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, follow these steps:
1. Start by setting up your viewing rectangle. The given range for x is [-4, 4], and the range for y is [-20, 20].
2. Plot the graph of f(x) = x^3 - 3x within the viewing rectangle. To do this, you can plot a few points and then connect them to form a smooth curve. You can choose various values of x within the range [-4, 4] and calculate the corresponding values of f(x) to plot the points.
3. To determine the secant line passing through the points (-3, -18) and (3, 18), you need to find the equation of this line. The equation of a secant line passing through two points (x₁, y₁) and (x₂, y₂) is given by:
y - y₁ = (y₂ - y₁) / (x₂ - x₁) * (x - x₁)
In our case, the coordinates of the two points are (-3, -18) and (3, 18). Substitute these values into the equation above and simplify to find the equation of the secant line.
4. Once you have the equation of the secant line, you can plot it on the same graph as the function f(x).
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. The Mean Value Theorem states that if a function is continuous on the interval [a, b] and differentiable on the interval (a, b), then there exists at least one number c in the interval (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
2. In this case, the interval is [-3, 3]. You need to find the value(s) of c that satisfy the conclusion of the Mean Value Theorem.
3. First, find the derivative of f(x) = x^3 - 3x. Use the power rule to differentiate each term: f'(x) = 3x^2 - 3.
4. Next, calculate f'(c) by substituting the values of c into the derivative function f'(x) = 3x^2 - 3.
5. Calculate (f(3) - f(-3)) / (3 - (-3)) to find the slope of the secant line passing through the points (-3, -18) and (3, 18).
6. Compare the calculated slope of the secant line to the slope of f'(x) = 3x^2 - 3. If they are equal, then there exists at least one number c in the interval (-3, 3) that satisfies the conclusion of the Mean Value Theorem.