The number of Internet users in China is projected to be...

The number of Internet users in China is projected to be $$ N(t)=94.5 e^{0.2 t} \quad(1 \leq t \leq 6) $$ where $N(t)$ is measured in millions and $t$ is measured in years, with $t=1$ corresponding to the beginning of 2005 . a. How many Internet users were there at the beginning of $2005 ?$ At the beginning of 2006 ? b. How many Internet users are there expected to be at the beginning of 2010 ? c. Sketch the graph of $N$.

The number of Internet users in China is projected to be
$$
N(t)=94.5 e^{0.2 t} \quad(1 \leq t \leq 6)
$$
where $N(t)$ is measured in millions and $t$ is measured in years, with $t=1$ corresponding to the beginning of 2005 .
a. How many Internet users were there at the beginning of $2005 ?$ At the beginning of 2006 ?
b. How many Internet users are there expected to be at the beginning of 2010 ?
c. Sketch the graph of $N$.

Key Concepts

Function Evaluation at Specific Points

Evaluating a function at specific points involves substituting given values into the function’s equation to calculate the output. In the context of exponential functions, this process is used to determine initial conditions, future forecasts, or any other point of interest by calculating A * e^(kt) for a particular value of t.

Graphing Exponential Functions and Domain Considerations

Graphing an exponential function involves plotting its continuous, curved line on a coordinate system with attention to its increasing or decreasing nature. It is important to consider the domain, which is the set of t-values for which the function is defined. This domain restriction is crucial in connecting the mathematical model to a real-world timeline or context.

Exponential Growth

Exponential growth describes a process where a quantity increases at a rate proportional to its current value. This concept is central to many real-world applications, allowing for modeling situations where growth accelerates over time. The general form is A * e^(kt), where A is the initial amount and k is a constant rate of growth.

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent. They are used to represent growth or decay processes, and their graphs typically exhibit rapid increases or decreases dependent on the sign of the exponent. These functions have the form f(t) = A * e^(kt), which allows for easy manipulation and prediction of future values.

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