Consider the function below.
g(x) = 2 - x^3
(a) Find g'(0).
(b) Use it to find an equation of the tangent line to the curve y = 2 - x^3 at the point (0,2).
To find the derivative of a function, we need to apply the power rule. The power rule states that for any term of the form x^n, the derivative is given by nx^(n-1).
(a) To find g'(0), we need to differentiate the function g(x) = 2 - x^3 with respect to x. Applying the power rule, we have:
g'(x) = 0 - 3x^2 = -3x^2
Now, plug in x = 0 into g'(x):
g'(0) = -3(0)^2 = 0
Therefore, g'(0) is equal to 0.
(b) To find the equation of the tangent line to the curve y = 2 - x^3 at the point (0,2), we need to use the derivative g'(0) that we found in part (a).
The equation of the tangent line can be expressed in the form y = mx + b, where m represents the slope of the line and b represents the y-intercept.
Since the derivative g'(0) represents the slope of the tangent line, we have m = g'(0) = 0.
Now, substitute the known values (0,2) into the equation y = mx + b:
2 = 0(0) + b
Simplifying, we find that b = 2.
Therefore, the equation of the tangent line to the curve y = 2 - x^3 at the point (0,2) is y = 2.