Question

Use Newton’s method to find solutions accurate to within 10−4 for x − 0.8 − 0.2 sin x = 0, x in[0, π/2]. (Choose ?0=π/4).

Answer #1

Q1: Use bisection method to ﬁnd solution accurate to within
10^−4 on the interval [0, 1] of the function f(x) = x−2^−x
Q3: Find Newton’s formula for f(x) = x^(3) −3x + 1 in [1,3] to
calculate x5, if x0 = 1.5. Also, ﬁnd the rate of convergence of the
method.
Q4: Solve the equation e^(−x) −x = 0 by secant method, using x0
= 0 and x1 = 1, accurate to 10^−4.
Q5: Solve the following system using the...

Use Newton’s method to find all solutions of the equation
correct to eight decimal places.
7? −?^2 sin ? = ?^2 − ? + 1

Use
Newton’s Method to approximate the real solutions of x^5 + x −1 = 0
to five decimal places.

Use Newton’s method to find all solutions of the equation
correct to six decimal places: ?^2 − ? = √? + 1

The function e^x −100x^2 =0 has three true solutions.Use
Newton’s method to locate the solutions with tolerance
10^(−10).

: Consider f(x) = 3 sin(x2) − x.
1. Use Newton’s Method and initial value x0 = −2 to approximate
a negative root of f(x) up to 4 decimal places.
2. Consider the region bounded by f(x) and the x-axis over the
the interval [r, 0] where r is the answer in the previous part.
Find the volume of the solid obtain by rotating the region about
the y-axis. Round to 4 decimal places.

Use Steffensen’s method with ?0 = 2 to compute an approximation
to √5 accurate to within 10−2.

Use Newton’s method to estimate the value of e. Use X0=2 and
find x4
Hint: e=x
In(e)=In(x)

Use Newton's method to find the absolute maximum value of the
function f(x) = 8x sin(x), 0 ≤ x ≤ π correct to
SIX decimal places.

Suppose that Newton’s method is applied to find the solution p
= 0 of the equation
e^x −1−x− (1/2)x^2 = 0. It is known that, starting with any p0
> 0, the sequence {pn} produced by the Newton’s method is
monotonically decreasing (i.e., p0 >p1 >p2 >···)and
converges to 0.
Prove that {pn} converges to 0 linearly with rate 2/3. (hint:
You need to have the patience to use L’Hospital rule repeatedly. )
Please do the proof.

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