1. The problem asks to estimate the values of the derivatives of the function $f$ at points $0$ through $7$ based on a given graph, and then sketch the graph of the 9th derivative $f^{(9)}$. 2. Since the original graph is not provided, we cannot numerically estimate the derivatives directly. However, the general approach is: - $f^{(0)}(x)$ is the original function value at $x$. - $f^{(1)}(x)$ is the slope (first derivative) at $x$. - $f^{(2)}(x)$ is the curvature (second derivative) at $x$. - Continue similarly for higher derivatives. 3. To sketch $f^{(9)}$, note that the 9th derivative is the derivative of the 8th derivative. Typically, higher derivatives of smooth functions tend to oscillate more and have smaller magnitudes unless the function is a polynomial of degree at least 9. 4. Without the graph or function, we cannot provide exact values or a precise sketch. The best we can do is explain the method: - Estimate slopes and curvatures at each point from the graph. - Use finite difference approximations for derivatives. - Sketch $f^{(9)}$ based on the pattern of changes in $f^{(8)}$. Final answer: Unable to estimate derivatives or sketch $f^{(9)}$ without the original graph or function data.