Problem 1 - Calculus Early Transcendentals

Question

Suppose \(z=f(x, y),\) where \(x\) and \(y\) are functions of \(t .\) How many dependent, intermediate, and independent variables are there?

Step-by-Step Solution

Answer

Answer: In the given function, there is 1 dependent variable (z), 2 intermediate variables (x and y), and 1 independent variable (t).

Step 1: Identify the dependent variable

A dependent variable is an output or response variable that relies on other variables. In our case, \(z\) depends on \(x\) and \(y\). So, \(z\) is the dependent variable.

Step 2: Identify the intermediate variables

Intermediate variables are those that depend on the independent variable(s) but also affect the dependent variable. In our case, \(x\) and \(y\) are both functions of \(t\) and affect \(z\). Therefore, \(x\) and \(y\) are intermediate variables.

Step 3: Identify the independent variable

An independent variable is one that can be manipulated without being affected by any other variable. In our case, \(t\) is the only variable that is not affected by any other variable, and both \(x\) and \(y\) depend on it. Thus, \(t\) is the independent variable.

Step 4: Count the number of dependent, intermediate, and independent variables

Based on our analysis, there is: - 1 dependent variable (\(z\)), - 2 intermediate variables (\(x\) and \(y\)), and - 1 independent variable (\(t\)).