Let's work through this step-by-step.

Given: $f(x) = x^2 + 3x - 5$

Part (a)

To estimate the instantaneous rate of change at $x = 1$, we need to find the derivative $f'(x)$ and evaluate it at $x = 1$. However, since this part asks for an estimation, let's use the definition of the derivative:

$f'(1) \approx \frac{f(1+h) - f(1)}{h}$ where $h$ is a small value. Let's use a small value for $h$ (like $h = 0.01$) for the estimation:

• $f(1) = 1^2 + 3 \cdot 1 - 5 = -1$
• $f(1 + 0.01) = (1 + 0.01)^2 + 3 \cdot (1 + 0.01) - 5$

Calculate the values and then approximate the rate of change.

Part (b)

We are asked to simplify the expression: $\frac{f(x+h) - f(x)}{h}$ Substitute $f(x) = x^2 + 3x - 5$:

1. $f(x+h) = (x+h)^2 + 3(x+h) - 5$
2. Expand $f(x+h)$: $f(x+h) = x^2 + 2xh + h^2 + 3x + 3h - 5$
3. Compute $f(x+h) - f(x)$: $f(x+h) - f(x) = (x^2 + 2xh + h^2 + 3x + 3h - 5) - (x^2 + 3x - 5)$
4. Simplify this expression by canceling terms: $f(x+h) - f(x) = 2xh + h^2 + 3h$
5. Factor out $h$: $\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + 3h}{h} = 2x + h + 3$

Part (c)

Examine the expression $2x + h + 3$ as $h \to 0$. As $h$ approaches 0, the expression simplifies to: $2x + 3$

Part (d)

Using the result from part (c), we find that the expression for the instantaneous rate of change (the derivative) at any point $x$ is: $f'(x) = 2x + 3$ To verify this with the result from part (a) for $x = 1$: $f'(1) = 2 \cdot 1 + 3 = 5$

Thus, the instantaneous rate of change at $x = 1$ is indeed 5, confirming our result from part (a).

Would you like any further details or have additional questions?

Here are 5 related questions to further expand on this topic:

1. What is the geometric interpretation of the instantaneous rate of change?
2. How would the rate of change change if the function were $f(x) = x^2 + 3x - 5 + c$, where $c$ is a constant?
3. How does the concept of a derivative connect to the slope of a tangent line?
4. What is the significance of the limit process in defining derivatives?
5. How would you find the second derivative of $f(x) = x^2 + 3x - 5$ and what does it represent?

Tip: When estimating the instantaneous rate of change, choosing a smaller $h$-value makes the estimation closer to the actual derivative value.