Question

find equations of the tangent line and normal lines to the curve f at the given point f(x)=(4x+1)x^1/3 at x=1

Answer 1

The equations of the tangent and normal lines to the curve f(x) = (4x+1)x^(1/3) at the point x=1 are y = 5x - 1 and y = -1/5x + 6/5, respectively.

Step-by-step explanation:

To find the equation of the tangent line, we use the derivative to find the slope. The derivative f'(x) is calculated as 5x^(-2/3) + 4. Substituting x=1 gives the slope of the tangent line as 5. Using the point-slope form (y-y₁) = m(x-x₁), where (x₁, y₁) is the given point, yields the tangent line equation y = 5x - 1. The normal line has a slope that is the negative reciprocal of the tangent's slope, resulting in y = -1/5x + 6/5.

Note: Equations are rounded to reasonable decimal places for simplicity.