Question

Find an equation for the tangent to the curve at the given point. f(x)=8 x −x+1,(64,1)

Answer 1

To find the tangent line equation at the point (64,1) for the curve f(x)=8x²-x+1, we first find the derivative, evaluate it at x=64 to get the slope, and then use the point-slope formula to write the equation as y = 1023x - 65471.

Step-by-step explanation:

To find an equation for the tangent to the curve f(x)=8x²-x+1 at the given point (64,1), we need to follow these steps:

1. Calculate the derivative of f(x) to find the slope of the tangent line at any point x.
2. Evaluate the derivative at x = 64 to get the slope m at the specific point where the tangent touches the curve.
3. Use the point-slope form of a line, y - y1 = m(x - x1), to write the equation of the tangent line using the point (64,1) and the slope m.

The equation of the derivative of f(x) is f'(x) = 16x - 1. Evaluating this at x = 64 gives us the slope m of the tangent line:

f'(64) = 16(64) - 1 = 1023.

The equation of the tangent line using the point-slope formula is:

y - 1 = 1023(x - 64).

Simplifying this equation gives us y = 1023x - 65471 as the final equation for the tangent line.