Problem 90 - Calculus Single Variable

Question

In each of Exercises 90-93, a function \(f\) is given. Calculate \(f^{\prime}\) and plot \(y=f^{\prime}(x)\) in a suitable viewing window. Use this plot to identify the points at which \(f\) has local extrema. $$ f(x)=x^{3}-x+\sin ^{2}(\pi x / 2),-1

Step-by-Step Solution

Answer

The local extrema occur where \( f'(x) = 0 \), inspect the plot for exact locations.

Step 1: Differentiate the Polynomial Part

Start by differentiating the polynomial part of the function \( f(x) = x^3 - x \). The derivative is found by applying the power rule: \( f'_1(x) = 3x^2 - 1 \). This represents the derivative for the polynomial components of \( f(x) \).

Step 2: Differentiate the Trigonometric Part

Next, differentiate the trigonometric part \( g(x) = \sin^2(\pi x / 2) \). Use the chain rule for \( (\sin(u))^2 \), where \( u = \pi x / 2 \). The derivative is \( g'(x) = 2 \sin(u) \cdot \cos(u) \cdot (\pi/2) = \pi \sin(\pi x / 2) \cos(\pi x / 2) \). This can be simplified using the identity \( \sin(2u) = 2 \sin(u) \cos(u) \) to \( g'(x) = \frac{\pi}{2} \sin(\pi x) \).

Step 3: Combine the Derivatives

Combine the derivatives from Steps 1 and 2 to form the derivative of \( f(x) \). Thus, \( f'(x) = 3x^2 - 1 + \frac{\pi}{2} \sin(\pi x) \). This expression represents the complete derivative of the given function.

Step 4: Plot \(f'(x)\) to Identify Local Extrema

Plot the derived function \( y = f'(x) = 3x^2 - 1 + \frac{\pi}{2} \sin(\pi x) \) over the interval \(-1 < x < 1\). Identify the points where the derivative changes sign: these indicate the local extrema of \( f(x) \). If \( f'(x) \) crosses from positive to negative, it typically indicates a local maximum; from negative to positive, a local minimum.

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus. They give us information about how a function changes. Geometrically, the derivative of a function at a point tells us the slope of the tangent line to the curve at that point.
To find a derivative, we often use rules like the power rule, product rule, and chain rule, depending on the function's composition.
For example, with a polynomial like \( x^3 - x \), we apply the power rule by multiplying the exponent by the coefficient and decreasing the exponent by one.

• Derivative of \( x^3 \) is \( 3x^2 \).
• Derivative of \( -x \) is \( -1 \).

These are simple rules that help simplify the differentiation process.

Finding Local Extrema

Local extrema refer to the highest or lowest points in a specific region of a function's graph. These are identified where the derivative changes signs.
When \( f'(x) \) crosses from positive to negative, it indicates a local maximum. Conversely, if it crosses from negative to positive, that's a local minimum.
To find these points, calculate the derivative and then determine where it equals zero. Solve \( f'(x) = 0 \) to find critical points. These points are potential candidates for local extrema, which can be confirmed by further analysis.

Using the Chain Rule

The chain rule is crucial for differentiating compositions of functions. It helps us differentiate a function like \( \sin^2(\pi x / 2) \) by breaking it into parts.
Let's consider \( u = \pi x / 2 \). We then differentiate \( \sin^2(u) \), which involves finding the derivative of \( \sin(u) \) and then multiplying by the derivative of \( u \):

• Derivative of \( \sin(u) \) is \( \cos(u) \).
• Multiply by \( 2\sin(u) \) due to the chain rule.
• Lastly, multiply by \( \pi/2 \) which is the derivative of \( u \).

This allows us to accurately compute the derivative of composite functions, crucial in complex differentiation.

Plotting Functions and Analyzing Graphs

Plotting functions is a powerful visual way to understand how they behave and interact. By graphing \( y = f'(x) = 3x^2 - 1 + \frac{\pi}{2} \sin(\pi x) \), we can easily see where the function's slope changes, implying potential extrema.
Use graphing tools to visualize \( f'(x) \) in the interval \(-1 < x < 1\). Look for points where the graph crosses the x-axis.
This crossing indicates a change from increasing to decreasing (or vice versa), signaling local extrema. Visual representation helps in comprehending the properties of a function more clearly.