Given the function $f(x) = x^2$, let's go through each part.

Part (A)

The slope of the secant line between two points on $f(x)$ is given by: $\text{slope} = \frac{f(a + h) - f(a)}{h}$

For $a = 1$:

1. When $h = 2$: $f(1 + 2) = f(3) = 3^2 = 9, \quad f(1) = 1^2 = 1$ $\text{slope} = \frac{f(3) - f(1)}{2} = \frac{9 - 1}{2} = \frac{8}{2} = 4$

2. When $h = 1$: $f(1 + 1) = f(2) = 2^2 = 4, \quad f(1) = 1^2 = 1$ $\text{slope} = \frac{f(2) - f(1)}{1} = \frac{4 - 1}{1} = 3$

So, the slopes of the secant lines for $h = 2$ and $h = 1$ are 4 and 3, respectively.

To graph $y = f(x) = x^2$ and the two secant lines, plot the curve $y = x^2$ and draw lines with these slopes passing through the point $(1,1)$.

Part (B)

Now, let’s find the slope of the secant line for $a = 1$ and any nonzero $h$.

1. Substitute $a = 1$ into the general formula: $\text{slope} = \frac{f(1 + h) - f(1)}{h}$
2. Substitute $f(1 + h) = (1 + h)^2$ and $f(1) = 1^2 = 1$: $\text{slope} = \frac{(1 + h)^2 - 1}{h}$
3. Expand $(1 + h)^2 = 1 + 2h + h^2$: $\text{slope} = \frac{1 + 2h + h^2 - 1}{h} = \frac{2h + h^2}{h} = \frac{h(2 + h)}{h} = 2 + h$

So, the slope of the secant line for $a = 1$ and any nonzero $h$ simplifies to $2 + h$.

Part (C)

To find the limit of the expression $2 + h$ as $h$ approaches 0: $\lim_{h \to 0} (2 + h) = 2$

Part (D)

The limit found in part (C) represents the slope of the tangent line to the curve $y = x^2$ at the point $x = 1$. Geometrically, as $h$ approaches 0, the secant line approaches the tangent line, so the slope of the secant line approaches the slope of the tangent line. This slope of 2 is the derivative $f'(1)$, which represents the instantaneous rate of change of $f(x) = x^2$ at $x = 1$.

Would you like to go over any specific part in more detail, or have any questions about the concepts here?

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Related Questions

1. How would the slope of the secant line change if $f(x) = x^3$?
2. What is the general formula for the slope of a secant line for a function $f(x)$?
3. How does the concept of a secant line help in understanding derivatives?
4. What happens if we choose other values for $a$ in this example?
5. How would you calculate the tangent line equation at $x = 1$?

Tip:

The slope of a secant line gives an average rate of change between two points, while the slope of a tangent line shows the instantaneous rate of change at a single point.