Problem 10 - Calculus Concepts

Question

a. use the limit definition of the derivative (algebraic method) to write an expression for the rate-of-change function of the given function. b. evaluate the rate of change as indicated. \(\quad m(p)=4 p+p^{2} ;\left.\frac{d m}{d p}\right|_{p=-2}\)

Step-by-Step Solution

Answer

The derivative is evaluated as 0 at \( p = -2 \).

Step 1: Understand the Limit Definition of the Derivative

The limit definition of the derivative for a function \( m(p) \) is given by \( \frac{dm}{dp} = \lim_{h \to 0} \frac{m(p+h) - m(p)}{h} \). This represents the rate of change of the function \( m(p) \) at any given point \( p \).

Step 2: Substitute m(p) in the Limit Definition

Start by substituting the given function \( m(p) = 4p + p^2 \) into the limit definition. Calculate \( m(p+h) \) by substituting \( p+h \) into the function to get \( m(p+h) = 4(p+h) + (p+h)^2 = 4p + 4h + p^2 + 2ph + h^2 \).

Step 3: Substitute into the Derivative Expression

Substitute \( m(p+h) \) and \( m(p) \) back into the limit definition: \( \frac{dm}{dp} = \lim_{h \to 0} \frac{(4p + 4h + p^2 + 2ph + h^2) - (4p + p^2)}{h} \). Simplify the expression to get \( \frac{4h + 2ph + h^2}{h} \).

Step 4: Simplify the Expression

Factor out \( h \) from the numerator: \( \frac{h(4 + 2p + h)}{h} \). After cancellation, the expression simplifies to \( 4 + 2p + h \).

Step 5: Evaluate the Limit as h Approaches 0

Take the limit as \( h \) approaches 0: \( \lim_{h \to 0} (4 + 2p + h) = 4 + 2p \). Thus, the derivative function is \( \frac{dm}{dp} = 4 + 2p \). This represents the rate of change function of \( m(p) \).

Step 6: Evaluate the Derivative at p = -2

Substitute \( p = -2 \) into the derivative: \( \frac{dm}{dp} = 4 + 2(-2) = 4 - 4 = 0 \). This is the rate of change of the function \( m(p) \) at \( p = -2 \).

Key Concepts

Rate of Change

The concept of rate of change is fundamental in understanding derivatives and calculus as a whole. In simple terms, it's how much one quantity changes in response to the change in another quantity.
For our problem, the rate of change function reflects how the function \( m(p) = 4p + p^2 \) changes as the variable \( p \) changes. This is expressed in terms of its derivative, \( \frac{dm}{dp} \).
Evaluating the rate of change at a specific point gives you the slope of the tangent line to the curve of the function at that point.

• The rate of change allows us to understand the behavior of functions and model real-world scenarios.
• It's particularly useful in physics and economics where it describes velocity, acceleration, and market trends.

In our example, after finding the derivative, we evaluated it at \( p = -2 \), determining that the rate of change at that point is zero. This means the function is neither increasing nor decreasing there.

Evaluating Derivatives

To evaluate a derivative means to find its value at a specific point. For the function \( m(p) \), we've already determined the expression for its derivative as \( 4 + 2p \).
However, evaluating a derivative requires substituting the given point into this expression.
Once we have our general derivative, \( \frac{dm}{dp} = 4 + 2p \), we plug in \( p = -2 \) to find the rate of change. This gives:
\[\frac{dm}{dp}\bigg|_{p=-2} = 4 + 2(-2) = 4 - 4 = 0\]

• This step-by-step evaluation tells us precisely how the function behaves at \( p = -2 \).
• By calculating this, we learn that there’s no change in the value of the function at this specific point.

Evaluating derivatives is crucial for understanding the specific behavior and granularity of functions, as opposed to just understanding their potential behavior in general terms.

Algebraic Method

The algebraic method for finding derivatives involves using the limit definition of the derivative, an essential formula in calculus. This method allows us to determine the derivative from first principles, giving us the most fundamental understanding of change in a function.
The limit definition is:\[\frac{dm}{dp} = \lim_{h \to 0} \frac{m(p+h) - m(p)}{h}\]
In our exercise, we applied this to the given function \( m(p) = 4p + p^2 \). Here’s how it breaks down:

• Calculate \( m(p+h) \), resulting in \( 4p + 4h + p^2 + 2ph + h^2 \).
• Substitute both \( m(p+h) \) and \( m(p) \) into the limit definition.
• Simplify the expression and factor out \( h \).
• Finally, evaluate the limit as \( h \to 0 \) to find \( \frac{dm}{dp} = 4 + 2p \).

This process shows how derivatives represent rates of change algebraically, using precise mathematical steps to arrive at the result. The algebraic method is central to deriving new functions and verifying calculations in calculus.