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Composite Function Evaluation is a technique that involves substituting one function into another to form a new function. This new function is called a composite function. Let's break down what this means practically.

In the given exercise, we have two functions: \(c(x) = 3x^2 - 2x + 5\) and \(x(t) = 2e^t\). To evaluate the composite function, we substitute \(x(t)\) into \(c(x)\). Essentially, wherever there's an \(x\) in \(c(x)\), we replace it with \(x(t)\).

This gives us \(c(x(t)) = 3(2e^t)^2 - 2(2e^t) + 5\), which simplifies to \(12e^{2t} - 4e^t + 5\). By doing so, we've effectively combined two separate functions into a single expression. Evaluating these composite functions often requires simplifying the expression and then substituting in any given values for additional calculations.

Function Substitution is the method of replacing a variable in one function with another function. This technique is essential for understanding composite functions and is a gateway to tackling more complex calculus problems.

Looking at our functions, \(x(t) = 2e^t\) is substituted for \(x\) in the function \(c(x) = 3x^2 - 2x + 5\). The substitution is straightforward—replace each occurrence of \(x\) with \(2e^t\).

This allows us to create a new equation: \(c(x(t)) = 3(2e^t)^2 - 2(2e^t) + 5\). From here, we continue simplifying the terms. Substitute and simplify each step to retain accuracy and reach the solution smoothly. This kind of substitution is not only foundational in creating composite functions but is also a useful skill in finding derivatives and integrals later on.

Calculus Problem Solving involves applying techniques like composite functions and substitutions to solve real-world problems involving rates of change, areas, and other variables.In this particular problem, after forming the composite function \(c(x(t)) = 12e^{2t} - 4e^t + 5\), our task was to evaluate it at a specific point, \(t = 2\). This step, known in calculus as evaluating the function, involves substituting \(t = 2\) into our simplified composite expression.

The steps are detailed but break down to simpler actions:

• First, plug \(t = 2\) into \(c(x(t)) = 12e^{2t} - 4e^t + 5\).
• Evaluate this by simplifying terms: \(c(x(2)) = 12e^4 - 4e^2 + 5\).
• Substitute known values of \(e\) to approximate results.

Approximations and understanding the exponential function \(e\) are key in calculus, as many functions involve these continual variable manipulations. By mastering these steps, students become adept at handling complex equations systematically.