Final answer:
To find the slope of the tangent to the parabola at (1, 7), we calculate the derivative of the parabola's equation, which yields a slope of 6. Then, using the point-slope form, we determine the equation of the tangent line to be y = 6x + 1.
Step-by-step explanation:
The student has asked about finding the slope of a tangent line to a parabola and then about finding the equation of that tangent line. The parabola in question is given by y = 8x - x2.
(a) To find the slope of the tangent at the point (1,7), we need to differentiate the equation of the parabola to find its derivative, which represents the slope of the tangent at any point x. The derivative of the equation y = 8x - x2 is dy/dx = 8 - 2x. Plugging in the x-value of 1, we get dy/dx = 8 - 2(1) = 6, so the slope of the tangent at the point (1, 7) is 6.
(b) To find the equation of the tangent line, we use the point-slope form of the equation of a line, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point on the line. Substituting the slope m = 6 and the point (1, 7), the equation of the tangent line is y - 7 = 6(x - 1). Simplifying, we get y = 6x + 1 as the equation of the tangent line.