Problem 58 - Calculus with Concepts in Calculus

Question

Plot the graph of \(f\). Using the zoom feature of the calculator, approximate to within \(0.1\) all values of \(c\) such that \(f(c)\) is a relative extreme value, and identify each as a relative maximum value or a relative minimum value. $$ f(x)=e^{x}+\sin x \text { for }-6 \leq x \leq 3 $$

Step-by-Step Solution

Answer

Plot and check points using a calculator; expect maxima and minima around points where derivative changes sign.

Step 1: Understand the Function

The function we need to analyze is given by \( f(x) = e^x + \sin x \). We need to investigate it on the interval \([-6, 3]\). This means we need to comprehend how the exponential function \(e^x\) and the sine function \(\sin x\) interact over this domain.

Step 2: Identify Critical Points

Critical points occur where the derivative of the function is zero or undefined. To find them, we first have to determine the derivative: \( f'(x) = e^x + \cos x \). We will set \( f'(x) = 0 \) and solve for \( x \). This is not straightforward analytically, so we'll use numerical methods or graphing methods.

Step 3: Use a Calculator for Graphing

Using a graphing calculator, plot \( f(x) = e^x + \sin x \) within the interval \([-6, 3]\). Adjust the zoom and scaling appropriately to clearly identify the turning points. Note these as our potential relative extreme values.

Step 4: Identify Relative Extrema

By closely examining the graph and looking at where the slope changes from positive to negative (and vice versa), you can identify relative maxima and minima. Note these \( x \)-values where such changes occur to a precision of \(0.1\).

Step 5: Confirm Extrema Classification

For each critical point \( c \) identified, observe \( f''(c) \) using your calculator (second derivative). If \( f''(c) > 0 \), then \( f(c) \) is a relative minimum; if \( f''(c) < 0 \), then \( f(c) \) is a relative maximum.

Key Concepts

Critical Points

A fundamental aspect of graphing functions involves finding the critical points of a function. Critical points are those input values where the derivative of the function is zero or undefined. - For a function like \[ f'(x) = e^x + \cos x \] you find critical points by setting \( f'(x) \) to zero and solving for \( x \). - These points could either be maximums, minimums, or points of inflection, but only a more detailed analysis can tell. In the exercise, the solution requires you to calculate these points within the interval \([-6, 3]\). Because this involves a transcendental equation, it's typically resolved using numerical methods or graphing tools where exact algebraic solutions are not feasible.

Relative Extrema

Once critical points are identified, the next step is classifying them to determine if they are relative extrema. Relative extrema refer to the points in the graph where the function peaks or troughs compared to nearby points. Relative maxima occur at peaks, while minima occur at troughs. To identify these: - Calculate the derivative and find where it changes sign. - A switch from positive to negative indicates a relative maximum, while a switch from negative to positive hints at a relative minimum. In the given function within the specified interval, you utilize graphing calculators to visually spot these sign changes in the slope \( f'(x) = 0 \), which provides a practical means of approximation.

Exponential and Trigonometric Functions

The interplay between exponential functions and trigonometric functions presents unique challenges and interesting results. In the function \( f(x) = e^x + \sin x \), the two components have distinct shapes and rates of growth or oscillation. - The exponential part \( e^x \) is known for its rapid growth, especially for positive \( x \). - The trigonometric part, \( \sin x \), oscillates between \(-1\) and \(1\), giving waves. These combined properties influence the overall graph appearance, causing complexity where \( \sin x \) affects the otherwise smooth, growing nature of \( e^x \). Understanding their characteristics helps explain the function's different behaviors, and ultimately its critical points and extrema.

Second Derivative Test

The second derivative test is an efficient tool for classifying the nature of critical points identified earlier. Once you have a critical point \( c \), the second derivative \( f''(c) \) aids in determining whether it's a relative maximum, minimum, or neither. - If \( f''(c) > 0 \), the function is concave up at \( c \), which means a relative minimum. - If \( f''(c) < 0 \), the function is concave down, indicating a relative maximum. - If \( f''(c) = 0 \), the test might be inconclusive; another test or inspection is needed. This process solidifies the nature of critical points, transforming initial assumptions from visual inspections into verified classifications.