Question

Energy is the capacity to do work, and power is the rate at which energy is used or consumed. Therefore, if $E(t)$ is the energy function for a system, then $P(t)=E^{\prime }(t)$ is the power function. A unit of energy is the kilowatt-hour ( $1\mathrm{kWh}$ is the amount of energy needed to light ten 100 -W lightbulbs for an hour); the corresponding units for power are kilowatts. The following figure shows the energy consumed by a small community over a 25 -hour period. a. Estimate the power at $t=10$ and $t=20\mathrm{hr}.$ Be sure to include units in your calculation. b. At what times on the interval [0,25] is the power zero? c. At what times on the interval [0,25] is the power a maximum?

Short answer

Based on the step by step solution, provide a short answer for each part of the question: a. For $t=10$ and $t=20$ hours, estimate the average rate of change of the energy function $E(t)$ over a small interval centered around these times using the graph. The estimated power at $t=10$ and $t=20$ hours, with units, are the average rates of change at these points. b. To find the times when the power is zero on the interval $[0,25]$, look for the critical points of the energy function where the slope changes, and note the corresponding time values. c. To find when the power is at a maximum on the interval $[0,25]$, analyze the graph of the power function and locate the points where the slope changes from positive to negative. Determine the corresponding time values when this occurs.

Step-by-step solution

a. Estimate the power at $t=10$ and $t=20$ hours

In order to estimate the power at $t=10$ and $t=20$ hours, we need to find the first derivative of the energy function $E(t)$ at these points. We haven't been given the specific formula for $E(t)$, only a graph. To find an estimate of the power at these times, we can find the average rate of change of energy over a small interval centered around these times. Then, this value will be the estimate of the derivative of $E(t)$ at these points. So, let's choose an interval of 1 hour centered around 10 hours and 20 hours. Calculate the average rate of change of energy for these intervals using the graph and use these values as the estimates for the power at the respective times.

b. Finding times when the power is zero on the interval $[0,25]$

The power function $P(t)$ is the first derivative of the energy function $E(t)$. In order to find when the power is zero on the interval $[0,25]$, we need to find when the first derivative is zero. This is equivalent to finding the critical points of the energy function. The critical points occur when there's a change in the function's slope (i.e., from positive to negative or vice versa). Determine the points on the graph of $E(t)$ where these changes in slopes occur, and note the corresponding time values.

c. Finding when the power is at a maximum on the interval $[0,25]$

To find when the power is at a maximum on the interval $[0,25]$, we need to determine when the derivative of the power function $P(t)$ is at a maximum. This is equivalent to finding when the second derivative of $E(t)$ is equal to zero, and the slope of the power function is changing from positive to negative. Examine the graph of the power function (found during estimating the power at $t=10$ and $t=20$ hours) and determine when the slope of the power function is changing from positive to negative on the interval $[0,25]$. Note the corresponding time values when this occurs.

Key concepts

Energy and Power

Energy and power are fundamental concepts in calculus and physics. Energy is the capacity to perform work, which can be measured in units such as kilowatt-hours (kWh). A kilowatt-hour is the energy required to power one 1,000-watt appliance for an hour. On the other hand, power is the rate at which this energy is used or consumed. It is measured in watts or kilowatts. So, when we talk about power, we're referring to the speed at which energy is being consumed.

If we have an energy function denoted by $E(t)$, the power function, $P(t)$, is simply the derivative of the energy function, $E^{\prime }(t)$. This derivative gives us the rate of change of energy with respect to time, providing insight into how quickly energy is being used at any given moment.

For instance, if you have a graph of energy consumption over time, you can estimate the power at a particular time by looking at the slope of the tangent line at that point. This tangent's slope is the derivative—hence the power—at that specific moment.

Rate of Change

The rate of change is a crucial concept in understanding derivatives, especially in the context of energy and power. A derivative, like $E^{\prime }(t)$, indicates the rate of change of one variable concerning another. Here, it shows how energy consumption alters over time; hence, it describes the power.

If the rate of change (or derivative) is high, energy consumption is changing rapidly. A low rate of change means it's altering slowly. Imagine monitoring the electricity usage in your home: a sudden jump in the energy graph indicates a high rate of change—perhaps you turned on an oven. On the flip side, a flat section of the graph suggests little to no change, possibly when most appliances are off.

To estimate the rate of change at specific times like $t=10$ and $t=20$, you would observe small intervals around these times on an energy graph. The slope, or rate of change, is given by the difference in energy divided by the time interval. This slope is what we call the derivative at a given time.

Critical Points

In calculus, critical points are points on a function's graph where the slope is zero or undefined—indicating a potential maximum or minimum. For the energy function $E(t)$, critical points happen when $E^{\prime }(t)=0$, suggesting the potential for power to be zero or attain extremum values.

When power is zero, it indicates that the energy consumption hasn't changed at that moment. To find when this occurs, look for where the tangent line's slope on the graph is flat, indicating no change in energy over time.

On the interval $[0,25]$, identifying maximum power involves determining where the energy function's derivative changes from increasing to decreasing—where it hits a peak. This is akin to climbing a hill and reaching the top. The second derivative can help confirm these points, showing changes in the rate of the rate of change (acceleration of the energy usage vector). At critical points, you carefully examine the transitions to spot where these peaks and troughs occur.