Question

Batting averages Batting averages in bascball 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 more if the batter fails to get a hit than it increases if the batter gets a hit? c. Does the answer to part (b) depend on the current batting average? Explain.

Short answer

Answer: The batting average is estimated to increase by approximately 0.0091 after the number of hits increases by 2 and the number of at-bats increases by 5. The average increases more when the batter gets a hit than it decreases when they fail to get one, and this conclusion holds true for any current batting average.

Step-by-step solution

Compute partial derivatives of A

Find the partial derivatives of the batting average function \(A(x,y) = \cfrac{x}{y}\) with respect to \(x\) and \(y\). \( \cfrac{\partial A}{\partial x} = \cfrac{1}{y}\) and \( \cfrac{\partial A}{\partial y} = -\cfrac{x}{y^2}\)

Step 2a: Estimate the change in A for part a

Using differentials, estimate the change in the batting average after the number of hits increases by 2 and the number of at-bats increases by 5. \(\Delta A \approx \cfrac{\partial A}{\partial x}\Delta x + \cfrac{\partial A}{\partial y}\Delta y = \cfrac{1}{175} \cdot 2 - \cfrac{60}{175^2} \cdot 5\) Now, we can compute the change in A: \(\Delta A \approx \cfrac{2}{175} - \cfrac{300}{175^2} \approx 0.0091\) So, the batting average is estimated to increase by approximately 0.0091.

Step 2b: Compare changes in A for part b

Compare the change in A when the batter gets a hit with the change when they fail to get one for a current batting average of 0.350. \(\Delta A_1 = \frac{\partial A}{\partial x} - \frac{\partial A}{\partial y} = \frac{1}{y} - \left(-\frac{x}{y^2}\right)\) \(\Delta A_2 = -\frac{\partial A}{\partial y} = \frac{x}{y^2}\) We know that the current average is \(A = 0.350 = \cfrac{x}{y}\). Let's assume that the batter has \(x_0\) hits and \(y_0\) at-bats. Then: \(\Delta A_1 = \cfrac{1}{y_0} + \cfrac{x_0}{y_0^2}\) \(\Delta A_2 = \cfrac{x_0}{y_0^2}\) Now, compare \(\Delta A_1\) and \(\Delta A_2\): \(\Delta A_1 - \Delta A_2 = \cfrac{1}{y_0}\) Since \(y_0 > 0\), we have \(\Delta A_1 - \Delta A_2 > 0\), meaning that the average increases more when the batter gets a hit than it decreases when they fail to get one.

Step 2c: Analyze dependence of part b on A

To determine if the conclusion in part b depends on the current batting average, note that: \(\Delta A_1 - \Delta A_2 = \cfrac{1}{y_0}\) Since \(y_0 > 0\), this difference is always positive, regardless of the value of \(A\). Thus, the conclusion that the average increases more when the batter gets a hit than it decreases when they fail to get one holds true for any current batting average A.

Key concepts

Differentials

Understanding differentials is crucial when estimating small changes in functions, as they often approximate changes in variables with greater ease than direct calculations. In the batting average problem, we use differentials to estimate the change in the batting average, defined as \( A = \frac{x}{y} \), when the number of hits \( x \) and the number of at-bats \( y \) are altered.

Differentials, denoted by \( dx \) and \( dy \), are infinitesimally small changes in the variables \( x \) and \( y \). In a practical sense, they represent how much \( x \) and \( y \) have changed, and thus, how much the function \( A(x,y) \) might change as a result - this is expressed as \( dA \). When we estimate the change in the batting average after an increase in the number of hits and at-bats, we calculate \( dA \) using the formula:
\[ dA \approx \frac{\partial A}{\partial x}dx + \frac{\partial A}{\partial y}dy \]
which simplifies to our estimated change in the batting average. This is a linear approximation, and while not exact, it provides a quick and intuitive measure of the impact of small changes.

Partial Derivatives

Partial derivatives come into play when dealing with functions of multiple variables, like the batting average \( A(x,y) \). They measure the rate of change of the function with respect to one variable while keeping the others constant.

In our example, we find the partial derivatives of the batting average function with respect to \( x \) and \( y \). The partial derivative with respect to \( x \), denoted as \( \frac{\partial A}{\partial x} \), tells us how the batting average changes as the number of hits changes, assuming the number of at-bats remains unchanged. Conversely, the partial derivative with respect to \( y \), denoted as \( \frac{\partial A}{\partial y} \), indicates how the average changes with the number of at-bats, assuming hits remain constant.

Calculating these for our function, we get:\[ \frac{\partial A}{\partial x} = \frac{1}{y} \] and \[ \frac{\partial A}{\partial y} = -\frac{x}{y^2} \]. This reveals that the batting average increases with more hits and decreases with more at-bats, which is intuitive for baseball fans.

Function Estimation

Function estimation is a valuable tool to approximate the values of functions at certain points, especially when a direct calculation is challenging. It uses the concepts of differentials and partial derivatives as its foundation.

In this calculus-based batting average problem, we use the partial derivatives and differentials to estimate the new batting average after a small change in the values of hits and at-bats. The method for estimating the change in the function \( A(x,y) \) involves using a linear approximation:

\[ \Delta A \approx \frac{\partial A}{\partial x}\Delta x + \frac{\partial A}{\partial y}\Delta y \]
This approximation assumes that over small changes, the function behaves almost linearly, which allows us to use the tangent plane at a point to estimate the new value of the function after those changes. The procedure simplifies complex relationships into a manageable estimation that provides insight without needing to recalculate the entire function.

Real Number Application

In the context of batting averages, the application of real numbers and their properties is evident. Batting averages are ratios that take values between 0 and 1, inclusive, representing a real number application within the bounds of this interval.

Our problem considers \( x \) and \( y \) as positive real numbers, embodying the real-world counts of hits and at-bats, respectively. As real numbers, they carry significant meaning: a higher number of hits \( x \) directly translates to a better batting performance, while more at-bats \( y \) without corresponding hits can negatively impact the average.

Furthermore, the question of whether a batting average fluctuates more with a miss or a hit draws on the properties of real numbers and their operations. Since the slopes of the tangent lines to the batting average function at a given point are dictated by the values of \( x \) and \( y \) (which are real numbers), we can make precise claims about the sensitivity of a batter's average to hits and misses. This demonstrates a concrete application of calculus concepts within the familiar frame of baseball statistics.