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Calculating definite integrals is a fundamental operation in calculus. It allows us to find the "net" area under a curve over a particular interval on the x-axis or, in this problem's context, the t-axis. When we speak about definite integrals, we focus on the interval \[ [a, b] \] and calculate the integral of the function over this range.

For example, given the integral \[ \int_{0}^{10} e^{0.01t} \, dt \],we are essentially summing up all the values of \( f(t) = e^{0.01t} \) from \( t = 0 \) to \( t = 10 \).The process of solving this involves finding the antiderivative of the function, then evaluating it at the boundaries given by the interval.

• First, find the antiderivative: \[ \int e^{0.01t} \, dt = \frac{1}{0.01} e^{0.01t} = 100 e^{0.01t} \].
• Next, apply the limits of integration: \[ \left[ 100 e^{0.01t} \right]_{0}^{10} = 100 e^{0.1} - 100 e^{0} \].

By computing this step, we determine the net value spanned by the function across the interval.

Exponential functions are widely used in mathematics and sciences because they can model growth and decay processes. In this context, the function \( f(t) = e^{0.01t} \) is an exponential function where the rate of growth is captured by the exponent \( 0.01t \).

The base of the exponent, \( e \), is a mathematical constant approximately equal to \( 2.71828 \).It's known as Euler's number and is fundamental to natural growth processes. In expressions like \( e^{0.01t} \), this base naturally models constant percentage growth or decay over time.

• The function tells us that as \( t \) increases, \( f(t) \) grows exponentially.
• The multiplier \( 0.01 \)suggests a relatively slow growth rate compared to a function like \( e^{t} \).

Understanding exponential functions helps us grasp phenomena where change accelerates over time, such as population growth or radioactive decay.

Interval notation is a concise way to describe a range of numbers, usually denoting the domain over which a function is evaluated. When we see \([0, 10]\),we're looking at an interval including all numbers from 0 to 10, with both endpoints included.

• When the brackets are square \(([\,])\),it signals that the endpoints are part of the interval.
• Conversely, round brackets \((\, )\)would mean the endpoints are not included.

In many calculus problems, identifying the correct interval is crucial to properly setting up and evaluating problems like finding the average value or solving a definite integral.

Recognizing interval notation is fundamental, ensuring you compute functions over their appropriate domains, gathering accurate results.