Problem 57 - Calculus Early Transcendentals

Question

Batting averages Batting averages in baseball are defined by \(A=x / y,\) where \(x \geq 0\) is the total number of hits and \(y>0\) is the total number of at- bats. Treat \(x\) and \(y\) as positive real numbers and note that \(0 \leq A \leq 1\) a. Use differentials to estimate the change in the batting average if the number of hits increases from 60 to 62 and the number of at-bats increases from 175 to 180 . b. If a batter currently has a batting average of \(A=0.350,\) does the average decrease if the batter fails to get a hit more than it increases if the batter gets a hit? c. Does the answer to part (b) depend on the current batting average? Explain.

Step-by-Step Solution

Answer

How does a hit or miss affect the player's average? Answer: The effect of getting a hit or not on a player's batting average does not depend on the player's current batting average. Regardless of their current batting average, the average will decrease more if the player misses than it will increase if the player gets a hit.

Step 1: (Part a - Step 1: Define the batting average function and find the differential)

First, we define the batting average function as: \(A(x, y) = \frac{x}{y}\). Then, we need to find the differential \(dA\). By using the multivariable version of the Chain Rule, we have: \(dA = \frac{\partial A}{\partial x} dx + \frac{\partial A}{\partial y} dy\), where \(\frac{\partial A}{\partial x}\) and \(\frac{\partial A}{\partial y}\) are the partial derivatives of A with respect to x and y respectively.

Step 2: (Part a - Step 2: Calculate the partial derivatives)

Now, we need to find the partial derivatives. We have: \(\frac{\partial A}{\partial x} = \frac{\partial}{\partial x}\left(\frac{x}{y}\right) = \frac{1}{y}\) and \(\frac{\partial A}{\partial y} = \frac{\partial}{\partial y}\left(\frac{x}{y}\right) = -\frac{x}{y^2}\).

Step 3: (Part a - Step 3: Estimate the change in the batting average with differentials)

Now we can use the differentials to approximate the change in the batting average when the number of hits increases from 60 to 62 and the number of at-bats increases from 175 to 180. We have: \(dx = 62 - 60 = 2\) and \(dy = 180 - 175 = 5\) Therefore, \(dA = \frac{1}{y} dx - \frac{x}{y^2} dy\) By plugging in the values x = 60 and y = 175, we get: \(dA = \frac{1}{175}(2) - \frac{60}{175^2}(5) \approx 0.00128\) The estimated change in the batting average given these changes in the number of hits and at-bats is approximately 0.00128.

Step 4: (Part b - Step 1: Define the average change in A for a hit and a miss)

Let's consider the cases where the batter gets a hit and when the batter misses. If the batter gets a hit, \(dx = 1\) and \(dy = 1\). If the batter misses, \(dx = 0\) and \(dy = 1\). Let's denote the average change in A for these two cases as: \(\Delta A_{hit} = \frac{1}{y} - \frac{x}{y^2}\) and \(\Delta A_{miss} = - \frac{x}{y^2}\)

Step 5: (Part b - Step 2: Compare the average change for a hit and a miss for A = 0.350)

Now, let's compare these two changes for a batter with a current batting average of A = 0.350. We have: \(0.350 = \frac{x}{y} \implies x = 0.350y\) So, we can rewrite the changes in A as: \(\Delta A_{hit} = \frac{1}{y} - \frac{0.350y}{y^2} = \frac{1}{y} - \frac{0.350}{y}\) and \(\Delta A_{miss} = - \frac{0.350y}{y^2} = - \frac{0.350}{y}\) Now, we compare these two expressions: \(\Delta A_{hit} - \Delta A_{miss} = \frac{1}{y} \ge 0\) Since \(\Delta A_{hit} \ge \Delta A_{miss}\), the average batting average will decrease more if the batter misses than it will increase if the batter gets a hit.

Step 6: (Part c - Step 1: Generalize the comparison for any batting average)

Now, let's analyze if the previous conclusion depends on the current batting average. To do this, let's consider any batting average A and rewrite the previous comparison as a function of A: \(\Delta A_{hit} - \Delta A_{miss} = \frac{1}{y} - A\) Since \(0 \le A \le 1\) and \(y > 0\), we have \(\Delta A_{hit} - \Delta A_{miss} \ge 0\) for any batting average A.

Step 7: (Part c - Step 2: Conclusion)

In conclusion, the answer to part (b) does not depend on the current batting average. Regardless of the batter's current batting average, it will decrease more if the batter misses than it will increase if the batter gets a hit.

Key Concepts

Partial Derivatives

Partial derivatives are an essential tool when dealing with functions of more than one variable. They help us understand how a function changes as one specific variable changes while keeping the others constant. In the context of our problem, the batting average is a function of two variables: the number of hits \(x\) and the number of at-bats \(y\).

To calculate the differential \(dA\) of the batting average \(A(x, y) = \frac{x}{y}\), we first find the partial derivatives with respect to \(x\) and \(y\). These are defined as \(\frac{\partial A}{\partial x}\) and \(\frac{\partial A}{\partial y}\), respectively.

For this specific function, we found:

• \(\frac{\partial A}{\partial x} = \frac{1}{y}\)
• \(\frac{\partial A}{\partial y} = -\frac{x}{y^2}\)

This means that as the number of hits changes, the batting average changes simply by \(\frac{1}{y}\) times the change in \(x\). Similarly, changes in \(y\) affect the average by \(-\frac{x}{y^2}\) times the change in at-bats. These derivatives make it easier to estimate changes without recalculating the full fraction every time.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of several variables. Unlike basic calculus with just one variable, here, we must consider how changes in one variable might affect others.

In the case of batting averages, both the number of hits \(x\) and the number of at-bats \(y\) influence the batting average \(A(x, y) = \frac{x}{y}\).

With the introduction of partial derivatives, we use these tools together to analyze how tiny changes in \(x\) and \(y\) will impact the overall batting average. The differential \(dA\) is computed as: \[dA = \frac{\partial A}{\partial x} \, dx + \frac{\partial A}{\partial y} \, dy\]

This expression allows for the approximation of the change in the function \(A\) for small variations of each variable. In practice, it's employed to estimate the effect on the average as a player hits or misses at the plate, enhancing our understanding of real-world applications where all variables won't always alter by large, discrete amounts.

Differential Approximation

Differential approximation is a concept used in calculus to estimate the change in a function's value given changes in its input variables. This is particularly useful when exact calculations are complex or unnecessary.

In our exercise, we employed differential approximation to estimate the change in the batting average \(\Delta A\) when the number of hits \(x\) changes from 60 to 62, and at-bats \(y\) from 175 to 180. The tiny increments \(dx = 2\) and \(dy = 5\) are plugged into the differential equation: \[dA = \frac{1}{y} \, dx - \frac{x}{y^2} \, dy\]

This approximation gives us a quick and efficient way to gauge the batting average's behavior without recalculating the entire average repeatedly. The resulting estimated change \(dA \approx 0.00128\) suggests a slight increase in the average based on the given changes.

• Useful for small changes where exact values aren't vital.
• Simplifies complex computations.

By applying these approximations, analysts and fans alike can make informed guesses about a player's performance trends with minimal effort.