In the viewing rectangle [-4, 4] by [-20, 20], graph the function f(x) = x3 - 3x and its secant line through the points (-3, -18) and (3, 18).
Find the values of the numbers c that satisfy the conclusion of the Mean Value Theorem for the interval [-3, 3].
To graph the function f(x) = x^3 - 3x and its secant line through the points (-3, -18) and (3, 18), you need to follow these steps:
1. Create a coordinate system by drawing two perpendicular lines, the x-axis (horizontal) and y-axis (vertical), intersecting at the origin point (0,0).
2. Set up the viewing rectangle [-4, 4] by [-20, 20] by using the x-axis range of -4 to 4 and the y-axis range of -20 to 20. Mark the intervals and labels on each axis accordingly.
3. Plot the function f(x) = x^3 - 3x on the coordinate plane. To do this, substitute various values of x into the function and plot the corresponding points. For example, substitute x = -4, -3, -2, -1, 0, 1, 2, 3, and 4 into the function to get the respective y-values. Plot these points on the graph.
4. To graph the secant line through the points (-3, -18) and (3, 18), draw a straight line passing through these points. This line represents the secant line.
Now, to find the values of the numbers c that satisfy the conclusion of the Mean Value Theorem for the interval [-3, 3], follow these steps:
1. Calculate the average rate of change of the function f(x) = x^3 - 3x over the interval [-3, 3]. This can be done by finding the difference in the y-values of the endpoints and dividing it by the difference in the x-values. In this case, the endpoints are (-3, -18) and (3, 18). So, the average rate of change is (18 - (-18))/(3 - (-3)) = 36/6 = 6.
2. Apply the Mean Value Theorem, which states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a). In this case, the interval is [-3, 3] and the average rate of change is 6.
3. To satisfy the conclusion of the Mean Value Theorem, find the derivative of the function f(x) = x^3 - 3x. The derivative will give you the instantaneous rate of change of the function at any point.
4. Solve the equation f'(c) = 6 for c to find the values of c that satisfy the conclusion of the Mean Value Theorem. Set the derivative f'(x) equal to 6 and solve for x. The solutions to this equation will give you the values of c.
5. Once you find the values of c, you can plot them on the graph to see where they lie in the interval [-3, 3].
Remember that the Mean Value Theorem guarantees the existence of at least one value c, but it does not provide a specific method for finding this value. Depending on the function, you may need to use numerical methods or other techniques to approximate the values of c.