Problem 56 - Essential Calculus Early Transcendentals

Question

\(53-56\) (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. $$f(x)=x-2 \cos x, \quad-2 \leqslant x \leqslant 0$$

Step-by-Step Solution

Answer

Max at \( x = 0 \), Min at \( x = -2 \).

Step 1: Setting up the Graph

First, we need to visualize the function on the given interval to estimate where the maximum and minimum might occur. Plot the graph of \( f(x) = x - 2\cos x \) over the interval \(-2 \leq x \leq 0\). Look at the plot to visually estimate the highest and lowest points.

Step 2: Estimating from the Graph

By examining the graph within the region \(-2 \leq x \leq 0\), identify an approximate location for the absolute maximum and minimum values of the function. Note the \( x \)-values where these extrema seem to occur.

Step 3: Finding the Derivative

For finding the exact extrema using calculus, calculate the derivative of the function. The derivative of \( f(x) = x - 2\cos x \) is \( f'(x) = 1 + 2\sin x \).

Step 4: Setting the Derivative to Zero

Solve for \( x \) where the derivative equals zero to find critical points: \( f'(x) = 1 + 2\sin x = 0 \). This simplifies to \( \sin x = -0.5 \). Use values in the interval \(-2 \leq x \leq 0\) to find \( x \) such that \( \sin x = -0.5 \).

Step 5: Evaluating Critical and Endpoint Values

Compute \( f(x) \) at all critical points as well as the endpoints of the interval, \(-2\) and \(0\). The critical points, derived from solving \( \sin x = -0.5 \), must also be calculated.

Step 6: Comparing to Find Extrema

Compare function values obtained for each critical point and endpoint to determine which is the maximum and which is the minimum. Conclusion should note the largest value as the absolute maximum and the smallest value as the absolute minimum.

Key Concepts

Understanding Critical Points

In calculus, critical points of a function are those points where the function's derivative is either zero or undefined. These points are crucial because they can indicate local maxima, minima, or points of inflection on the graph of the function. To find critical points, we first compute the derivative of the function, then set the derivative equal to zero, and solve for the variable. For example, considering the function \( f(x) = x - 2\cos x \), the derivative is \( f'(x) = 1 + 2\sin x \). Setting this equal to zero, \( 1 + 2\sin x = 0 \), helps identify the points where the behavior of the function might change, revealing potential extreme values. - Critical points are where the derivative is zero or undefined.- They require solving \( f'(x) = 0 \) or finding where \( f'(x) \) does not exist.- These points help locate possible local maxima or minima.

The Role of Derivative

The derivative of a function represents the rate at which the function's value changes as its input changes. In the context of finding extrema, the derivative tells us about the slope of the tangent to the graph of the function. A derivative of zero denotes a flat tangent, which is associated with potential maxima or minima. In our example with the function \( f(x) = x - 2\cos x \), the derivative is given by \( f'(x) = 1 + 2\sin x \). Solving \( 1 + 2\sin x = 0 \) helps identify where the slope is flat. - Derivatives indicate the rate of change.- Zero derivatives imply flat slopes and potential extrema.- Calculate derivatives to find critical points.

Graphical Estimation Techniques

Graphical estimation is a technique used to visually approximate where the maximum and minimum values of a function might be. By plotting the graph of the function, you can look for the highest and lowest points on the curve within a given interval. For the function \( f(x) = x - 2\cos x \) over the interval \(-2 \leq x \leq 0\), you might observe the curve and estimate that the max and min occur around certain \( x \)-values. This initial examination provides a helpful visual guide before diving into calculus methods for precise calculations.- Visualize the function's graph over the relevant interval.- Estimate high and low points (extrema) visually.- Use the graph as a guide before calculating exact values.

Calculus Methods for Finding Extrema

Using calculus to find the exact maximum and minimum values of a function involves several steps. First, compute the function's derivative and find critical points by setting the derivative to zero. Next, evaluate the function at these critical points as well as at the endpoints of the interval. For our function \( f(x) = x - 2\cos x \), critical points arise where \( 1 + 2\sin x = 0 \), alongside evaluations at \( x = -2 \) and \( x = 0 \). Compare the function values at all these points to determine which is the absolute maximum and which the absolute minimum. - Solve for and evaluate critical points and endpoints.- Compare function values to identify absolute extrema.- Use calculus steps to confirm initial graphical estimations.