Final answer:
To find the equations of the tangent line and normal line to the curve y = x⁴ at the point (1, 0.2), we first need to find the derivative of the curve. The derivative of y = x⁴ is dy/dx = 4x³. The slope of the tangent line is 4, and the equation of the tangent line is y = 4x - 3.8. The slope of the normal line is -1/4, and the equation of the normal line is y = -0.25x + 0.45.
Step-by-step explanation:
To find the equations of the tangent line and normal line to the curve y = x⁴ at the point (1, 0.2), we first need to find the derivative of the curve. The derivative of y = x⁴ is dy/dx = 4x³. At the point (1, 0.2), the derivative is dy/dx = 4(1)³ = 4. This slope represents the slope of the tangent line.
Using the point-slope form of a linear equation, we can write the equation of the tangent line as y - y₁ = m(x - x₁), where (x₁, y₁) is the point of tangency and m is the slope. Substituting the values, we get y - 0.2 = 4(x - 1). Simplifying the equation gives y = 4x - 3.8, which is the equation of the tangent line.
The normal line is perpendicular to the tangent line and has a slope equal to the negative reciprocal of the slope of the tangent line. So the slope of the normal line is -1/4. Using the point-slope form again, the equation of the normal line can be written as y - 0.2 = -1/4(x - 1). Simplifying gives y = -0.25x + 0.45, which is the equation of the normal line.