Question

Find the slope and the equation of the tangent-line to the graph of the function at The slope of the tangent ine is the given value of x (Smplify your answet.) f(x)=x^{4}-20 x^{2}+64, x=1

Answer 1

To find the slope of the tangent line to the graph of f(x) at x=1, one must differentiate f(x), then evaluate the derivative at x=1. The slope at x=1 is -36, and the equation of the tangent line is y - 45 = -36(x - 1).

Step-by-step explanation:

The question pertains to finding the slope of the tangent line to the function f(x) = x^4 - 20x^2 + 64 at the point where x = 1.

To find this slope, we will first find the derivative of the function, which gives us the slope of the tangent at any point x.

1. Find the derivative of f(x): f'(x) = 4x^3 - 40x.
2. Evaluate the derivative at x = 1: f'(1) = 4(1)^3 - 40(1) = 4 - 40 = -36.
3. Write the equation of the tangent line using the point-slope form: y - f(1) = -36(x - 1).
4. Calculate f(1) to find the point: f(1) = 1^4 - 20(1)^2 + 64 = 1 - 20 + 64 = 45.
5. Plug the point into the equation: y - 45 = -36(x - 1).

This gives us the equation of the tangent line at x = 1 for the given function.