Problem 15 - Applied Calculus

Question

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

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: Identifying the Inside Function (a)

For the function \(y = 2^{3x-1}\), identify the inside function. We see that the expression \(3x-1\) is inside the exponential function. Thus, we set \(u = 3x - 1\).

Step 2: Expressing (a) as a Composite Function

Now that we have \(u = 3x - 1\), substitute this into the equation for \(y\). The function becomes \(y = 2^u\), showing that \(y\) is a function of \(u\).

Step 3: Identifying the Inside Function (b)

For \(P = \sqrt{5t^2 + 10}\), identify the inside function, which is the expression under the square root. Set \(u = 5t^2 + 10\).

Step 4: Expressing (b) as a Composite Function

Substitute \(u = 5t^2 + 10\) into the equation for \(P\) to express it as a function of \(u\). The function becomes \(P = \sqrt{u}\).

Step 5: Identifying the Inside Function (c)

For \(w = 2 \ln(3r + 4)\), identify the inside function, which is the argument of the logarithm. Set \(u = 3r + 4\).

Step 6: Expressing (c) as a Composite Function

With \(u = 3r + 4\), rewrite the function \(w\) as \(w = 2 \ln(u)\), expressing \(w\) in terms of \(u\).

Key Concepts

Exponential Function

The exponential function is a mathematical operation involving an exponent, meaning it raises a constant base, usually denoted as \(e\), to the power of a variable. However, in general contexts, any constant value can be used as the base. The general form is \(f(x) = a^x\), where \(a\) is a positive constant. For example, in the original problem, we have \(y = 2^{3x-1}\), an exponential function where 2 is the base. The important part here is the exponent \(3x-1\), which acts as the 'inside function.' The notion of an inside function is crucial for understanding composite functions, as it affects the exponent and ultimately changes the output.

Some key characteristics of exponential functions include:

• The function grows (or decays) rapidly as the exponent increases (or decreases).
• An exponential function's graph is always a curve, never crossing the x-axis, and it approaches zero.
• Exponential functions are used to model phenomena that grow or shrink at constant percentage rates, like population growth or radioactive decay.

Square Root Function

The square root function is another fundamental mathematical operation. It is often denoted as \(f(x) = \sqrt{x}\), which extracts the square root of a number or expression. In the provided problem, we look at the function \(P = \sqrt{5t^2 + 10}\). Here, \(5t^2 + 10\) is the expression under the square root, which serves as the inside function in a composite context. Understanding how to separate the inside function helps simplify complex calculations and makes analyzing changes in the output easier.

Key points about square root functions include:

• The function's domain is limited to non-negative numbers because square roots of negative numbers are not real.
• The graph of a square root function is also a curve, typically starting from the origin and rising gradually.
• Square root functions are used in geometry for calculating distances or in physics for deriving formulas involving area and velocity.

Logarithmic Function

A logarithmic function is the inverse of an exponential function. It answers the question of which power the base must be raised to in order to achieve a given number. The general form of a logarithmic function is \(f(x) = \log_a{(x)}\), where \(a\) is the base of the logarithm. In our function \(w = 2 \ln(3r + 4)\), \(\ln\) denotes the natural logarithm, which has the base \(e\). The argument \(3r + 4\) is the inside function here. By changing this inside function, we manipulate which values \(w\) takes on.

Key features of logarithmic functions include:

• They increase slowly compared to exponential functions, which is why they are more stretched along the x-axis.
• The domain includes all positive real numbers since one cannot take a logarithm of zero or a negative number.
• Logarithmic functions are widely used in calculations involving rates of change, like measuring sound intensity or earthquake magnitudes.