Problem 246 - Calculus Volume 1

Question

\(\begin{array}{ll}& \text { [T] The graph below plots the quadratic }\end{array}\) \(p(t)=6.48 t^{2}-80.31 t+585.69\) against the data in preceding table, normalized so that \(t=0\) corresponds to 1963\. Estimate the average number of bald eagles per year present for the 37 years by computing the average value of \(p\) over [0,37]

Step-by-Step Solution

Answer

Compute the average value of the function over [0, 37] using integration.

Step 1: Understand the Problem

The problem requires us to find the average value of the quadratic function \(p(t) = 6.48t^2 - 80.31t + 585.69\) over the interval from \(t = 0\) to \(t = 37\). This represents the average number of bald eagles per year over these 37 years.

Step 2: Formula for Average Value of Function

The average value of a continuous function \(f(t)\) over the interval \([a, b]\) is given by \(\frac{1}{b-a} \int_{a}^{b} f(t) \, dt\). Here, \(f(t)\) is the quadratic \(p(t)\), \(a=0\), and \(b=37\).

Step 3: Set Up the Integral

Substitute \(p(t)\) into the average value formula: \[ \text{Average} = \frac{1}{37-0} \int_{0}^{37} (6.48t^2 - 80.31t + 585.69) \, dt. \]

Step 4: Calculate the Integral

Compute the definite integral: \[ \int (6.48t^2 - 80.31t + 585.69) \, dt = 6.48\frac{t^3}{3} - 80.31\frac{t^2}{2} + 585.69t \] Evaluating from 0 to 37.

Step 5: Evaluate from 0 to 37

Substitute 37 into the antiderivative:\[ 6.48\frac{37^3}{3} - 80.31\frac{37^2}{2} + 585.69 \times 37. \]Subtract the result of substituting 0 into the antiderivative (which is 0).

Step 6: Divide by Interval Length

Calculate the average by dividing the integral's value by 37:\[ \frac{1}{37} \left( 6.48\frac{37^3}{3} - 80.31\frac{37^2}{2} + 585.69 \times 37 \right). \] Compute this to get the average value.

Step 7: Compute Final Result

Calculate the result using the calculator or software for precision. The numerical result of the calculations in the previous step provides the average number of bald eagles observed per year.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that can be represented in the form \( ax^2 + bx + c \). It describes a parabola when plotted on a graph. The general shape is a "U" that can either open upwards or downwards depending on the sign of the leading coefficient \(a\).
Quadratic functions have several important properties. For example:

• The vertex of the parabola represents its highest or lowest point. This can be found using the vertex formula \(x = -\frac{b}{2a}\).
• The axis of symmetry is a vertical line that passes through the vertex, given by \(x = -\frac{b}{2a}\).
• The direction of the parabola (opening up or down) depends on the sign of \(a\): if \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards.

In the context of the given problem, \(p(t) = 6.48t^2 - 80.31t + 585.69\) is a quadratic function, detailing a parabola that gives the population trend of bald eagles over time.

Integral Calculus

Integral calculus is a fundamental part of calculus that focuses on accumulations and areas under curves. It helps us find things like total quantity, area, and in this case, the average value of functions.
Central concepts in integral calculus include:

• Indefinite integral: Represents a family of functions and is written using the integral sign \(\int\) without specific limits.
• Definite integral: Has limits and provides a specific numerical value.
• The Fundamental Theorem of Calculus: Connects differentiation and integration, establishing that applying the antiderivative to the boundaries of functions gives us the definite integral.

Understanding integral calculus is critical when computing the average value of functions, as it allows us to evaluate areas over which a function acts.

Definite Integral

The definite integral is a specific calculation within integral calculus that measures the area accumulated under the curve of a function between two points on the x-axis. It is denoted as:\[ \int_{a}^{b} f(t) \, dt \]where \(a\) and \(b\) are the bounds of integration, and \(f(t)\) is the function.
This integral gives a precise value that represents the total accumulation from point \(a\) to point \(b\). In our problem, we compute the definite integral of the quadratic function \(p(t)\) from \(t = 0\) to \(t = 37\), which helps us in finding the total representation of bald eagles over the given period.
Once calculated, the definite integral provides the numerator in the average value formula.

Antiderivative

The antiderivative is the reverse process of differentiation and is central to finding the integral of a function. It represents a function whose derivative gives back the original function. Finding the antiderivative of a function is an essential step in calculating integrals, especially definite integrals.
For polynomial functions like our quadratic \(p(t)\), the antiderivative can be straightforwardly determined. For instance, the antiderivative of \(6.48t^2\) is \(6.48\frac{t^3}{3}\), and similarly for other terms:

• \(6.48t^2 \rightarrow 6.48\frac{t^3}{3}\)
• \(-80.31t \rightarrow -80.31\frac{t^2}{2}\)
• \(585.69 \rightarrow 585.69t\)

This full antiderivative is used to evaluate the function's value at the boundaries of the integral \([0, 37]\), ultimately leading us to the definite integral's value to then find the average. Understanding antiderivatives is key to solving integrals and finding average values of functions over an interval.