of the
following times using the tangent with base point t = 0, the tangent with base point t = 1, and the secant connecting times t = 0 and t = 1. Graph each of the relevant tangents and secants. Which method is best for what? t = 0.1.
Get solution
3.7.44
Consider a declining population following the formula ...
(measured in millions). Approximate the population at each of the
following times using the tangent with base point t = 0, the tangent with base point t = 1, and the secant connecting times t = 0 and t = 1. Graph each of the relevant tangents and secants. Which method is best for what? t = 0.5.
Get solution
3.7.45
Consider a declining population following the formula ...
(measured in millions). Approximate the population at each of the
following times using the tangent with base point t = 0, the tangent with base point t = 1, and the secant connecting times t = 0 and t = 1. Graph each of the relevant tangents and secants. Which method is best for what? t = 0.9.
Get solution
3.7.46
Consider a declining population following the formula ...
(measured in millions). Approximate the population at each of the
following times using the tangent with base point t = 0, the tangent with base point t = 1, and the secant connecting times t = 0 and t = 1. Graph each of the relevant tangents and secants. Which method is best for what? t = 1.1.
Get solution
3.7.47 Consider the following table giving mass as a function of age. ... The data follow the equation ....
Estimate each of the following using the tangent line approximation and
the secant line approximation. Which approximation is closer to the
exact answer? Which method would be best if you did not know the formula
for M(a)? M (1.25)
Get solution
3.7.48 Consider the following table giving mass as a function of age. ... The data follow the equation ....
Estimate each of the following using the tangent line approximation and
the secant line approximation. Which approximation is closer to the
exact answer? Which method would be best if you did not know the formula
for M(a)? M (1.45)
Get solution
3.7.49 Consider the following table giving temperature as a function of time. ... The data follow the equation ...but
there is some noise in each of the measurements, so that the values are
not exactly on the curve. Estimate each of the following using the
tangent line approximation and a secant line approximation. Which method
do you think deals best with the noise? Estimate T (1) using the values at t = 0 and t = 2.
Get solution
3.7.50 Consider the following table giving temperature as a function of time. ... The data follow the equation ...but
there is some noise in each of the measurements, so that the values are
not exactly on the curve. Estimate each of the following using the
tangent line approximation and a secant line approximation. Which method
do you think deals best with the noise? Estimate T (3) using the values at t = 2 and t = 4.
Get solution
3.7.51 Even
after they have been controlled, diseases can affect many individuals.
Suppose a medication has been introduced, and each person infected with a
new variety of influenza infects only r people on average, and those people infect r people
themselves and so forth. Suppose ...people are initially infected. Find
the total number of people infected by a disease in the following
cases. Suppose r = 0.5.
Get solution
3.7.53 Find the Taylor polynomials for cos(x) and sin(x) with the base point x =0 up to degree 10. Can you see the pattern? Graph the functions, ...on domains around 0 that get larger and larger. What happens to the approximation for values of x far from 0?
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3.7.53 Find the Taylor polynomials for cos(x) and sin(x) with the base point x =0 up to degree 10. Can you see the pattern? Graph the functions, ...on domains around 0 that get larger and larger. What happens to the approximation for values of x far from 0?
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3.7.54 Find the Taylor polynomial for the function defined by ... with the base point x = 0 up todegree 10 (you will have to take the limit as x → 0 to compute the derivatives). Can you see the pattern? Graph the function on the domain −1≤ x ≤ 1. What happens to the approximation for values of x far from 0? Do the Taylor polynomials make sense? What is the Taylor series for this function?
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3.7.55 A simple equation that is impossible to solve algebraically is ...
a. Graph the two sides and convince yourself there is a solution.
b. Replace ... with its tangent line at x = 0, and try to solve for the point where the tangent line is equal to x + 2. This is an approximate solution. What goes wrong in this case?
c. Replace ...with its quadratic approximation at x = 0, and solve for the point where it is equal to x + 2.
d. Replace ... with its tangent line at x = 1 and solve.
e. Replace ... with its quadratic approximation at x = 1 and solve.
f. How close are these solutions to the exact answer?
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