To address the exercises from the uploaded image, I will walk through some solutions. Let me know which problems you'd like to focus on, or I can proceed with detailed explanations for all exercises.

Part 1: Average Rate of Change

The formula for the average rate of change of a function $f(x)$ over an interval $[a, b]$ is: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}.$

Example Calculation:

1. $f(x) = x^3 + 1$, Interval: $[2, 3]$:

   • $f(2) = 2^3 + 1 = 9$,
   • $f(3) = 3^3 + 1 = 28$, $\text{Rate of Change} = \frac{28 - 9}{3 - 2} = 19.$

2. $g(x) = x^2$, Interval: $[-1, 1]$:

   • $g(-1) = (-1)^2 = 1$,
   • $g(1) = (1)^2 = 1$, $\text{Rate of Change} = \frac{1 - 1}{1 - (-1)} = 0.$

Would you like me to calculate other rates of change explicitly?

Part 2: Slope of a Curve at a Point

The slope of a curve at a point is found using the derivative of the function. For $y = f(x)$, the derivative is: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}.$

The equation of the tangent line at a point $P(x_0, y_0)$ is: $y - y_0 = f'(x_0)(x - x_0).$

Example Calculation:

7. $y = x^2 - 3$, Point $P(2, 1)$:
   • Derivative: $f'(x) = 2x$,
   • Slope at $x = 2$: $f'(2) = 2(2) = 4$,
   • Tangent Line: $y - 1 = 4(x - 2)$ or $y = 4x - 7$.

Would you like more tangent line equations worked out?

Let me know your preferred focus or if you'd like further details for all problems. Here are 5 questions to expand on this:

1. Would you like step-by-step solutions for other intervals in Part 1?
2. Do you need detailed derivative calculations for Part 2?
3. Would graphing any of these functions or tangent lines be helpful?
4. Should I cover the trigonometric or square root-based problems next?
5. Are there specific problem-solving tips you'd like for this section?

Tip: For derivatives, practice differentiating basic functions (polynomials, trigonometric, square root) to build intuition for more complex problems.