`f(x) = (x^3 + 4x - 1)(x - 2), (1,-4)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.
The function `f(x) = (x^3+4x-1)*(x-2)` . The slope of the tangent at point x = a, is given by f'(a).
`f'(x) = ((x^3+4x-1)*(x-2))'`
= `(x^3+4x-1)'*(x-2)+ (x^3+4x-1)*(x-2)'`
= `(3x^2+4)*(x-2)+ (x^3+4x-1)`
= `3x^3 + 4x - 6x^2 - 8 + x^3 + 4x - 1`
= `4x^3 + 8x - 6x^2 - 9`
`f'(1) = 4 + 8 - 6 - 9 = -3`
The equation of the tangent to the curve f(x) = (x^3+4x-1)*(x-2) at (1, -4) is
`(y + 4)/(x - 1) = -3`
y + 4 = -3x + 3
y = -3x - 1
The graph of the curve and the tangent at (1, -4) is: