Question

The tangent line to the graph of \(y=g(x)\) at the point (5,2) passes through the point (9,0) . Find \(g(5)\) and \(g^{\prime}(5)\).

Short answer

The function value at x=5, \(g(5) = 2\) and the derivative of the function at x=5, \(g^{\prime}(5) = -\frac{1}{2}\).

Step-by-step solution

Calculate the slope of the tangent line

The slope of the tangent line can be calculated using the formula \(\frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1, y_1, x_2, y_2\) are the coordinates of the two given points. Substituting the given points (5,2) and (9,0) into this formula gives us \(\frac{0 - 2}{9 - 5} = -\frac{1}{2}\). Thus, the slope of the tangent line is -1/2.

Find \(g^{\prime}(5)\)

Since the derivative of a function at a given point equals the slope of the tangent line at that point, we have \(g^{\prime}(5) = -\frac{1}{2}\). This is the derivative of the function \(g(x)\) at x=5.

Find \(g(5)\)

The function \(g(x)\) passes through the point (5,2). This means that when x=5, y=2. Hence, \(g(5)\) = 2.