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Differentiation is a key concept in calculus that involves finding the derivative of a function. It measures how a function changes as its input changes. In this context, differentiation helps us determine the function \( f(x) \) given the equation \( 4 + \int_{a}^{x} f(t) \, dt = e^{2x} \). By differentiating both sides of the equation with respect to \( x \), we use the Fundamental Theorem of Calculus which states that \( \frac{d}{dx} \int_{a}^{x} f(t) \, dt = f(x) \). This allows us to find \( f(x) \), the derivative of the given integral, which results in \( f(x) = 2e^{2x} \).Understanding differentiation:

• The process involves finding the slope or the rate of change of a function.
• For exponential functions like \( e^{2x} \), differentiation involves using the chain rule.
• The chain rule states that if you have a function inside another function, you differentiate the outer function and multiply it by the derivative of the inner function.

A definite integral calculates the accumulation of quantities, such as areas under a curve, from one point to another. In our equation, \( \int_{a}^{x} f(t) \, dt \), we are integrating the function \( f(t) \) from the lower limit \( a \) to the upper limit \( x \).What does this mean and how do you solve it?- It represents the total accumulation of the function \( f(t) \) between these limits.- In our example, this sum, when added to the constant \( 4 \), equals \( e^{2x} \).- By setting \( x = a \), we evaluate the integral from \( a \) to \( a \), which equals zero.- This fact helps establish the constant \( 4 = e^{2a} \), leading to finding the value of \( a \).Thus, definite integrals play a vital role in linking the accumulation of a function with algebraic expressions.

Exponential functions are functions where the variable exponent is the independent variable. They have the form \( e^{u(x)} \), where \( e \) is approximately 2.71828, and \( u(x) \) is a function of \( x \).Understanding exponential functions in our example:

• Exponential functions exhibit rapid growth, which means small changes in \( x \) can lead to large changes in the value of the function.
• In our original equation \( 4 + \int_{a}^{x} f(t) \, dt = e^{2x} \), the right side, \( e^{2x} \), is an exponential function whose differentiation gives \( 2e^{2x} \).
• This differentiation reveals the underlying function \( f(x) = 2e^{2x} \), showing how the rate of change corresponds with the behavior of exponential growth.

Thus, exponential functions are pivotal in modeling scenarios where quantities grow or decay at rates proportional to their current size, making them common in natural and financial contexts.