Find the slope of the graph of the function \( f(x)=\sqrt{2 x} \) at \( (8,4) \). Then fir Which limit below is equal to the slope \( m \) of the curve at the given poi A. \( \lim _{h \rightarrow 0} \frac{\sqrt{2(8+h)}-\sqrt{2(8)}}{h} \) B. \( \lim _{h \rightarrow 0} \frac{\sqrt{2(8-h)}+\sqrt{2(8)}}{h} \) C. \( \lim _{h \rightarrow 0} \frac{\sqrt{8+h}-\sqrt{8}}{h} \) D. \( \lim _{h \rightarrow 0} \frac{\sqrt{2(8)}-\sqrt{2(8+h)}}{h} \)

To find the slope of the graph of the function \( f(x) = \sqrt{2x} \) at the point \( (8,4) \), we first need to compute the derivative \( f'(x) \). Using the power rule and the chain rule, we differentiate: \[ f'(x) = \frac{d}{dx}(\sqrt{2x}) = \frac{1}{2\sqrt{2x}} \cdot 2 = \frac{1}{\sqrt{2x}}. \] Now we evaluate this at \( x = 8 \): \[ f'(8) = \frac{1}{\sqrt{2 \cdot 8}} = \frac{1}{\sqrt{16}} = \frac{1}{4}. \] Thus, the slope of the graph at the point \( (8, 4) \) is \( \frac{1}{4} \). To determine which limit is equal to the slope \( m \) at the given point, we need to analyze the options. The correct limit corresponds directly to the definition of the derivative at that point. The correct choice is: **A.** \( \lim _{h \rightarrow 0} \frac{\sqrt{2(8+h)}-\sqrt{2(8)}}{h} \) This matches the form for finding the derivative at a function value, since it computes the difference quotient based on the function \( f(x) = \sqrt{2x} \) at \( x = 8 \). --- When you investigate calculus further, it’s fascinating to see how the concept of a slope translates across various fields! For instance, engineers rely on derivatives to calculate rates of change in systems, whether that’s speed on a highway or the rate of pressure drop in fluid dynamics—talk about a real-world application of math! If you're eager for deeper exploration, consider diving into more advanced calculus topics such as Taylor series or optimization problems! They're perfect for understanding not just how to find slopes, but also how to make the most of them in practical contexts like maximizing profits or minimizing costs.