Final answer:
To find the equation for the tangent to the curve y=8x/x² at the point (1,4), use the derivative to find the slope of the tangent line and then use the point-slope form of a line to find the equation of the tangent.
Step-by-step explanation:
To find the equation for the tangent to the curve y=8x/x² at the point (1,4), we first need to find the derivative of the curve. Using the quotient rule, we have:
dy/dx = (8x² - 8x(2x))/(x²)² = (8x² - 16x²)/(x⁴) = (-8x²)/(x⁴)
Next, we substitute x=1 into the derivative to find the slope of the tangent line at the point (1,4):
m = (-8(1)²)/(1⁴) = -8/1 = -8
Now, we can use the point-slope form of a line to find the equation of the tangent:
y - 4 = -8(x - 1)
y - 4 = -8x + 8
y = -8x + 12
Therefore, the equation for the tangent to the curve y=8x/x² at the point (1,4) is y = -8x + 12.