Question

let f be the function definded by f(x) = 4x^3 -5x + 3. which of the following is an equation of the line tangent to the graph of f at the point where x = -1?

Answer 1

To find the equation of the tangent line at x = -1 for the function f(x) = 4x^3 -5x + 3, first calculate the slope using the derivative of f(x), then find the y-coordinate at x = -1, and finally use the point-slope form to write the equation as y - 4 = 7(x + 1).

Step-by-step explanation:

To find the equation of the line tangent to the graph of f at the point where x = -1, we need to:

1. Calculate the derivative of f(x) to find the slope of the tangent line at x = -1.
2. Plug in x = -1 into the function f(x) to find the y-coordinate of the point of tangency.
3. Use the point-slope form of a line to write the equation of the tangent line.

The derivative of f(x) = 4x3 -5x + 3 is f'(x) = 12x2 - 5. Plugging in x = -1 gives us f'(-1) = 12(-1)2 - 5 = 7, which is the slope of the tangent line at x = -1.

Now, we evaluate f(-1) = 4(-1)3 -5(-1) + 3 = 4, which gives us the y-coordinate of the point of tangency.

With a slope of 7 and passing through the point (-1, 4), the equation of the tangent line can be written using the point-slope form: y - 4 = 7(x + 1).