Question

- Find the root of the function
*f(x)**= 8 - 4.5 ( x -*sin*x )*in the interval*[2,3]*. Exhibit a numerical solution using Bisection method.

Answer #1

Let f(x) = x^3 + x - 4
a. Show that f(x) has a root on the interval [1,4]
b. Find the first three iterations of the bisection method on f
on this interval
c. Find a bound for the number of iterations needed of bisection
to approximate the root to within 10^-4

Find the root of f(x) = exp(x)sin(x) - xcos(x) by the
Newton’s method
starting with an initial value of xo = 1.0.
Solve by using Newton’method until satisfying the tolerance
limits of the followings;
i. tolerance = 0.01
ii. tolerance = 0.001
iii. tolerance= 0.0001
Comment on the results!

Let
f(x)=sin(x)+x^3-2. Use the secant method to find a root of f(x)
using initial guesses x0=1 and x1=4. Continue until two consecutive
x values agree in the first 2 decimal places.

1. Use the Intermediate Value Theorem to show that
f(x)=x3+4x2-10 has a real root in the
interval [1,2]. Then, preform two steps of Bisection method with
this interval to find P2.

Consider the function on the interval (0, 2π). f(x) = sin(x)
cos(x) + 4. (A) Find the open interval(s) on which the function is
increasing or decreasing. (Enter your answers using interval
notation.) (B) Apply the First Derivative Test to identify all
relative extrema.

Use Newton's method to derive root of f(x) = sin(x) +
1. What is the order of convergence?

Find an interval of length 0.2 that contains a root of the equation
sin(x)= (x)^2 - x

Consider the function on the interval (0, 2π).
f(x) =
sin(x)/
2 + (cos(x))2
(a) Find the open intervals on which the function is increasing
or decreasing. (Enter your answers using interval notation.)
increasing
decreasing
(b) Apply the First Derivative Test to identify the relative
extrema.
relative maximum
(x, y) =
relative minimum
(x, y) =

1) find the
absolute extrema of function f(x) = 2 sin x + cos 2x on the
interval [0, 2pi]
2)
is f(x) = tanx
concave up or concave down at x = phi / 6

Find the tangent to the function f(x) = sin(2x) at x = π.

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