Find the coefficients a, b, c, and d so that the curve shown in the accompanying figure is the g

Find the coefficients a, b, c, and d so that the curve shown...

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving a sum of terms where each term is a constant multiplied by a variable raised to a non-negative integer power. They serve as a basic building block in algebra and calculus, representing a wide range of curves and behaviors depending on the degree and coefficients involved.

Cubic Functions

Cubic functions are a specific type of polynomial function where the highest power of the variable is three. They typically have the form y = ax³ + bx² + cx + d and can exhibit complex behaviors such as turning points and an inflection point, making them useful in modeling scenarios where non-linear dynamics are present.

Polynomial Interpolation

Polynomial interpolation involves finding a polynomial that exactly fits a set of data points. In the case of cubic functions, knowing four distinct points on the curve allows one to set up equations based on the general form, leading to a uniquely determined cubic polynomial that passes through these points.

Systems of Linear Equations

When determining the coefficients of a polynomial from given points, each point generates a linear equation in terms of the unknown coefficients. These equations form a system of linear equations that can be solved by various algebraic methods, such as substitution, elimination, or using matrix operations, to yield the values of the coefficients.

Transcript

00:01 Hi, so for this exercise we have here a polynomial of degree 3 that is given by a x, q, b x squared square c x and d but we don't know the expression of this point but we know that it passed through four points so the points are defined here p1 p2 p3 and b4 and based on this information that we know that this point nominal pass through these four points then we can find the solution for the coefficients.

00:33 And actually we are going to obtain a linear system.

00:36 So let's start.

00:38 So let's pick p1.

00:42 So p1 is the point 010 and that is equivalent to say that y evaluated at 0 is equal to 10.

00:53 So from that we have that, let me write here, we have a polynomial that is a, x cubed plus b x squared plus c x plus d so all the values of x are going to be equals to zero so we obtain this is equivalent to say that 10 is equal to this will cancel out this will cancel out this as well so we obtained that the value of d it's going to be 10 then we take the second point the point one point one seven so the second point is the point one seven so this is equivalent to say that why evaluated at one it's going to be equals to seven so by replacing the data on the equation we obtain that there's equivalent to say that seven it's going to be equal to a plus b plus c plus d but we know already the value of d.

02:04 So this means that the equation that we have is a plus b plus c equals to minus 3.

02:13 Then we have the third point is the point 3 minus 11.

02:21 So this is equivalent to say that y evaluated at 3 is going to be equals to minus 11.

02:28 So from this, after replacing the data, we think here minus 11, equals to 27 a plus 9b plus 3c and plus d again but d is equal to 10 so we can put directly here plus 10 but this results into the equation right below here to the equation 27a plus 9b plus 3c equals to minus 21 and the last point is b4 so the p4 is evaluated at 4 result in minus 14 so this is equivalent to say that y evaluated at 4 is equals to minus 14 and this is equivalent to say that minus 14 is equals to 64a plus 16v plus 4c equal plus 10 but this is equivalent to having the equation 64a plus 16b plus 4c equals to minus 24 so this is the fourth equation so here we have all three equations actually are four but one of the those equations say a value of one of the constants that we need to find so we just have three equations with three unknowns so let's write this linear system so after evaluating this points on the polynomial we obtain a system of linear equations given by a plus b plus c equals to three 27a plus 9b plus 3 plus 3c equals to minus 21, 64a plus 16b plus 4c equals to minus 24.

04:45 So the point is that from here we can obtain a similar system.

04:50 What i'm going to do is taking this, the second equation and divide by three, and the third equation and divide by four.

04:59 So by that we obtain this system, a plus b plus c, to 3, 9 times a plus b, 3 times b, plus c equals to minus 7.

05:19 And the last one is 16 a plus 4b plus c equals to minus 6.

05:27 Okay, so now we have this system, this system of linear equations and we can represent it in the matrix form.

05:35 We can solve this applying the gaussian elimination...

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