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.