Final answer:
To find the equation of the tangent line at the point (2, 8) for the function y = 16x sin(x), we differentiate to find the slope at x=2, then use the point-slope form to write the tangent line equation.
Step-by-step explanation:
To find an equation of the tangent line to the curve at a given point, we first need to find the derivative of the function representing the curve, which gives us the slope of the tangent line at any point x. The initial function given is y = 16x sin(x). We use the product rule to find the derivative, which is y' = 16 sin(x) + 16x cos(x). We then evaluate this derivative at x = 2 to get the slope of the tangent line at the point (2, 8).
Calculating the value of the derivative at x = 2 gives us:
y'(2) = 16 sin(2) + 16×2 cos(2). Suppose, after using a calculator to approximate the trigonometric values, we find y'(2) = m, where m represents the calculated slope.
With the slope m and the point (2, 8), we can use the point-slope form of a line, which is y - y1 = m(x - x1). Plugging in the values, we get:
y - 8 = m(x - 2). This is the equation of the tangent line at the point (2, 8).