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Given y=1/5x^5+4x^3+36x-1000, what starting points would make Newton's method f…
Related Topics
Wize University Calculus 1 Textbook > Applications of Differentiation
Newton's Method
5 Activities
Given
y
=
1
5
x
5
+
4
x
3
+
36
x
−
1000
y=\frac{1}{5}x^5+4x^3+36x-1000
y
=
5
1
x
5
+
4
x
3
+
36
x
−
1000
, what starting points would make Newton's method fail? Find a formula for the iterate
x
n
+
1
x_{n+1}
x
n
+
1
in term of
x
n
x_n
x
n
.
Newton's method fail at x=0,
x
n
+
1
=
x
n
−
1
5
x
n
5
+
4
x
n
3
+
36
x
n
+
1000
x
n
4
+
12
x
n
2
−
36
x_{n+1}=x_n-\frac{\frac{1}{5}x_n^5+4x_n^3+36x_n+1000}{x_n^4+12x_n^2-36}
x
n
+
1
=
x
n
−
x
n
4
+
12
x
n
2
−
36
5
1
x
n
5
+
4
x
n
3
+
36
x
n
+
1000
Newton's method fail at x=1,
x
n
+
1
=
x
n
−
1
5
x
n
5
+
4
x
n
3
+
36
x
n
−
1000
x
n
4
+
12
x
n
2
−
36
x_{n+1}=x_n-\frac{\frac{1}{5}x_n^5+4x_n^3+36x_n-1000}{x_n^4+12x_n^2-36}
x
n
+
1
=
x
n
−
x
n
4
+
12
x
n
2
−
36
5
1
x
n
5
+
4
x
n
3
+
36
x
n
−
1000
Newton's method fail at x=-1,
x
n
+
1
=
x
n
−
1
5
x
n
5
+
4
x
n
3
+
36
x
n
+
1000
x
n
4
+
12
x
n
2
+
36
x_{n+1}=x_n-\frac{\frac{1}{5}x_n^5+4x_n^3+36x_n+1000}{x_n^4+12x_n^2+36}
x
n
+
1
=
x
n
−
x
n
4
+
12
x
n
2
+
36
5
1
x
n
5
+
4
x
n
3
+
36
x
n
+
1000
Never fail,
x
n
+
1
=
x
n
−
1
5
x
n
5
+
4
x
n
3
+
36
x
n
−
1000
x
n
4
+
12
x
n
2
+
36
x_{n+1}=x_n - \frac{\frac{1}{5}x_n^5+4x_n^3+36x_n-1000}{x_n^4+12x_n^2+36}
x
n
+
1
=
x
n
−
x
n
4
+
12
x
n
2
+
36
5
1
x
n
5
+
4
x
n
3
+
36
x
n
−
1000
I don't know
Check Submission
More Newton's Method Questions:
Newton's Method: Estimate Intersection Point
Use Newton's method to estimate the point of intersection of
y
=
e
−
x
2
\displaystyle y=e^{-x^{2}}
y
=
e
−
x
2
and
y
=
x
y=x
y
=
x
.
Newton's Method: Inverse Trigonometric Function
Starting from
x
0
=
1
x_0=1
x
0
=
1
, Newton's method is used to approximate the solutions of the equation
arctan
x
−
1
2
x
2
=
0
\displaystyle \arctan x-\frac{1}{2x^2}=0
arctan
x
−
2
x
2
1
=
0
. Find the next approximation
x
1
x_1
x
1
.
Newton's Method: Estimate Intersection Point
Use Newton's method to estimate the point of intersection of
y
=
e
−
x
2
\displaystyle y=e^{-x^{2}}
y
=
e
−
x
2
and
y
=
x
y=x
y
=
x
.
Practice: Approximating a Ln
Q.
\textbf{Q.}
Q.
Approximate
ln
2
\ln{2}
ln
2
to 4 decimal places using Newton's method.
Newton's Method
Use Newton's Method to approximate a solution to
f
(
x
)
=
x
4
−
x
=
10
f(x)=x^4-x=10
f
(
x
)
=
x
4
−
x
=
10
Starting with
x
1
=
1
x_1=1
x
1
=
1
,
x
2
=
?
x_2=?
x
2
=
?
Newton's Method
Using
x
0
=
1
x_0=1
x
0
=
1
use Newton's Method to approximate a solution to
cos
x
=
x
\cos x=x
cos
x
=
x
.
Newton's Method
Using Newton's Method with
x
0
=
0
x_0=0
x
0
=
0
, find
x
1
,
…
,
x
4
x_1, \dots, x_4
x
1
,
…
,
x
4
for
f
(
x
)
=
2
x
3
−
x
+
1
f(x)=2x^3-x+1
f
(
x
)
=
2
x
3
−
x
+
1
.
Newton's Method
Find
2
3
\sqrt[3]{2}
3
2
to three decimal places.
Newton's Method
Take
x
0
=
0
x_{0}=0
x
0
=
0
and find
x
1
,
x
2
,
x
3
x_1,x_2,x_3
x
1
,
x
2
,
x
3
for
f
(
x
)
=
x
3
−
3
x
2
+
x
+
1
f(x)=x^{3}-3x^{2}+x+1
f
(
x
)
=
x
3
−
3
x
2
+
x
+
1
. If necessary, round your answer to the fourth decimal place.
Given
y
=
1
5
x
5
+
4
x
3
+
36
x
−
1000
y=\dfrac{1}{5}x^5+4x^3+36x-1000
y
=
5
1
x
5
+
4
x
3
+
36
x
−
1000
, what starting points would make Newton's Method fail? Find a formula for
x
n
+
1
x_{n+1}
x
n
+
1
in terms of
x
n
x_n
x
n
.
Practice: Approximating a Ln
Q.
\textbf{Q.}
Q.
Approximate
ln
2
\ln{2}
ln
2
to 4 decimal places using Newton's method.
Newton's Method: Inverse Trigonometric Function
Starting from
x
0
=
1
x_0=1
x
0
=
1
, Newton's method is used to approximate the solutions of the equation
arctan
x
−
1
2
x
2
=
0
\displaystyle \arctan x-\frac{1}{2x^2}=0
arctan
x
−
2
x
2
1
=
0
. Find the next approximation
x
1
x_1
x
1
.
Newton's Method: Estimate Intersection Point
Use Newton's method to estimate the point of intersection of
y
=
e
−
x
2
\displaystyle y=e^{-x^{2}}
y
=
e
−
x
2
and
y
=
x
y=x
y
=
x
.
Use one iteration of Newton's method to approximate
9
3
\sqrt[3]{9}
3
9
where the initial position is x=2.
Newton's Method
Which formula should be used in order to approximate
arctan
(
0.1
)
\arctan(0.1)
arctan
(
0.1
)
using Newton's Method?
Newton's Method
Which of the following points on the following function should you NOT use in order to use Newton's Method?
f
(
x
)
=
x
3
−
2
x
+
2
f(x) = x^3 - 2x + 2
f
(
x
)
=
x
3
−
2
x
+
2
Choose all that apply.
Newton's Method
Consider the curve generated by the equation
x
=
−
5
x
2
x = - \frac{5}{x^2}
x
=
−
x
2
5
Which polynomial would you use Newton's Method on to estimate a solution to the above equation, and what integer should you choose as a starting point to best estimate the solution?