Problem 36 - Single Variable Calculus

Question

Plot the graph of \(f\), and find (a) the approximate intervals where the graph of \(f\) is concave upward and where it is concave downward and (b) the approximate coordinates of the point \((s)\) of inflection accurate to 1 decimal place. $$ f(x)=\cos (\sin x) \quad-2

Step-by-Step Solution

Answer

The graph of \(f(x) = \cos(\sin(x))\) in the interval \(-2 < x < 2\) is concave upward approximately in the intervals \(-1.2 < x < 1.2\), and it is concave downward approximately in the intervals \(-2 < x < -1.2\) and \(1.2 < x < 2\). The approximate coordinates of the points of inflection are \(s \approx (-1.2, 0.8)\) and \(s \approx (1.2, 0.8)\).

Step 1: 1. Find the first derivative of \(f(x)\)

To find the first derivative of \(f(x)\), we need to use the Chain Rule for derivatives: $$ f'(x) = \frac{d}{dx}(\cos(\sin(x))) $$ Applying the chain rule: $$ f'(x) = -\sin(\sin(x)) \cdot \cos(x) $$

Step 2: 2. Find the second derivative of \(f(x)\)

Now, let's find the second derivative, which will help us determine the concavity of the function. To find the second derivative of \(f(x)\), differentiate the first derivative again: $$ f''(x) = \frac{d}{dx}(-\sin(\sin(x)) \cdot \cos(x)) $$ Applying the product rule and the chain rule: $$ f''(x) = -\cos(\sin(x)) \cdot \cos^2(x) + \sin(\sin(x)) \cdot \sin(x) \cdot \cos(x) $$

Step 3: 3. Determine the concavity

To determine the concavity of the function, we need to analyze the sign of the second derivative, \(f''(x)\). (a) We have: Concave upward: \(f''(x) > 0\) Concave downward: \(f''(x) < 0\) Since the exact sign of the second derivative is complicated to determine, we can approximate the intervals using a graphing calculator or online plotting tool.

Step 4: 4. Approximate intervals and inflection points

(b) A point of inflection occurs where the concavity of the function changes, meaning that the second derivative changes its sign. To find the approximate coordinates of the point(s) of inflection, we can examine the graph of the function and look for the points where the concavity change occurs: Using a graphing calculator or an online graphing tool, plot the function \(f(x) = \cos(\sin(x))\) over the interval \(-2 < x < 2\). Approximately, the graph appears to be: Concave upward: \(-1.2 < x < 1.2\) Concave downward: \(-2 < x < -1.2\) and \(1.2 < x < 2\) Inflection point(s): \(s \approx (-1.2, 0.8)\) and \(s \approx (1.2, 0.8)\)

Key Concepts

Derivative

In calculus, the derivative measures how a function changes as its input changes. For example, if you have a function that describes how quickly a car moves, its derivative would tell you the car's speed. In this exercise, we find the first derivative of the function\[ f(x) = \cos(\sin(x)) \]using the chain rule, which we will discuss later. The derivative helps us understand important features of the function, such as whether it is increasing or decreasing.

• The derivative of a function at a specific point can be imagined as the slope of the tangent line at that point.
• When the derivative is positive, the function is increasing.
• When it is negative, the function is decreasing.
• A zero derivative indicates a possible minimum, maximum, or an inflection point (which we will explore further later).

In this specific problem, we find that the derivative of the function is \[ f'(x) = -\sin(\sin(x)) \cdot \cos(x). \]By analyzing this derivative, we can better understand how the function behaves and moves.

Chain Rule

The chain rule is a powerful technique in calculus used to find the derivative of composite functions. A composite function is a function that is made up of other functions nested inside each other. For instance, in our problem, the function \[ f(x) = \cos(\sin(x)) \]is a composite of the cosine and sine functions.When you take the derivative of such functions, the chain rule tells you how to "chain" these functions together. The rule states:\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x). \]Here is how the chain rule works step by step in our function:

• First, take the derivative of the outer function while keeping the inside part unchanged. So, for the outer function cos, its derivative is -sin.
• Multiply this by the derivative of the inside function. The inside function is sin(x), and its derivative is cos(x).

Applying these steps gives us:\[ f'(x) = -\sin(\sin(x)) \cdot \cos(x). \]This process helps us obtain the derivative correctly and is essential for analyzing the behavior of more complicated functions like in our exercise.

Concavity

Concavity describes the way a graph curves. It shows us whether a function is bending upwards like a cup or downwards like an umbrella. In mathematical terms:

• If the second derivative \( f''(x) \) is positive, the function is concave upward (like a cup) in that interval.
• Conversely, if the second derivative is negative, the function is concave downward (like an umbrella).

For this exercise, finding the second derivative \[ f''(x) = -\cos(\sin(x)) \cdot \cos^2(x) + \sin(\sin(x)) \cdot \sin(x) \cdot \cos(x) \]allows us to determine the concavity of \( f(x) = \cos(\sin(x)) \).Understanding the concavity is crucial because it helps us identify how the function looks graphically and pinpoints intervals where the function curves in different directions. This information is often utilized to locate areas that may contain important points, such as maximums, minimums, or inflection points (explained next).

Inflection Point

An inflection point is where a function changes its concavity. At these particular points, a graph of a function will switch from being concave upward to concave downward or vice versa. Inflection points are significant in understanding the overall shape and behavior of a graph.To find inflection points, you look for where the second derivative of the function changes signs. This indicates a reversal in the direction of the bending of the function.In this problem, the approximate inflection points were determined by observing the graph of the function \( f(x) = \cos(\sin(x)) \)and noting where the function's concavity shifted. The approximate coordinates of the inflection points are

• \( (-1.2, 0.8) \)
• \( (1.2, 0.8) \)

Understanding inflection points provides insights into the dynamics of the function’s graph, reflecting points where the behavior of the function contextually shifts.