Final answer:
Calculate the derivative of y = 7x² - x³ to find the slope of the tangent, substitute x = 1 to get the slope at the given point (1,6). Use the slope and the point to write the equation of the tangent line using the point-slope form. The equation is y = 11x - 5.
Step-by-step explanation:
To find an equation of the tangent line to the curve at the given point (1,6) for the function y = 7x² - x³, you first need to calculate the derivative of the function, which represents the slope of the tangent line at any point x. The function's derivative is dy/dx = 14x - 3x². Substituting x = 1 gives the slope m = 14(1) - 3(1)² = 11. Now, we have a point and a slope, so we can use the point-slope form to write the equation of the tangent line: y - y₁ = m(x - x₁). Substituting the slope we found and the point (1,6), we get y - 6 = 11(x -1). Simplifying, the equation of the tangent line is y = 11x - 5. To illustrate this by graphing, imagine a curve representing y = 7x² - x³ and a straight line with the equation y = 11x - 5. The curve will be parabolic opening downwards (since the coefficient of x³ is negative), and the tangent line will intersect this curve precisely at the point (1,6), touching without crossing at that exact point.