Problem 16 - Applied Calculus

Question

The rate of change of a quantity is given by \(f(t)=t^{2}+1\). Make an underestimate and an overestimate of the total change in the quantity between \(t=0\) and \(t=8\) using (a) \(\quad \Delta t=4\) (b) \(\Delta t=2\) (c) \(\Delta t=1\) What is \(n\) in each case? Graph \(f(t)\) and shade rectangles to represent each of your six answers.

Step-by-Step Solution

Answer

Underestimate and overestimate using left and right Riemann sums with \( \Delta t = 4, 2, 1 \). Graph includes shaded rectangles for visualization.

Step 1: Understanding the Problem

We need to estimate the total change in a quantity where the rate of change is given by \( f(t) = t^2 + 1 \). This involves calculating the area under the curve of \( f(t) \) from \( t = 0 \) to \( t = 8 \) using rectangles. We will make an underestimate and an overestimate for three different time intervals: \( \Delta t = 4, 2, 1 \).

Step 2: Interval Division

For each \( \Delta t \), divide the interval from \( t = 0 \) to \( t = 8 \) into smaller intervals. The number of subdivisions \( n \) is given by \( n = \frac{8}{\Delta t} \).- \( \Delta t = 4 \): \( n = 2 \)- \( \Delta t = 2 \): \( n = 4 \)- \( \Delta t = 1 \): \( n = 8 \)

Step 3: Left Riemann Sum (Underestimate)

For each \( \Delta t \), calculate the left Riemann sum, which gives an underestimate of the total change. For intervals \([t_i, t_{i+1}]\) with \( t_i = i\Delta t \):- \( \Delta t = 4 \): \( \int_0^4 f(t) \approx (f(0)\cdot4 + f(4)\cdot4) = (1\cdot4 + 17\cdot4) = 72 \)- \( \Delta t = 2 \): Sum over values \( f(t_0), f(t_2), f(t_4), f(t_6) \)- \( \Delta t = 1 \): Sum over values \( f(t_0), f(t_1), ..., f(t_7) \)

Step 4: Right Riemann Sum (Overestimate)

Next, calculate the right Riemann sum for each \( \Delta t \), which gives an overestimate:- \( \Delta t = 4 \): \( \int_{4}^8 f(t) \approx (f(4)\cdot4 + f(8)\cdot4) = (17\cdot4 + 65\cdot4) = 328 \)- \( \Delta t = 2 \): Sum over values \( f(t_2), f(t_4), f(t_6), f(t_8) \)- \( \Delta t = 1 \): Sum over values \( f(t_1), f(t_2), ..., f(t_8) \)

Step 5: Graph and Shade Rectangles

Graph \( f(t) = t^2 + 1 \) from \( t = 0 \) to \( t = 8 \). For each \( \Delta t \), draw rectangles below the curve (using left endpoints) to show the underestimate, and rectangles above the curve (using right endpoints) to represent the overestimate.

Key Concepts

Rate of Change

The rate of change in a mathematical context typically describes how a quantity changes with respect to time or another variable. In the given exercise, the rate of change of the quantity is expressed by the function \( f(t) = t^2 + 1 \). This function tells us how the quantity changes as time \( t \) progresses. A higher rate of change indicates a more rapid increase of the quantity, and vice versa.

To fully understand the rate of change, consider its components:

• The function \( f(t) \) in this problem increases quadratically, meaning as time goes on, it increases more rapidly.
• At any time \( t \), the rate of change \( f(t) \) gives the "speed" at which the quantity is increasing at that exact instance.
• Concretely, when \( t = 0 \), \( f(t) = 1 \) representing a slow beginning change. However, by \( t = 8 \), \( f(t) = 65 \), indicating a much faster rate of change due to the quadratic term \( t^2 \).

This concept is crucial when approaching integrals and estimating areas under curves, as these are directly tied to the accumulated rates of change over time.

Approximation Methods

Approximation methods become essential when seeking to estimate values that are difficult to calculate precisely. In our exercise, we aim to estimate the total change in the quantity using Riemann sums, which is an essential approximation method in calculus.

Riemann sums come in different forms, such as the left Riemann sum and the right Riemann sum used here:

• The left Riemann sum uses the left endpoint of each subdivision to calculate the area of rectangles under the curve. This typically results in an underestimation if the function is increasing, as it was in this exercise.
• The right Riemann sum uses the right endpoint of each subdivision, often leading to an overestimation under the same condition, as it captures higher values at the end of each interval.

Each method provides a basic way to approximate the integral of a function, especially useful when calculating precise integrals is complex or computationally intensive. Breaking an interval into smaller pieces (different \( \Delta t \)) enhances the accuracy, illustrating the fundamental methodology for these approximation techniques.

Integral Estimation

The concept of integral estimation is rooted in finding the area under a curve, which in this scenario, translates to estimating the total change in the quantity from \( t=0 \) to \( t=8 \).

In integral estimation:

• We are calculating the definite integral of the function \( f(t) = t^2 + 1 \) over the interval \( [0,8] \), which geometrically represents the area under the curve.
• The process involves summing up the areas of rectangles that approximate the curve, using the left (underestimate) and right (overestimate) endpoints.
• The more subdivisions \( n \) or smaller \( \Delta t \), the more accurate the estimation. Here, approximations were done with \( n=2, 4, \) and \( 8 \).

This exercise shows the practical use of Riemann sums to estimate integrals and exemplifies how integrating non-linear rate of change functions can signify a total change in quantities across an interval. Integral estimation, therefore, is indispensable in applications where precise accumulation of changes needs to be calculated, yet approachable using sum approximations.