Question

Approximate the solution to the equation above using three iterations of successive approximation. Use the graph below as a starting point.
x = 27/16
x = 13/8
x = 25/16
x = 7/4

Answer 1

Answer:

x=25/16

Step-by-step explanation:

Approximate Roots of Functions:

We are required to find the root of  

$\displaystyle 5^{-x}+7=\ 2x+4$

Let's construct a function

$\displaystyle f(x)=\ 5^{-x}+7-2x-4$

Reducing:

$\displaystyle f(x)= 5^{-x}-2x+3$

The equation will have a solution when f(x)=0. We'll start from the approximate crossing point given in the graph (x=1.5)

$\displaystyle f(1.5)=5^{-1.5}-2(1.5)+3\ =0.089$

We'll move up on x=1.52  

$\displaystyle f(1.52)=5^{-1.52}-2(1.52)+3=\ 0.046$

We are closer to the solution, let's try some more up

$\displaystyle f(1.54)=5^{-1.54}-2(1.54)+3=\ 0.0038$

This value x=1.54 is close enough to the solution of the original equation

From the options given, $\frac{25}{16}$ is the closest to our solution

$\boxed{Answer:\ x=25/16}$