Questions: a. Find an equation for the line that is tangent to the curve y=x^3-4x at the point (2,0). b. Graph the curve and tangent line together. c. The tangent line intersects the curve at another point. Use your calculator to find the point's coordinates by solving the equations for the curve and tangent line simultaneously. a. The equation of the tangent line is y=8x-16 (Type an equation.) b. Choose the correct graph of the curve and tangent line below All graphs are shown in the window [-7,7,1] by [-60,60,6] A. B. C. D.

Solution Steps

The given curve is \( y = x^3 - 4x \). To find the equation of the tangent line, we first need to find the derivative of the curve, which gives us the slope of the tangent line at any point \( x \).

\[ y' = \frac{d}{dx}(x^3 - 4x) = 3x^2 - 4 \]

We need the slope of the tangent line at the point \( (2, 0) \). Substitute \( x = 2 \) into the derivative:

\[ y'(2) = 3(2)^2 - 4 = 3 \cdot 4 - 4 = 12 - 4 = 8 \]

So, the slope of the tangent line at \( (2, 0) \) is 8.

The point-slope form of the equation of a line is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency. Here, \( m = 8 \) and \( (x_1, y_1) = (2, 0) \).

\[ y - 0 = 8(x - 2) \] \[ y = 8x - 16 \]

Final Answer