Question

Use the definition of derivative, ′(x) = lim(h→0) [f(x + h) - f(x)] / h to differentiate the following, and then find the equation of the tangent line to y = f(x) at the point x = -1:

f(x) = √3; x = -1
a) f'(x) = 0; y = 0
b) f'(x) = 1/√3; y = 1/√3(x + 1)
c) f'(x) = 1/(2√3); y = 1/(2√3)(x + 1)
d) f'(x) = 2√3; y = 2√3(x + 1)

Answer 1

The derivative of the constant function f(x) = √3 is zero, and the equation of the tangent line at x = -1 is y = √3, which is a horizontal line at the height of the constant function.

Step-by-step explanation:

To answer the student's question, we need to find the derivative of the function f(x) using the definition of the derivative and then find the equation of the tangent line to y = f(x) at the point x = -1. The function given is f(x) = √3, which is a constant function. The derivative of a constant is always 0, and thus f'(x) = 0.

Now we need to find the equation of the tangent line. Since the derivative represents the slope of the tangent line at a given point, and the slope is zero, the tangent line is horizontal. The equation of a horizontal line is simply y = c, where c is the y-value of the function at the point in question. Since f(x) is always √3, the equation of the tangent line at any point, including x = -1, is y = √3.

Thus, the correct answer to the student's question is a) f'(x) = 0; y = √3, indicating that the slope of the tangent line is zero and that the tangent line is a horizontal line at the constant value of the function.