Question

Analyzing Derivative Data Assume that \(f\) is continuous on \([-2,2]\) and differentiable on \((-2,2) .\) The table gives some values of \(f^{\prime}(x)\) $$ \begin{array}{cccc}\hline x & {f^{\prime}(x)} & {x} & {f^{\prime}(x)} \\\ \hline-2 & {7} & {0.25} & {-4.81} \\ {-1.75} & {4.19} & {0.5} & {-4.25} \\\ {-1.5} & {1.75} & {0.75} & {-3.31} \\ {-1.25} & {-0.31} & {1} & {-2}\end{array} $$ $$ \begin{array}{rrrr}{-1} & {-2} & {1.25} & {-0.31} \\ {-0.75} & {-3.31} & {1.5} & {1.75} \\ {-0.5} & {-4.25} & {1.75} & {4.19}\end{array} $$ $$ \begin{array}{cccc}{-0.25} & {-4.81} & {2} & {7} \\ {0} & {-5}\end{array} $$ $$ \begin{array}{l}{\text { (a) Estimate where } f \text { is increasing, decreasing, and has local }} \\ {\text { extrema. }} \\ {\text { (b) Find a quadratic regression equation for the data in the table }} \\ {\text { and superimpose its graph on a scatter plot of the data. }} \\ {\text { (c) Use the model in part (b) for } f^{\prime} \text { and find a formula for } f \text { that }} \\ {\text { satisties } f(0)=0 .}\end{array} $$

Short answer

The function \(f(x)\) is increasing on \([-2,-1.25]\) and \([1.5,2]\), and decreasing on \([-1.25,1.5]\). It has a local maximum at \(x=-1.25\) and a local minimum at \(x=1.5\). We can use quadratic regression on the given data to find an equation \(f'(x)\), and then integrate this equation from 0 to \(x\) to find the formula for \(f(x)\) that satisties \(f(0)=0\).

Step-by-step solution

Identify Increasing and Decreasing Intervals

Examine the given table values of \(f'(x)\). We find that \(f'(x)>0\) on intervals \([-2,-1.25]\) and \([1.5,2]\), and \(f'(x)<0\) on intervals \([-1.25,1.5]\). This tells us that \(f(x)\) is increasing on intervals \([-2,-1.25]\) and \([1.5,2]\) and decreasing on interval \([-1.25,1.5]\).

Identify Local Extrema

Local extrema occurs where the derivative changes sign. The derivative changes sign from positive to negative at \(x=-1.25\), making this a local maximum. The derivative changes sign from negative to positive at \(x=1.5\), hence this a local minimum.

Finding Quadratic Regression Equation

Use statistical software or a calculator to perform quadratic regression on the given data. This will give us a quadratic equation that best fits the data points.

Formulate Function Formula

Using the model from quadracti regression equation and knowing that \(f(0)=0\), you can integrate the equation from \(0\) to \(x\) to find \(f(x)\).

Key concepts

Increasing and Decreasing Intervals

Understanding increasing and decreasing intervals of a function is fundamental in calculus, particularly when analyzing derivative data. When we look at the derivative of a function, denoted as f', we're essentially exploring the slope at various points of the original function. Positive values of f' indicate that the function f is on the rise, or increasing, as its slope is positive. Similarly, when f' is negative, the function f is decreasing, since the slope of the graph is negative at those points.

For the given exercise, we detect intervals of increase and decrease by examining the sign of f' in each interval. As stated in the solution, f is increasing when f' is above zero, and decreasing when it is below zero. This analysis is essential for visualizing the function's behavior and lays the foundation for finding local extrema, which often occur at the boundaries between increasing and decreasing intervals.

Local Extrema

Local extrema refer to the high and low points on the graph of a function, known as local maximums and minimums, respectively. These points represent the 'peaks' and 'valleys' where the function changes direction. The derivative of a function can tell us where these extrema occur, due to a key property: a local extremum is typically found where the derivative changes sign.

In the solved exercise, we identify a local maximum at x = -1.25, where f' switches from positive to negative, indicating the function f halts its upwards trend and starts to decrease. Conversely, a local minimum is spotted at x = 1.5, where f' turns from negative to positive. Recognizing these points is critical because they can signify critical moments in applied contexts, such as maximum profits in economics or minimum material stress in engineering.

Quadratic Regression

Quadratic regression is a type of polynomial regression that fits a quadratic equation, typically in the form y = ax² + bx + c, to a set of data. This method helps predict the relationship between an independent variable x and a dependent variable y. It is particularly useful when the data exhibits a parabolic trend, suggesting that the rate of change is not constant.

In our exercise, quadratic regression is applied to derive a model that describes how the derivative of f, or f', behaves. By finding the best-fitting quadratic curve, we are given an equation that we can use to estimate values of f' for inputs x not listed in our table. Moreover, this equation becomes the cornerstone for reconstructing the original function f through the process of integration, addressed in the next section.

Integration of Functions

Integration is the reverse operation of differentiation and plays a major role in reconstructing a function from its derivative. In essence, integration provides us with the accumulated area under the curve of the derivative function, which corresponds to the original function's values. In practice, when integrating a function, we are adding up an infinite number of infinitely small pieces to find the whole.

As per the exercise, once the quadratic regression provides us with a model for f', we integrate this model starting from a known value to find f. Since the exercise specifies that f(0) = 0, we use this condition as our starting point. The result of integrating the model from zero to any point x will yield the formula for f(x). This allows us to determine the original function's values for any point in the domain, bringing full circle our analysis of derivative data.