Final answer:
The equation of the line through the point (-4,5) and perpendicular to the curve y = x⁴ - 1 is found by first getting the derivative of the curve to determine the slope of the tangent line. Then, take the negative reciprocal of this slope to find the slope of the perpendicular line. Finally, use the point-slope form to get the perpendicular line equation, which is y = (1/256)x + 261/64.
Step-by-step explanation:
To find the equation of the line that goes through the point (-4,5) and is perpendicular to the curve given by y = x⁴ - 1, we first need to determine the slope of the tangent to the curve at the point where the perpendicular line will intersect. Let's find the derivative of the curve, which gives us the slope of the tangent line at any point on the curve. The derivative of y = x⁴ - 1 is dy/dx = 4x³. We can then find the slope of the tangent line at x = -4, which will be 4*(-4)³ = -256. The slope of the line perpendicular to the tangent is the negative reciprocal of the tangent slope, therefore, the slope of our perpendicular line is 1/256.
Now, we use the point-slope form of the line equation, y - y1 = m(x - x1), where (x1, y1) is the point the line passes through (in this case (-4,5)) and m is the slope of the line. Substituting the given point and the perpendicular slope into the equation, we get y - 5 = (1/256)(x + 4). Simplifying this equation, we find the equation of the line through (-4,5) that is perpendicular to the curve: y = (1/256)x + 261/64.