Problem 247 - Calculus Volume 1

Question

ITI The graph below plots the cubic \(p(t)=0.07 t^{3}+2.42 t^{2}-25.63 t+521.23 \quad\) against the data in the 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

Calculate the definite integral of the polynomial and divide by 37.

Step 1: Understanding the Function

The function given is a cubic polynomial of the form \(p(t) = 0.07 t^3 + 2.42 t^2 - 25.63 t + 521.23\). This function models the number of bald eagles over time, starting from \(t=0\), which corresponds to the year 1963.

Step 2: Determine the Time Interval

The average is to be calculated over the interval \([0, 37]\), which calculates from the year 1963 to 2000 (1963 + 37 years).

Step 3: Formula for Average Value

The average value of a continuous function \(p(t)\) over an interval \([a, b]\) can be calculated using the formula:\[\text{Average value} = \frac{1}{b-a} \int_{a}^{b} p(t)\, dt\]Here, \(a = 0\) and \(b = 37\).

Step 4: Calculate the Definite Integral

To find the average, calculate the definite integral of \(p(t)\) from 0 to 37:\[\int_{0}^{37} \left(0.07 t^3 + 2.42 t^2 - 25.63 t + 521.23\right)\, dt\]Evaluate this integral:1. Integrate each term: - \(\int 0.07 t^3 \, dt = 0.07 \frac{t^4}{4} = 0.0175 t^4\) - \(\int 2.42 t^2 \, dt = 2.42 \frac{t^3}{3} = 0.8067 t^3\) - \(\int -25.63 t \, dt = -25.63 \frac{t^2}{2} = -12.815 t^2\) - \(\int 521.23 \, dt = 521.23 t\)2. Substitute \(t = 0\) and \(t = 37\) into the antiderivative and subtract the values.

Step 5: Evaluate the Terms at the Bounds

Substitute \(t = 37\) and \(t = 0\) into the integrated formula:\[0.0175 (37)^4 + 0.8067 (37)^3 - 12.815 (37)^2 + 521.23(37)\]Evaluate each term, and then substitute \(t = 0\) which yields 0 since all terms have \(t\) multiplied into them.

Step 6: Calculate the Average

Once the integration evaluation results are calculated, insert the difference into the average formula:\[\text{Average value} = \frac{1}{37-0} \times \text{(result from the integral)}\]Calculate this division to find the average number of bald eagles per year.

Key Concepts

Cubic Polynomial Basics

A cubic polynomial is an algebraic function of degree three. This means it has a variable raised to the third power, represented in a general form as follows: \( ax^3 + bx^2 + cx + d \). In the cubic polynomial given in our exercise, \( p(t) = 0.07t^3 + 2.42t^2 - 25.63t + 521.23 \), each term plays a role in the shape of the polynomial's graph and the values it produces.

• The \( t^3 \) term is the leading term, which determines the polynomial's end behavior.
• The \( t^2 \) term affects the curvature of the graph.
• The \( t \) term contributes to the slope of the graph.
• The constant term is the y-intercept when \( t = 0 \).

These elements together create a complex curve that can model real-world phenomena, like the population of bald eagles over a given time period.

Understanding Integration

Integration is a fundamental concept in calculus, essentially the reverse of differentiation. It is used to calculate areas under curves and total accumulated quantities. When dealing with problems involving functions like cubic polynomials, integration helps us determine the aggregate value over an interval. The process of integration, or finding an integral, involves finding a function whose derivative is the given function. This process calculates the sum of infinitely small areas under the curve of the function, which in the exercise becomes the total number of bald eagles over 37 years. Integration in this context helps us see not just instant changes but the overall behavior and impact of a function over time.

The Role of Definite Integrals

Definite integrals are used to calculate the net area between the graph of a function and the x-axis over a specific interval. In other words, it provides the total accumulation from point a to point b. The notation \( \int_{a}^{b} f(x) \, dx \) represents this.In our bald eagles example, we apply the definite integral to find the area under the cubic polynomial \( p(t) \) from 0 to 37, representing the years 1963 to 2000. Each component of the polynomial is integrated separately, resulting in an antiderivative that calculates this total aggregate value. Once we evaluate the integrated polynomial at these bounds, we determine the sum of bald eagles over the years in question.

Calculating the Average

Determining the average value of a function over an interval involves both integration and division. After you calculate the definite integral, you divide this value by the interval's length.The formula for average value is:\[ \text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \]In our exercise, this finds the mean number of bald eagles per year between 1963 and 2000. Using the approximate values from the definite integral, dividing by 37 years yields an average rate. This step is crucial in contexts where you want to understand long-term averages or consistent trends over time, like in ecological studies or economic forecasts.