Question

2. (a) For the equation e^x = 3 - 2 x , find a function, f(x), whose x-intercept is the solution of the equation (i.e. a function suitable to use in Newton’s Method), and use it to set up xn+1 for Newton’s Method.

(b) Use Newton's method to find x3 , x4 and x5 using the initial guess x1 = 0 . How many digits of accuracy are you certain of from these results?

(c) Use x1+ ln 2 and show that Newton’s Method fails in this case. Explain why this happens in terms of the the tangent lines of f(x).

Answer #1

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Approximate the zero for f(x) = (x^3)+(4x^2)-3x-8 using newton's
method
Use x1 = -6
A)Find x2,x3,x4,x5,x6
B)Based on the result, you estimate the zero for the function to
be......?
C)Explain why choosing x1 = -3 would have been a bad idea?
D) Are there any other bad ideas that someone could have chosen
for x1?

Newton's method: For a function ?(?)=ln?+?2−3f(x)=lnx+x2−3
a. Find the root of function ?(?)f(x) starting with
?0=1.0x0=1.0.
b. Compute the ratio |??−?|/|??−1−?|2|xn−r|/|xn−1−r|2, for
iterations 2, 3, 4 given ?=1.592142937058094r=1.592142937058094.
Show that this ratio's value approaches
|?″(?)/2?′(?)||f″(x)/2f′(x)| (i.e., the iteration converges
quadratically). In error computation, keep as many digits as you
can.

17. I am using Newton’s method to ﬁnd the negative root of f(x)
= 3−x2.
(a) What would be a good guess for x1? Draw the line
tangent to f(x) at your x1 and explain why using
Newton’s method would lead to the negative root of the
function.
(b) What would be a bad guess for x1? Draw the line
tangent to f(x) at your x1 and explain why using
Newton’s method would not lead to the negative root of...

3.8/3.9
5. Use Newton's Method to approximate the zero(s) of the
function. Continue the iterations until two successive
approximations differ by less than 0.001. Then find the zero(s) to
three decimal places using a graphing utility and compare the
results.
f(x) = 3 − x + sin(x)
Newton's Method: x=
Graphing Utility: x=
6. Find the tangent line approximation T to the graph
of f at the given point. Then complete the table. (Round
your answer to four decimal places.)...

Use Newton's method with the specified initial approximation
x1 to find x3, the third
approximation to the root of the given equation.
x3 + 5x − 2 =
0, x1 = 2
Step 1
If
f(x) =
x3 + 5x − 2,
then
f'(x) = _____ x^2 + _____
2- Use Newton's method to find all roots of the
equation correct to six decimal places. (Enter your answers as a
comma-separated list.)
x4 = 5 + x
.

Find the equation of the tangent line to the function f(x) =
ln(7x) at x=4.
(Use symbolic notation and fractions where needed. Let y = f(x)
and express the equation of the tangent line in terms of y and
x.)
equation:

1). Consider the following function and point.
f(x) = x3 + x + 3; (−2,
−7)
(a) Find an equation of the tangent line to the graph of the
function at the given point.
y =
2) Consider the following function and point. See Example
10.
f(x) = (5x + 1)2; (0, 1)
(a) Find an equation of the tangent line to the graph of the
function at the given point.
y =

1.) Find the equation of the tangent line to the graph of the
function f(x)=5x-4/2x+2 at the point where x=2
2.) Find the derivative: r(t)=(ln(t^3+1))^2

Part A.
Consider the nonlinear equation
x5-x=15
Attempt to find a root of this equation with Newton's method
(also known as Newton iteration).
Use a starting value of x0=4 and apply Newton's
method once to find x1
Enter your answer in the box below correct to four
decimal places.
Part B.
Using the value for x1 obtained in Part A, apply
Newton's method again to find x2
Note you should not round x1 when computing
x2

Calculate two iterations of Newton's Method to approximate a
zero of the function using the given initial guess. (Round your
answers to four decimal places.)
f(x) = cos x, x1 = 0.8
n
xn
f(xn)
f '(xn)
f(xn)
f '(xn)
xn −
f(xn)
f '(xn)
1
2

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