Use the Intermediate Value Theorem and a graphing utility to find intervals of length 1 in which the polynomial is guaranteed to have a zero. Use the root feature of a graphing utility to approximate the zeros of the function.
h(x)=x^4-10x^2+2
The intermediate value theorem states that:
"for each value between the least upper bound and greatest lower bound of the image of a continuous function there is a corresponding point in its domain that the function maps to that value".
This implies that when the function evaluated at the limits of an interval have opposing signs, a zero of the function exists between the limits.
By graphing the function, you will find that all four real roots fall between -4 and +4. Use the roots features available in most graphing calculators to find the "exact" roots.
Hint: you can check the results by substituting y=x² which transforms the function to a quadratic function in y that can be solved exactly. x is simply ±√(y).
e^x=x^2
To find the intervals of length 1 in which the polynomial is guaranteed to have a zero, we can use the Intermediate Value Theorem. According to the theorem, if a function is continuous on a closed interval [a, b], and if f(a) and f(b) have opposite signs, then there must be at least one zero in the interval (a, b).
First, let's plot the graph of the polynomial using a graphing utility:
h(x) = x^4 - 10x^2 + 2
The graphing utility will help us visualize the function and locate the zeros.
Upon graphing the function, we can see that it intersects the x-axis at multiple points. We will look for intervals of length 1 where the polynomial has zeros.
Let's divide the x-axis into intervals of length 1 and find the sign changes of the function within each interval. If there is a sign change within an interval, it means there is a zero within that interval.
Interval 1: [-4, -3]
h(-4) = (-4)^4 - 10(-4)^2 + 2 = 258 < 0
h(-3) = (-3)^4 - 10(-3)^2 + 2 = -4 > 0
Since there is a sign change within this interval, there is a zero within the interval [-4, -3].
Interval 2: [-3, -2]
h(-3) = -4 < 0
h(-2) = 24 > 0
Since there is a sign change within this interval, there is a zero within the interval [-3, -2].
Interval 3: [-2, -1]
h(-2) = 24 > 0
h(-1) = -9 < 0
Since there is a sign change within this interval, there is a zero within the interval [-2, -1].
Interval 4: [-1, 0]
h(-1) = -9 < 0
h(0) = 2 > 0
There is no sign change within this interval, so there may or may not be a zero within the interval. We will need to use a root finding method or the graphing utility's root feature to approximate the zero within this interval.
Interval 5: [0, 1]
h(0) = 2 > 0
h(1) = -7 < 0
Since there is a sign change within this interval, there is a zero within the interval [0, 1].
Using the root feature of a graphing utility, we can approximate the zeros of the function:
The zeros of the function h(x) = x^4 - 10x^2 + 2 are approximately:
x ≈ -3.162
x ≈ -1.725
x ≈ -0.618
x ≈ 3.162
Please note that these are approximate values and may not be exact.
To find intervals of length 1 in which the polynomial is guaranteed to have a zero, we can make use of the Intermediate Value Theorem. The Intermediate Value Theorem states that if a function f(x) is continuous over an interval [a, b], and if f(a) and f(b) have opposite signs (one is positive and the other is negative), then there must exist at least one value c between a and b for which f(c) = 0.
To apply this theorem to the polynomial h(x) = x^4 - 10x^2 + 2, we can use a graphing utility. Here's how you can use a graphing utility:
1. Graph the function h(x) = x^4 - 10x^2 + 2 on the graphing utility.
2. Zoom in on the x-axis to see the details of the graph.
3. Look for intervals where the graph crosses the x-axis or changes sign.
4. Zoom in further to get a more precise view of the intervals.
By following these steps, you should see that the polynomial intersects the x-axis at multiple points. These points correspond to the zeros of the polynomial.
To approximate the zeros of the function more accurately, you can use the root-finding feature of the graphing utility. This feature allows you to find the x-values at which the function equals zero.
Here's how you can use the root feature of a graphing utility:
1. Open the root-finding feature of your graphing utility.
2. Enter the function h(x) = x^4 - 10x^2 + 2.
3. Specify the interval over which you want to find the zeros. For example, you can specify an interval of length 1 that you identified before.
4. Execute the root-finding feature, and it will provide you with the approximate zeros of the function within the specified interval.
By following these steps, you can use the Intermediate Value Theorem and a graphing utility to identify intervals of length 1 in which the polynomial is guaranteed to have a zero and approximate the zeros of the function more accurately using the root-finding feature.