Mathematics, 29.12.2019 01:31 HannaTheGurls

40 pts ! approximate the solution to the equation above using three iterations of successive approximation. use the graph below as a starting point.

40 pts ! approximate the solution to the equation above using three iterations of successive approx

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Answer from: brobles9118

Option B. 7/16

Step-by-step explanation:

Approximation of Roots of Equations

Newton-Raphson's method is widely used to calculate approximate values of the roots of equations which cannot be solved with algebraic methods.

Let's suposse we need to solve the equation

f(x)=0

To find the approximate value of x, we use the following iterative formula

$\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}$

where f' is the first derivative of f

Each value will determine the next one which should be closer to the solution f(x)=0. We need an initial value to start the iterations

We want to solve this equation:

$4^{-x}+5=3^x+4$

We construct a function f(x) which can be equated to zero and find its roots. Thus we define

$f(x)=4^{-x}+5-3^x-4$

$f(x)=4^{-x}+1-3^x$

To solve the original equation, it's the same as finding the roots of f(x)=0

The starting point x_1 will be obtained from the graph provided in the figure

x_1=0.5

Lets find the derivative

$f'(x)=-ln4\ 4^{-x}-ln3\ 3^x$

Evaluating in the point x_1=0.5

$f(0.5)=4^{-0.5}+1-3^{0.5}=-0.23205$

$f'(0.5)=-ln4\ 4^{-0.5}-ln3\ 3^{0.5}=-2.596$

We find the next value:

$\displaystyle x_{2}=0.5-\frac {-0.23205}{-2.596}$

x_2=0.4106

Let's perform another iteration

Evaluating in the point x_2=0.4106

$f(0.4106)=4^{-0.4106}+1-3^{0.4106}=-0.00408$

$f'(0.4106)=-ln4\ 4^{-0.4106}-ln3\ 3^{0.4106}=-2.50946$

We find the next value

$\displaystyle x_{3}=0.4106-\frac {-0.00408}{-2.50946}$

x_3=0.409

This is a very good approximation since

f(0.409)=-0.0000378281

We need to pick one of the options and find none of them is close enough to 0.409. So we'll just choose the closest

Option B. 7/16=0.4375

Answer from: Quest

i need more information to answer

Answer from: Quest

step-by-step explanation:

a. since the profit made in 2012 is equal to the profit made in 2014, the profit is symmetric about the line y = 120,000.

step-by-step explanation:

given that megan owns a clothing boutique.

the profits of 2010, 11, are recorded and shown as follows:

years since 2010 0 1 2 3 4 5

profits made in 1000's 39 84 111 120 111 84

we find that profit gradually increases form 2010 to 2013 upto 120, then starts decreasing.

the profits for 2012 and 2014 are the same while that of 2011 and 2015 are also the same.

this shows that there is symmetry about the line profit = 120000

hence answer is:

a. since the profit made in 2012 is equal to the profit made in 2014, the profit is symmetric about the line y = 120,000.

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40 pts ! approximate the solution to the equation above using three iterations of successive approx...

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