Questions: Find an equation of the line tangent to the graph of f(x)=-2 x^3 at (2,-16). The equation of the tangent line is y= (Type an expression using x as the variable.)

Solution Steps

To find the equation of the tangent line to the graph of a function at a given point, we need to determine the slope of the tangent line at that point. This is done by finding the derivative of the function and evaluating it at the given x-coordinate. Once we have the slope, we can use the point-slope form of a line to write the equation of the tangent line.

To find the slope of the tangent line to the function \( f(x) = -2x^3 \), we first compute its derivative: \[ f'(x) = -6x^2 \]

Next, we evaluate the derivative at the point \( x = 2 \): \[ f'(2) = -6(2^2) = -6 \cdot 4 = -24 \] Thus, the slope of the tangent line at the point \( (2, -16) \) is \( -24 \).

Using the point-slope form of the line, which is given by \( y - y_1 = m(x - x_1) \), we substitute \( m = -24 \), \( x_1 = 2 \), and \( y_1 = -16 \): \[ y - (-16) = -24(x - 2) \] Simplifying this, we get: \[ y + 16 = -24x + 48 \] \[ y = -24x + 32 \]

Final Answer