Let's break down the problem and analyze the statement.

We are given the function:

$f(x) = x^2 - 9x + 2$

Step 1: Slope of the Secant Line

The secant line passes through the points $(-2, f(-2))$ and $(b, f(b))$.

• First, compute $f(-2)$:

$f(-2) = (-2)^2 - 9(-2) + 2 = 4 + 18 + 2 = 24$

So the point $(-2, f(-2)) = (-2, 24)$.

The slope of the secant line between $(-2, 24)$ and $(b, f(b))$ is given by:

$\text{slope of secant} = \frac{f(b) - f(-2)}{b - (-2)} = \frac{f(b) - 24}{b + 2}$

Step 2: Slope of the Tangent Line at $x = -2$

To find the slope of the tangent line, we need the derivative of $f(x)$:

$f'(x) = \frac{d}{dx}(x^2 - 9x + 2) = 2x - 9$

Now, evaluate the derivative at $x = -2$:

$f'(-2) = 2(-2) - 9 = -4 - 9 = -13$

So, the slope of the tangent line at $x = -2$ is $-13$.

Step 3: Analyzing the Limit of the Secant Slope as $b$ Approaches $-2$

We want to check if the slope of the secant line approaches the slope of the tangent line as $b$ gets close to $-2$. This means we need to evaluate:

$\lim_{b \to -2} \frac{f(b) - 24}{b + 2}$

Using L'Hopital's Rule (since both the numerator and denominator approach 0 as $b \to -2$):

1. Differentiate the numerator: $f'(b) = 2b - 9$
2. Differentiate the denominator: $\frac{d}{db}(b + 2) = 1$

So, applying L'Hopital's Rule:

$\lim_{b \to -2} \frac{f(b) - 24}{b + 2} = \lim_{b \to -2} (2b - 9) = 2(-2) - 9 = -13$

Conclusion

The slope of the secant line does indeed approach the slope of the tangent line ($-13$) as $b$ approaches $-2$. Therefore, the statement is True.

Would you like further details or explanations on any of these steps? Here are some related questions for practice:

1. What is the geometric interpretation of the derivative in terms of secant and tangent lines?
2. How is L'Hopital's Rule used to evaluate limits that result in an indeterminate form?
3. What is the significance of the derivative of a function at a point?
4. How does the limit definition of the derivative relate to the slopes of secant and tangent lines?
5. How can the concept of limits be applied to real-world situations involving rates of change?

Tip: When computing limits of functions, L'Hopital's Rule is very useful when both the numerator and denominator approach 0 or $\infty$.