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Author: nsharp1
No. Questions: 3
Created: Sep 25, 2018
Last Modified: 6 years ago

Approximation of Intersection Two Functions

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Simone sketches the functions

$f(x) = x^3 + 4x^2 + 2x - 9$ and

$g(x) = e^(x-5) - 7.$ She sees that one of the intersection points occurs between

$x=-4$ and

$x=-2.$ She also notices that, near this intersection point, to the left of the intersection point,

$f(x)$ is below

$g(x)$ on the graph, and to the right of the intersection point

$f(x)$ is above

$g(x)$ .

Grade 10 Functions and Relations CCSS: HSA-REI.D.11

A.

In order to get a more precise location of the point of intersection, she decides to try the value of

$x$ in the middle of the interval. What are the values of each function at this point, rounded to two decimal places?

$f(-3) = -6.00, \ \ g(-3)=-7.00$
$f(-3) = -78.00, \ \ g(-3)=-7.00$
$f(-3) = -78.00, \ \ g(-3) = -0.39$
$f(-3)= 0.00, \ \ g(-3) = -6.86$

Grade 11 Functions and Relations CCSS: HSA-REI.D.11

B.

Given the values of the functions at

$x=-3$ , and the relative values of

$f(x)$ and

$g(x)$ near this point, should Simone choose a value greater than or less than

$x=-3$ to get a more precise value of the point of intersection?

Neither, $x=-3$ is the point of intersection.
Less than, since $f(-3) > g(-3)$ .
Greater than, since $f(-3) > g(-3)$ .
Either, since it is still just an approximation.

Grade 11 Functions and Relations CCSS: HSA-REI.D.11

C.

If Simone continues to choose values of

$x$ in the middle of each new interval (for example, she would choose

$x=-2.5$ if she looks in the interval

$-3 < x < -2$ ), what would be a reasonable criterion for believing that she has found an accurate point of intersection and why?

$|f(x)|$ is very small, because the $x$ value is very precise.
$|g(x)|$ is very small, since $x$ is near the middle of the interval.
$(|f(x)+g(x)|)/2$ is near zero, as you've averaged the absolute value of the functions and the functions should cancel each other out.
$|f(x)-g(x)|$ is very small, since these values should be almost equal near the point of intersection.