Problem 45 - Advanced Engineering Mathematics

Question

Find a function \(y=f(x)\) whose graph at each point \((x, y)\) has the slope given by \(8 e^{2 x}+6 x\) and has the \(y\) -intercept \((0,9)\).

Step-by-Step Solution

Answer

The function is \( f(x) = 4e^{2x} + 3x^2 + 5 \).

Step 1: Understand the Problem

The problem asks for a function \( y = f(x) \) which has a graph such that the slope of the tangent at any point \((x, y)\) is given by \( 8 e^{2x} + 6x \). Additionally, this function passes through the point \((0, 9)\), which is the y-intercept.

Step 2: Set Up the Differential Equation

A function's slope at a point is given by its derivative. Therefore, we need to find a function \( f(x) \) such that its derivative \( f'(x) = 8 e^{2x} + 6x \).

Step 3: Integrate to Find the General Solution

To find \( f(x) \), we integrate the expression for \( f'(x) \):\[ f(x) = \int (8e^{2x} + 6x) \, dx\].

Step 4: Integrate Each Component

Separate the integral into two components to solve:\[ \int 8e^{2x} \, dx + \int 6x \, dx \].First, integrate \( 8e^{2x} \, dx \):- Use substitution \( u = 2x \), \( du = 2 \, dx \), thus \( dx = \frac{du}{2} \).- Modify the integral: \( 8 \cdot \frac{1}{2} \int e^{u} \, du \) results in \( 4e^{2x} \).Now, integrate \( 6x \, dx \):- Use the power rule: \( \int 6x \, dx = 3x^2 \).Combining the results gives:\[ f(x) = 4e^{2x} + 3x^2 + C\].

Step 5: Solve for the Constant Using the Initial Condition

We use the point \((0, 9)\) to solve for the constant \( C \):\[ f(0) = 4e^{2 \cdot 0} + 3(0)^2 + C = 9\].Since \( e^0 = 1 \), it simplifies to \( 4 + C = 9 \), hence \( C = 5 \).

Step 6: Write the Specific Solution

Substitute \( C = 5 \) into the general solution to get the specific function:\[ f(x) = 4e^{2x} + 3x^2 + 5\].

Key Concepts

Slope of Tangent Line

The slope of a tangent line is a measure of how steep a graph is at a given point. For any function, the slope at a point \((x, y)\) is represented by the derivative of that function at that point, which we'll denote as \(f'(x)\). For example, if the derivative is given as \(8e^{2x} + 6x\), this means that at any point on the graph, the slope of the tangent line will be computed using this expression.

• When the value of \(x\) is substituted into the equation, it gives the exact rate of change at that specific point.
• This value indicates how steeply the function is rising or falling at the point \((x, f(x))\).

Understanding how the slope changes helps us understand the behavior of the function throughout its domain.This concept is foundational in calculus, as it allows us to analyze and understand how different functions behave through rate of change.

Integral Calculus

Integral calculus is a branch of calculus focused on finding the accumulated quantity, or the "integral," depending on a range of values from a specific function. In this problem, we are tasked with finding the function \(f(x)\), given its derivative \(f'(x)\). The process of integration is essentially the reverse of differentiation.

• We integrated \(f'(x) = 8e^{2x} + 6x\) to find \(f(x)\).
• Breaking it down, we calculated separate integrals: \(\int 8e^{2x} \, dx\) and \(\int 6x \, dx\).

Integration often involves applying rules and techniques such as substitution or power rules, making it an essential component for solving problems that involve determining functions from their rates of change.

Initial Value Problem

In calculus, an initial value problem involves finding a specific solution to a differential equation that satisfies a given condition. Here, we have a condition that the function passes through a particular point on the graph, \((0, 9)\). This specific point is crucial as it allows us to determine the constant of integration \(C\) in our solution.

• The general solution of \(f(x)\) from the integral was \(4e^{2x} + 3x^2 + C\).
• Using the point \((0, 9)\), we substitute \(x = 0\) and \(f(x) = 9\) into the equation to solve for \(C\).

This results in \(C = 5\), so the specific solution \(f(x) = 4e^{2x} + 3x^2 + 5\). Establishing an initial condition helps narrow down infinite possible solutions to one that is tailored to specific scenarios.

Exponential Functions

Exponential functions are mathematical functions of the form \(e^{kx}\) where \(e\) is Euler's number. They describe a process of exponential growth or decay. In the given differential equation, an exponential term \(8e^{2x}\) is part of the function's slope.

• The base \(e\) is approximately equal to 2.718, a fundamental constant in mathematics.
• This function rapidly changes as \(x\) increases or decreases, contributing significantly to the function's overall shape and growth rate.

Exponential functions are common in modeling real-world phenomena, including population growth, radioactive decay, and interest calculations. They are crucial in any calculus problem that involves exponential rates of change or forms part of the function's derivative.