Question

**Let f(x)=sin(x), where x is measured in radians.
Calculate f^' (x=0.9) using h=0.1 h=0.01 h=0.001 Calculate the
error using the value of cos(x=0.9). Use f^'
(x)≈(f(x+h)-f(x-h))/2h.**

Answer #1

Find the root of the function: f(x)=2x+sin(x)-e^x,
using Newton Method and initial value of 0. Calculate the
approximate error in each step. Use maximum 4 steps (in case you do
not observe a convergence).

Use the three-point center-difference formula to approximate f ′
(0), where f(x) = e x , for (a) h=0.1; (b) h=0.01; (c) h=0.001.

Calculate the average rate of change of the given function
f over the intervals [a, a + h]
where h = 1, 0.1, 0.01, 0.001, and 0.0001.
f(x) = 8x2; a = 0
h = 1
h = 0.1
h = 0.01
h = 0.001
h = 0.0001

Calculate the average rate of change of the given function f
over the intervals [a, a + h] where h = 1, 0.1, 0.01, 0.001, and
0.0001. (Technology is recommended for the cases h = 0.01, 0.001,
and 0.0001.) HINT [See Example 4.] (Round your answers to seven
decimal places.)
f(x) = x^2/4 ; a = 1
h = 1
h = 0.1
h = 0.01
h = 0.001
h = 0.0001

Calculate the average rate of change of the given function f
over the intervals [a, a + h] where h = 1, 0.1, 0.01, 0.001, and
0.0001. (Technology is recommended for the cases h = 0.01, 0.001,
and 0.0001.) HINT [See Example 4.] (Round your answers to five
decimal places.) f(x) = 3 x ; a = 4

Calculate the average rate of change of the given function
f over the intervals [a, a + h]
where h = 1, 0.1, 0.01, 0.001, and 0.0001. (Technology is
recommended for the cases h = 0.01, 0.001, and 0.0001.)
HINT [See Example 4.] (Round your answers to five decimal
places.)
f(x)=7/x; a=2

Let f(x,y)=e^(−5x)sin(3y).
(a) Using difference quotients with Δx=0.1 and Δy=0.1, we
estimate
fx(3,2)≈
fy(3,2)≈
(b) Using difference quotients with Δx=0.01 and Δy=0.01, we find
better estimates:
fx(3,2)≈
fy(3,2)≈

Let
x be fixed. Use the method of undetermined coefficients to find a
higher order differentiation formula for computing an approximation
of the second order derivative f′′(x) based on the values
f(x−2h),f(x−h),f(x),f(x+h),f(x+2h). What is the order of the
error?

1. Let f : R2 → R2, f(x,y) = ?g(x,y),h(x,y)? where g,h : R2 → R.
Show that
f is continuous at p0 ⇐⇒ both g,h are continuous at p0

Please show all steps, thank you.
Using the Taylor formula for f(x+h) and f(x-h) with f ''' in the
error term, find the error of the approximate formula f '' (x) =
(f(x+h)+f(x-h)-2f(x))/(h^2) in terms of f ''' (eta), for some point
eta between x-h and x+h. Then give an upper bound for the absolute
error assuming that |f ''' (t)| =< M for t between x-h and
x+h.

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