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Exponential functions play a vital role in mathematics, characterized by their consistent rate of change. To graph these functions, we look at their forms. Here, we have two specific functions: \(f(x) = 10(0.85^x)\) and \(g(x) = 6(0.75^x)\). Both are examples of exponential decay because their bases (0.85 and 0.75) are less than one. This means these functions decrease as \(x\) increases.
To graph these functions:

• Identify the point where each graph starts by evaluating the function at \(x = 0\). \(f(0) = 10\) and \(g(0) = 6\).
• Observe that \(f(x)\) begins higher than \(g(x)\) and declines less steeply, as 0.85 is closer to 1 than 0.75 is.
• The interval of interest is from \(x = 2\) to \(x = 10\), where we sketch these functions to see how they behave in this range.
• Shade the area between the two curves to highlight the region of interest.

By analyzing their equations and sketching their graphs, you get a deeper insight into how exponential functions reduce over an interval, especially with different bases.

Integration is the mathematical process of finding the area under a curve. For exponential functions like \(f(x) = 10(0.85^x)\) and \(g(x) = 6(0.75^x)\), the integration process provides the cumulative area from one point to another along the \(x\)-axis.
Exponential functions have a specific property: The integral of \(a^x\) is given by \(\frac{a^x}{\ln(a)}\), where \(\ln(a)\) is the natural logarithm of \(a\). Applying this to our functions:

• The integral of \(f(x) = 10(0.85^x)\) is \(\frac{10(0.85^x)}{\ln(0.85)}\).
• For \(g(x) = 6(0.75^x)\), the integral is \(\frac{6(0.75^x)}{\ln(0.75)}\).

When performing these integrations over a specific interval, say from \(x = 2\) to \(x = 10\), we evaluate each at the upper and lower bounds, then subtract. Using integration formulas for exponential functions is handy because it standardizes the process and minimizes errors. This makes it easier to calculate the desired area.

The area between curves is a common problem in calculus that requires both graphing and integration skills. When you're asked to find the area between two curves—like with \(f(x) = 10(0.85^x)\) and \(g(x) = 6(0.75^x)\) from \(x = 2\) to \(x = 10\)—you first find the difference between the two functions in terms of integration.
To set this up, the integral for the area \(A\) is:

• \[ A = \int_{2}^{10} [f(x) - g(x)] \, dx = \int_{2}^{10} [10(0.85^x) - 6(0.75^x)] \, dx \]

The steps to calculate this include:

• Individually integrate each function over the given bounds.
• Evaluate these integrals at \(x = 10\) and \(x = 2\), then subtract the lower bound from the upper bound value.
• Finally, subtract the integrated area under \(g(x)\) from that under \(f(x)\) to find the total area between the two curves.

Calculating areas between curves is essential for understanding differences in growth patterns, which is widely applicable in fields such as economics and environmental sciences.