Working with area functions Consider the function ƒ and the points a, b, and c.
(b) Graph ƒ and A.
ƒ(𝓍) = eˣ ; a = 0 , b = ln 2 , c = ln 4

Verified step by step guidance

1. First, understand that the function given is \(f(x) = e^x\), which is an exponential function that grows as \(x\) increases.
2. Next, identify the points \(a = 0\), \(b = \ln 2\), and \(c = \ln 4\). These points will be important for defining intervals on the \(x\)-axis.
3. The area function \(A(x)\) typically represents the area under the curve \(f(t)\) from a fixed point \(a\) to a variable upper limit \(x\). So, define \(A(x)\) as the integral \(A(x) = \int_{0}^{x} e^t \, dt\).
4. To graph \(f(x) = e^x\), plot the exponential curve starting at \(f(0) = 1\) and increasing rapidly. Mark the points \(x = 0\), \(x = \ln 2\), and \(x = \ln 4\) on the \(x\)-axis to show the intervals of interest.
5. To graph the area function \(A(x)\), recognize that \(A(x)\) is the accumulation of the area under \(f(t)\) from \$0\( to \)x\(. Since the integral of \)e^t\( is \)e^t\(, \)A(x)\( can be expressed as \)A(x) = e^x - e^0 = e^x - 1\(. Plot this function alongside \)f(x)$ to compare the original function and its area function.

Key Concepts

Area Functions and the Definite Integral

An area function A(x) represents the accumulated area under the curve of a function ƒ from a fixed point a to a variable point x. It is defined as A(x) = ∫_a^x ƒ(t) dt, linking the concept of integration to area calculation under ƒ. Understanding this helps in interpreting A graphically and analytically.

Exponential Functions and Their Properties

The function ƒ(x) = e^x is an exponential function with base e, characterized by its continuous growth and the unique property that its derivative equals itself. Knowing how to evaluate and graph e^x, especially at points like ln 2 and ln 4, is essential for plotting ƒ and understanding the behavior of the area function.

Natural Logarithm and Its Relationship to Exponentials

The natural logarithm ln x is the inverse of the exponential function e^x, meaning ln(e^x) = x. Points like b = ln 2 and c = ln 4 correspond to x-values where e^x equals 2 and 4, respectively. This relationship is crucial for interpreting the given points and accurately graphing both ƒ and the area function A.