Questions: For the given function, (a) find the slope of the tangent line to the graph at the given point; (b) find the equation of the tangent line. f(x)=x^2-8 at x=2 (a) The slope of the tangent line at x=2 is . (b) The equation of the tangent line is .

Solution Steps

To solve this problem, we need to follow these steps:

(a) To find the slope of the tangent line at \( x = 2 \), we need to compute the derivative of the function \( f(x) \) and then evaluate it at \( x = 2 \).

(b) To find the equation of the tangent line, we use the point-slope form of the equation of a line, which is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope found in part (a) and \( (x_1, y_1) \) is the point on the graph.

To find the slope of the tangent line at \( x = 2 \), we first need to compute the derivative of the function \( f(x) = x^2 - 8 \). The derivative is given by: \[ f'(x) = 2x \]

Next, we evaluate the derivative at \( x = 2 \): \[ f'(2) = 2 \cdot 2 = 4 \] Thus, the slope of the tangent line at \( x = 2 \) is \( 4 \).

We also need the value of the function at \( x = 2 \): \[ f(2) = 2^2 - 8 = 4 - 8 = -4 \] So, the point on the graph at \( x = 2 \) is \( (2, -4) \).

Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point on the graph, we have: \[ y - (-4) = 4(x - 2) \] Simplifying this, we get: \[ y + 4 = 4x - 8 \] \[ y = 4x - 12 \]

Final Answer