Problem 13 - Applied Calculus

Question

Use the variable \(u\) for the inside function to express eac of the following as a composite function: (a) \(y=2^{3 x-1}\) (b) \(P=\sqrt{5 t^{2}+10}\) (c) \(w=2 \ln (3 r+4)\)

Step-by-Step Solution

Answer

(a) \(y = 2^u\), \(u = 3x - 1\); (b) \(P = \sqrt{u}\), \(u = 5t^2 + 10\); (c) \(w = 2\ln(u)\), \(u = 3r + 4\).

Step 1: Identify the Outside Function for Part (a)

For the expression \(y = 2^{3x-1}\), the outside function is the exponential function. We can set it as \(f(u) = 2^u\).

Step 2: Identify the Inside Function for Part (a)

The inside function, which will be represented by \(u\), is \(u = 3x - 1\). The composite function is \(y = f(u) = 2^u\).

Step 3: Express as a Composite Function for Part (a)

Combine the outside and inside functions: \(y = 2^{3x-1}\) is expressed as \(y = f(u) = 2^u\) where \(u = 3x - 1\).

Step 4: Identify the Outside Function for Part (b)

For the expression \(P = \sqrt{5t^2 + 10}\), the outside function is the square root function. We can set it as \(g(u) = \sqrt{u}\).

Step 5: Identify the Inside Function for Part (b)

The inside function is \(u = 5t^2 + 10\). The composite function is \(P = g(u) = \sqrt{u}\).

Step 6: Express as a Composite Function for Part (b)

Combine the outside and inside functions: \(P = \sqrt{5t^2 + 10}\) is expressed as \(P = g(u) = \sqrt{u}\) where \(u = 5t^2 + 10\).

Step 7: Identify the Outside Function for Part (c)

For the expression \(w = 2 \ln(3r + 4)\), the outside function is the multiplication by 2 and the natural logarithm function. We can set it as \(h(u) = 2\ln(u)\).

Step 8: Identify the Inside Function for Part (c)

The inside function is \(u = 3r + 4\). The composite function is \(w = h(u) = 2\ln(u)\).

Step 9: Express as a Composite Function for Part (c)

Combine the outside and inside functions: \(w = 2 \ln(3r + 4)\) is expressed as \(w = h(u) = 2\ln(u)\) where \(u = 3r + 4\).

Key Concepts

Exponential Functions

Exponential functions are fundamental in algebra and calculus. They involve expressions where a constant base is raised to a variable exponent. A common base for exponential functions is the number 2, as in the expression \( y = 2^{3x-1} \). In this form, replacing the expression \(3x-1\) with a single variable like \(u\) simplifies the function. Thus, we have \( f(u) = 2^u \). In this way, you see that transformations on the exponent can be imagined as separate manipulations within this simplified function.

• **Positive exponents** increase the value as the exponent grows larger.
• **Negative exponents** result in fractional numbers, indicating division.

Exponential functions grow quickly. This rapid increase makes them useful in modeling situations like population growth and radioactive decay. They show how one quantity can rise sharply over time based on previous values.

Square Root Functions

Square root functions are represented mathematically as \( \sqrt{\text{expression}} \). They are the inverse of squaring functions. For example, in the equation \( P = \sqrt{5t^2 + 10} \), the square root is applied to the expression inside the radical, \(5t^2 + 10 \). By introducing the composite function concept, we set \(g(u) = \sqrt{u}\), and this expression inside the root \(5t^2 + 10\) becomes the inside function \(u\).

• **Square roots** are only defined for non-negative numbers in the real number system.
• **Graphs of square root functions** curve upwards in a gentle arc, starting from the origin point \((0, 0)\) when expressed simply as \(y = \sqrt{x}\).

Square root functions are crucial in many contexts, including physics, engineering, and even finance, where they can describe elements like standard deviations and variance.

Natural Logarithms

Natural logarithms are represented with the ln notation and come into play when dealing with logarithmic functions, particularly in continuous compounding scenarios. For example, \( w = 2 \ln(3r + 4) \) involves the logarithm of \(3r + 4\) multiplied by 2. In composite terms, this translates to \(h(u) = 2\ln(u)\) with \(u = 3r + 4\) representing the expression inside the logarithm.

• **Natural logarithms** are the inverse operations of exponential functions with base \(e\).
• **The constant \(e\)** is approximately \(2.718\) and is a fundamental constant similar to \(\pi\).

Natural logarithms are extremely useful for solving problems involving exponential growth processes. They appear frequently in mathematics and science, particularly in the solution of differential equations, financial modeling, and in problems involving growth and decay rates.