Certainly! Here is the transcription suitable for an educational website:---**19.** The figure shows the graph of a function \( f \). Suppose that Newton's method is used to approximate the root \( r \) of the equation \( f(x) = 0 \) with initial approximation \( x_1 = 1 \).**(a)** Draw the tangent lines that are used to find \( x_2 \) and \( x_3 \), and estimate the numerical values of \( x_2 \) and \( x_3 \).**(b)** Would \( x_1 = 5 \) be a better first approximation? Explain.---**Graph Explanation:**The graph below illustrates a function \( f(x) \) with the x and y axes labeled. The curve begins above the x-axis, descends towards and crosses the x-axis at a point \( r \), and then ascends again. The initial approximation point \( x_1 = 1 \) is marked on the x-axis. The root of the function is labeled as \( r \).- **Grid**: The grid is composed of evenly spaced horizontal and vertical lines, representing a standard Cartesian plane.- **Curve**: The curve is continuous and smooth, suggesting a polynomial-like behavior.- **Label \( s \)**: A point \( s \) is labeled further along the x-axis, beyond the root \( r \).To apply Newton's method, one would draw tangent lines at the chosen points and observe where these tangents intersect the x-axis to approximate the roots \( x_2 \) and \( x_3 \).The question also asks whether starting with \( x_1 = 5 \) would serve as a better initial approximation for finding the root. This involves examining the curve's behavior near these points and the nature of the tangent line's intersections with the x-axis.

The give figure is, If we apply Newton's method to the function then, (a) We will have to draw tangent lines that are used to find second and third approximations if initial approximation is x1=1. (b) To Examine: whether x1=5 would be a better approximation or not. (a) Initial approximation be x1=1, The black line represents the tangent line at x1=1, and the point at which this line cuts the x-axis is the second approximation. The blue line represents the tangent line at x2, and the point at which this blue line cuts the x-axis is the third approximation. Now, Red line represents the tangent line at x3 which cuts the x-axis very close to r. So, x3 will be a good approximated root of the function. estimated numerical values of x2, and x3: x1=1 is given. So, 2<x2<3. from the graph x2≈2.4. Also, x3≈3.(b) If we take x1=5, then we will have to perform more iterations (at least four). see from the tangent lines in the diagram below, So, x1=5 would not be a better approximation.