Question

Multiple Choice What is the average value of the cosine function on the interval [ 1,5 ] ? \(\begin{array} { l l } { \text { (A) } - 0.990 } & { ( \text { B) } - 0.450 } \\\ { \text { (D) } 0.412 } & { ( \text { E) } 0.998 } \end{array}\)

Short answer

The average value of the cosine function on the interval [1,5] is \( \frac{1}{4} \) times \( \sin(5) - \sin(1) \).

Step-by-step solution

Average Value Formula

Remember, the average value of a function on an interval [a,b] is given by \( \frac{1}{b-a} \) times the integral from a to b of the function. In this case, a=1, b=5, and the function is the cosine function.

Calculate the Integral

Calculate the definite integral from 1 to 5 of the cosine function. The integral of cosine is sine, so this will result in \( \sin(5) - \sin(1) \).

Calculate Average Value

Substitute a=1 and b=5 into the average value formula, and multiply it by the result from the definite integral. The calculation is \( \frac{1}{5 - 1} \) times \( \sin(5) - \sin(1) \). Simplify this to get your final result.

Key concepts

Definite Integral

The definite integral is a fundamental concept in calculus that quantifies the area under a curve. When we integrate a function between two points, say from point 'a' to point 'b', we are effectively finding the total accumulation of the function’s values across that interval. This process is directly linked to the idea of finding the average value over an interval as it provides the total sum that will later be adjusted by the interval's length.

To compute a definite integral, we use the integral symbol along with the lower limit 'a' and the upper limit 'b'. The basic notation looks like this: \[ \int_{a}^{b} f(x) \, dx \] where \( f(x) \) is the function we’re integrating. If the function represents a physical quantity like distance, the definite integral can be interpreted as total distance traveled between time 'a' and time 'b'.

Cosine Function

The cosine function is one of the primary trigonometric functions and is periodic with a period of \(2\pi\), which means it repeats its values every \(2\pi\) units along the horizontal axis. It’s defined for all real numbers and fluctuates between 1 and -1. The function can be written as \(cos(x)\), where 'x' usually represents an angle in radians.

The cosine graph has a peak at \(0\) and troughs at multiples of \(\pi\). When we integrate the cosine function over a period, the area above the axis cancels the area below the axis, resulting in a net area of zero. However, the definite integral over an interval that is not aligned with the cosine function’s period doesn’t necessarily cancel out and must be evaluated for the specific limits.

Average Value Formula

The average value formula is central to understanding how to find the typical or mean value of a function over a certain interval. It is defined as the total sum of function values divided by the length of the interval.

Mathematically, if we want to find the average value of a function \(f(x)\) over the interval [a,b], we use the formula: \[ \text{Average Value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \] This formula is particularly useful as it takes into account the function's behavior over the entire interval, rather than just at discrete points. In the context of periodic functions like the cosine function, the average value over a full period is always zero, but over any other interval, it needs to be computed specifically using this formula.

Integration

Integration is the process of finding the integral of a function. In the broad sense, it can be seen as the reverse of differentiation. While differentiation is about finding the rate at which something changes, integration is about finding the accumulation over an interval.

For example, if you are given a velocity function and you integrate it over time, you will obtain the total distance traveled. In practice, integration can be used to solve a variety of problems including area under a curve, the average value of functions, and the total growth given a rate of growth. There are several methods for integration, such as substitution, integration by parts, and partial fraction decomposition, but the basic approach for simple functions involves finding an antiderivative and applying the limits of integration.