Problem 15 - College Algebra and Calculus: An Applied Approach

Question

A point is moving along the graph of \(y=x^{2}\) such that \(d x / d t\) is 2 centimeters per minute. Find \(d y / d t\) for each value of \(x\). (a) \(x=-3\) (b) \(x=0\) (c) \(x=1\) (d) \(x=3\)

Step-by-Step Solution

Answer

(a) For \(x=-3\), \(dy/dt = -12\) cm/min. (b) For \(x=0\), \(dy/dt = 0\) cm/min. (c) For \(x=1\), \(dy/dt = 4\) cm/min. (d) For \(x=3\), \(dy/dt = 12\) cm/min.

Step 1: Setup the Equation

We first write down the derivative of \(y\) with respect to \(x\), which is given as \(dy/dx = 2x\). This derivative is obtained by applying the power rule for differentiation to \(y=x^2\).

Step 2: Apply the Chain Rule

Using the chain rule of derivatives, we can express \(dy/dt\) as \(dy/dt = dy/dx * dx/dt\). Since \(dx/dt\) is given as 2 cm/min, the expression simplifies to \(dy/dt = 2x * 2\).

Step 3: Compute the Values

For each given value of \(x\), we can now compute the corresponding value of \(dy/dt\). Remember, \(dy/dt\) represents the speed at which \(y\) is changing for a given value of \(x\). (a) For \(x=-3\), \(dy/dt = 2*(-3) * 2 = -12\) cm/min. (b) For \(x=0\), \(dy/dt = 2*0 * 2 = 0\) cm/min. (c) For \(x=1\), \(dy/dt = 2*1 * 2 = 4\) cm/min. (d) For \(x=3\), \(dy/dt = 2*3 * 2 = 12\) cm/min.

Key Concepts

Derivatives

In mathematics, a derivative represents how a function is changing at any given point. It's like looking at the slope or incline of a curve at a specific position. When we talk about the derivative of a function such as \(y = x^2\), we are trying to determine how the value of \(y\) changes in relation to changes in \(x\).

• The derivative of \(y = x^2\) with respect to \(x\) is \(dy/dx = 2x\).
• This tells us that for each unit increase in \(x\), \(y\) changes by \(2x\) units.
• Derivatives help us understand the rate of change, which is crucial in related rates problems.

Recognizing the role of derivatives in understanding motion and change is key in solving many calculus problems, including the one we are examining here.

Chain Rule

The chain rule is a powerful technique in calculus used for finding the derivative of composite functions. This rule helps when we have a situation where one variable changes with respect to another and then this second variable changes with respect to a third.

• For a situation where \(y\) is a function of \(x\) and \(x\) is a function of \(t\) (time), the chain rule comes in handy.
• The chain rule states that \(dy/dt = (dy/dx) \times (dx/dt)\).
• In our problem, since \(dx/dt\) is 2 cm/min, we can multiply it by \(dy/dx\) to find \(dy/dt\).

This approach breaks down the problem into more manageable parts, making it easier to determine how fast one quantity is changing in relation to another.

Differentiation

Differentiation is the process of finding a derivative. It involves applying rules and techniques to calculate how a function changes. When dealing with functions of a single variable, differentiation is straightforward when you're familiar with the basic rules.

• In our example, differentiating \(y = x^2\) gives us \(dy/dx = 2x\).
• This process uses differentiation rules like the power rule, which we'll elaborate on next.
• Understanding differentiation is essential for solving problems where you need to find rates of change, like related rates problems.

Differentiation connects with many real-world problems, enabling us to model and solve them by understanding how different quantities interact and change over time.

Power Rule

The power rule is a basic differentiation rule used to find the derivative of functions that are expressed as powers of \(x\). This rule makes it simple and fast to differentiate polynomial functions.

• If you have a function of the form \(x^n\), the derivative is \(nx^{n-1}\).
• For \(y = x^2\), applying the power rule gives us \(dy/dx = 2x^{2-1} = 2x\).
• This is a straightforward application that simplifies finding the derivative.

Using the power rule can save time and reduce errors, especially when dealing with complex polynomials. It is a crucial tool in the calculus toolbox for efficiently dealing with differentiation tasks.