Problem 113 - University Calculus: Early Transcendentals

Question

Solve the initial value problems in Exercises. Find the curve \(y=f(x)\) in the \(x y\) -plane that passes through the point (9,4) and whose slope at each point is \(3 \sqrt{x}\)

Step-by-Step Solution

Answer

The curve is \(y = 2x^{3/2} - 50\).

Step 1: Understand the Problem

The given problem involves finding a function \(y = f(x)\) whose rate of change or slope is determined by the equation \(\frac{dy}{dx} = 3 \sqrt{x}\) and passes through the specific point (9,4).

Step 2: Identify the Differential Equation

We need to solve the differential equation \(\frac{dy}{dx} = 3 \sqrt{x}\). This equation expresses the relationship between \(x\) and its derivative \(y\).

Step 3: Integrate Both Sides

To find \(y\), we integrate both sides of the differential equation. On the left, we integrate \(dy\), and on the right, we integrate \(3 \sqrt{x} \, dx\):\[y = \int 3 \sqrt{x} \, dx = \int 3 x^{1/2} \, dx. \]

Step 4: Solve the Integral

Calculate the integral using the power rule. The integral of \(x^{1/2}\) is \(\frac{x^{3/2}}{3/2}\). Therefore:\[y = 3 \cdot \frac{x^{3/2}}{3/2} + C = 2x^{3/2} + C,\]where \(C\) is the constant of integration.

Step 5: Use the Initial Condition

Utilize the initial condition \(y(9) = 4\) to find \(C\). Plug \(x = 9\) and \(y = 4\) into the equation:\[4 = 2 \cdot 9^{3/2} + C.\]Evaluate \(9^{3/2} = (9^{1})^{3/2} = 27\), yielding:\[4 = 2 \cdot 27 + C = 54 + C.\]Solve for \(C:\)\[C = 4 - 54 = -50.\]

Step 6: Write the Final Equation

Substitute \(C = -50\) back into the solution to get the specific curve:\[y = 2x^{3/2} - 50.\]

Key Concepts

Initial Value Problem

An initial value problem is a type of differential equation that comes with a specified value, often called an initial condition. This condition is given at a certain point and allows us to find a specific solution among the infinite possibilities of functions that satisfy the differential equation. In our exercise, the differential equation is \(\frac{dy}{dx} = 3 \sqrt{x}\), and the initial condition is the point (9, 4).
This point tells us that when \(x = 9\), the corresponding \(y\) value should be 4. This condition helps us find the constant of integration later. The overall goal is to find a function \(y = f(x)\) that not only satisfies the differential equation but also passes through this specific point. Initial value problems are essential for determining unique solutions that fit certain initial scenarios.

Integration

Integration is the process of finding a function with a given derivative. In solving differential equations, integration helps us find an original function given its rate of change. In our exercise, we started with the differential equation \(\frac{dy}{dx} = 3 \sqrt{x}\).
To find \(y\), we integrate both sides of this equation. When integrating, the notation changes the derivative \(dy/dx\) back to the original function \(y\). Specifically, we need to integrate \(3 \sqrt{x} \ dx\) to find the function that represents how \(y\) changes with respect to \(x\). Integration is a fundamental tool in calculus, allowing us to reverse-engineer the behavior of functions from their derivatives.

Power Rule

The power rule is a basic rule in calculus used for integrating and differentiating expressions involving powers of variables. When it comes to integration, the power rule states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\), provided \(neq -1\).
In our problem, we applied the power rule to integrate \(3x^{1/2}\). Using the power rule, we found the integral of \(x^{1/2}\) to be \(\frac{x^{3/2}}{3/2}\).
Thus, the integral \(\int 3x^{1/2} \, dx = 3 \times \frac{x^{3/2}}{3/2} = 2x^{3/2}\). The power rule simplifies the integration process, making it easier to find functions that have derivatives of given forms.

Constant of Integration

When we integrate a function, we must include a constant of integration, commonly denoted as \(C\). This constant accounts for the family of possible solutions arising from the indefinite integration. Essentially, different values of \(C\) represent different vertical shifts of the same curve.
In our exercise, after integrating, we arrived at the expression \(y = 2x^{3/2} + C\). To determine the specific value of \(C\), we used the initial condition \(y(9) = 4\).
Substituting \(x = 9\) and \(y = 4\) into this equation helped us solve for \(C\), finding that \(C = -50\). The constant of integration is crucial because it allows us to align our integrated curve with real-world data or specified conditions.