Problem 5 - Calculus Early Transcendentals

Question

Given the rate of change of a quantity \(Q\) and its initial value \(Q(0)\) explain how to find the value of \(Q\) at a future time \(t \geq 0\)

Step-by-Step Solution

Answer

Question: Given the rate of change of a quantity Q with respect to time as R(t), and the initial value of Q at t=0, determine the value of Q at any future time t ≥ 0. Answer: To determine the value of Q at any future time t ≥ 0, follow these steps: 1. Determine the rate of change function R(t). 2. Integrate R(t) with respect to time to find Q(t), including the constant of integration 'C': Q(t) = ∫R(t) dt + C. 3. Use the initial value Q(0) to find the constant of integration 'C': C = Q(0) - ∫R(0) dt. 4. Substitute the value of 'C' into the Q(t) function: Q(t) = ∫R(t) dt + C. 5. Plug in the desired value of t into the Q(t) function to find the value of Q at that specific time.

Step 1: Determine the rate of change function

Given the rate of change of the quantity \(Q\), let's denote it with a function \(R(t)\), where \(R(t)\) represents the rate of change at time \(t\). It is important to have a clear understanding of what \(R(t)\) represents so we can proceed to the next steps.

Step 2: Integration of the rate of change function

To find the value of \(Q\) at a future time \(t \geq 0\), we need to integrate the rate of change function with respect to time: $$\int R(t) dt$$ The result of the integration will give us a function that represents the accumulated change in quantity \(Q\) over time, which we can denote as \(Q(t)\). This function may also contain a constant of integration, let's denote it as 'C'.

Step 3: Find the constant of integration 'C' using the initial value

We are given the initial value of \(Q(0)\). Using this information, we can determine the value of the constant. We plug in \(t=0\) in the function we found in Step 2: $$Q(0) = \int R(0) dt + C$$ Solve for 'C': $$C = Q(0) - \int R(0) dt$$ Now we have the value of the constant of integration 'C', and we can find the complete function \(Q(t)\).

Step 4: Find the value of Q(t)

In the previous step, we found the value of 'C'. Now, we can substitute 'C' into the function we found in Step 2. The function \(Q(t)\) is now given by: $$Q(t) = \int R(t) dt + C$$

Step 5: Find the value of Q at the future time t

Now that we have the function \(Q(t)\), we can find the value of \(Q\) at any future time \(t \geq 0\) by simply plugging in the value of \(t\) into the function \(Q(t)\).

Key Concepts

Rate of Change

When discussing the rate of change in calculus, we're exploring how a quantity varies over time. It's essentially a function that describes the speed or velocity at which something is happening. For example, if you're tracking the amount of water flowing into a tank, the rate of change would be the flow rate at any given moment. This is usually expressed as a function, denoted as \( R(t) \), where \( t \) represents time. The rate of change tells us how fast or slow a certain quantity increases or decreases, making it a crucial concept for predicting future behavior. It's like checking your speedometer to see how fast you're going right now! Understanding this concept helps us prepare to move further into solving the problem at hand.

Initial Value Problem

An initial value problem in calculus tells us about the state of a system at the beginning. Imagine it as setting the origin point for your calculations. In any real-world scenario, the initial value represents the condition of the system at time zero, \( Q(0) \). This value is essential because it helps anchor the mathematical model we're creating.

• It provides a starting point for integration.
• It helps in determining the constant (which we'll get to in the next section).

This problem usually comes with a differential equation describing how the system changes, plus the initial condition, which is \( Q(0) \) in our case. With these tools, we can solve for the complete function describing the system's behavior over time.

Constant of Integration

After performing integration on a rate of change function, we encounter an unknown constant termed as 'constant of integration'. This constant \( C \) arises because the process of integration could have been derived from any number of potential original functions, each slightly shifted on the vertical axis. Thus, an undetermined add-on appears in the solution, which we call \( C \).

To find this mysterious constant, we employ the initial value. By knowing \( Q(0) \), the initial condition, we substitute \( t = 0\) into the integrated function \( Q(t) \). This allows us to solve for \( C \) precisely.

• We balance the integrated equation with the initial condition.
• We solve for \( C \) giving us the particular solution to our integration problem.

Once found, \( C \) is plugged back into the function \( Q(t) \) to yield a specific solution that satisfies both the rate of change and initial conditions. This completes the model, allowing the function \( Q(t) \) to describe accurately how the system evolves with time.