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Function evaluation involves finding the output value for a specific input in a given function. It's like putting a number into a machine and seeing what comes out. For the function \(f(x) = \sqrt{3x}\), we want to know the value \(f(x)\) when \(x\) takes different values.
The first step involves calculating the initial output when \(x = 1\). We substitute \(x\) into the function:

• \(f(1) = \sqrt{3 \times 1} = \sqrt{3}\)

This process gives us the baseline result of the function for the specific input. Function evaluation is a crucial first step in understanding how a function behaves as its inputs change.

The change in output refers to the difference in the function's value after a change in the input value. This concept helps us understand how sensitive the function is to changes in its input.
Our task here involves finding the function value for a new input, \(1 + \Delta x = 1.01\). We substitute this new \(x\) value into the function:

• \(f(1.01) = \sqrt{3 \times 1.01}\)

Now that we have both the initial output \(f(1) = \sqrt{3}\) and the output after the input change \(f(1.01)\), the change in output \(\Delta y\) is:

• \(\Delta y = f(1.01) - f(1)\)

This difference \(\Delta y\) tells us how much the function's output has changed due to a small change in the input.

Understanding small changes in input, denoted by \(\Delta x\), is key in differential calculus. It helps us analyze how a function responds as its input slightly varies. Here \(\Delta x = 0.01\) represents a very small increase from the original input, altering \(x\) from \(1\) to \(1.01\).
This small change might not seem significant, but its effect is critical. By computing the change in output as described earlier, we can observe the function's sensitivity to variations in input. This subtlety underlines the foundation of many calculus concepts, including derivatives. Such calculations help us visualize how a small shift can impact results, crucial for insights and predictions across different applications in mathematics and science.