Using the Intermediate Value Theorem, (a) use the...

Using the Intermediate Value Theorem, (a) use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. (b) Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results. $$ g(x)=3 x^{4}+4 x^{3}-3 $$

Using the Intermediate Value Theorem, (a) use the Intermediate Value Theorem
and the table feature of a graphing utility to find intervals
one unit in length in which the polynomial function
is guaranteed to have a zero. (b) Adjust the table to
approximate the zeros of the function. Use the zero or root
feature of the graphing utility to verify your results.
$$
g(x)=3 x^{4}+4 x^{3}-3
$$

Key Concepts

Table of Values

Using a table of values involves calculating the function's output at a series of input points, which helps in detecting intervals where the function changes sign. This method is commonly used in conjunction with the Intermediate Value Theorem to identify intervals that likely contain a root.

Graphing Utility

A graphing utility is a computational tool that allows for the visualization of functions, facilitating the analysis of their properties such as roots, extrema, and intervals of increase or decrease. It is particularly useful for approximating zeros and analyzing the behavior of functions graphically.

Polynomial Functions

Polynomial functions are algebraic expressions involving sums of powers of a variable with constant coefficients. They are inherently continuous and differentiable, which makes them particularly amenable to analysis using techniques like the Intermediate Value Theorem and graphing methods for zero-finding.

Continuous Function

A continuous function is one in which small changes in the input lead to small changes in the output, without any abrupt jumps or breaks. The concept of continuity is essential for applying the Intermediate Value Theorem and ensuring that properties like the existence of roots hold true within an interval.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on an interval [a, b] and takes on different values at the endpoints, then it must take on every value between those endpoints at least once. This property is crucial for guaranteeing the existence of zeros when the function values at the endpoints have opposite signs.

Zero of a Function

A zero (or root) of a function is a point at which the function evaluates to zero. Finding zeros is a fundamental aspect in many areas of mathematics and is critical for solving equations and understanding the behavior of functions.

Transcript

00:03 Here we are going to use the intermediate value theorem to find an interval where this function is going to cross the x -axis.

00:14 We're going to approximate those zeros with a graphing calculator.

00:18 We use the table function to help get that approximation.

00:20 And then we're going to use the zero function to get an even better approximation.

00:25 There are several graphing calculators out there, so i'll kind of explain the basic process a little bit.

00:30 So you can apply it to whatever you're using.

00:32 And the first thing i did for this function is i did graph it, and i noticed it looked a little something like this.

00:39 So there were two intervals we were really focused on.

00:42 We did want to focus on that interval.

00:45 I'll use interval notation to write this between negative 2 and negative 1.

00:48 And we also want to focus on this other interval between 0 and 1.

00:53 So what i did is using that table function, i actually had my table start at negative 2, and then i had it count by just one -tenth so i could find a really tiny interval to help with the approximation.

01:06 So on that x -y table, when i did count by one -tenth, i noticed that as it went from negative 1 -6 -tenth to negative 1 -5 -tenths for the input, the function value went from above the x -axis to below the x -axis.

01:24 So with the intermediate value theorem, we know that somewhere in there, since we're going to, it's continuous, it must have crossed zero.

01:32 Since that, you know, approximately, you know, 2 ,800s that you see right there, since that's a little bit closer to the zero than the negative 1 in 3 tenths you see down below, we're going to approximate that the 0 is a little closer to the negative 1 in 6 tenths.

01:47 So why don't we say for an initial estimation, let's say like negative 1 and 5 ,800s, we'll go closer to negative 1 in 6 tenths than closer to negative 1 in 5 now on the other interval, i did the same thing.

02:00 I started my table at zero...

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