sympy import symbols, limit, sqrt

Define symbols

x = symbols('x')

Problem 1

f1 = x**3 x1 = 3 instant_rate_f1 = limit((f1 - f1.subs(x, x1))/(x - x1), x, x1)

Problem 2

f2 = 2 * sqrt(x) x2 = 4 instant_rate_f2 = limit((f2 - f2.subs(x, x2))/(x - x2), x, x2)

Problem 3

f3 = 6 / x x3 = -3 instant_rate_f3 = limit((f3 - f3.subs(x, x3))/(x - x3), x, x3)

instant_rate_f1, instant_rate_f2, instant_rate_f3Here are the instantaneous rates of change for the given functions:

1. For $f(x) = x^3$ at $x = 3$, the instantaneous rate of change is 27.
2. For $y = 2\sqrt{x}$ at $x = 4$, the instantaneous rate of change is 1/2.
3. For $f(x) = \frac{6}{x}$ at $x = -3$, the instantaneous rate of change is -2/3.

Let me know if you'd like further explanations or have any questions.

Here are some related questions to consider:

1. How can the concept of limits be applied to real-world problems involving rates of change?
2. What are the similarities and differences between using limits and derivatives to calculate instantaneous rates of change?
3. How does the choice of function type (polynomial, radical, rational) affect the computation of limits?
4. Could these results be obtained using derivative rules instead of limits? How?
5. What is the interpretation of the instantaneous rate of change for each of these functions in a real-world context?

Tip: When using limits to find the instantaneous rate of change, always ensure the denominator approaches zero carefully to avoid undefined expressions.