Final answer:
The student is asked to compute the derivative of the function g(x) = 7 - 2x³ using the definition of the derivative, which is the limit of (g(x+h) - g(x)) / h as h approaches zero. This process involves algebraic manipulation and taking a limit, representing the rate of change of the function g at a point x.
Step-by-step explanation:
The question posed involves finding the limit of a function as h approaches zero, which is a fundamental concept in calculus. Specifically, we are asked to evaluate the limit of (g(x+h) - g(x)) / h as h approaches zero for the function g(x) = 7 - 2x³. This limit represents the derivative of g(x) at x.
To solve this, we first compute g(x + h), which is 7 - 2(x + h)³. Then we subtract g(x) from this, yielding 7 - 2(x + h)³ - (7 - 2x³). We simplify this expression and divide by h. After simplifying, we should end up with an expression for which the limit as h approaches zero can be found. The resulting value is the derivative of g(x) at the point x, which tells us the slope of the tangent line to the graph of g(x) at that point.