Problem 71 - Thomas Calculus

Question

In Exercises \(71-74,\) let \(F(x)=\int_{a}^{x} f(t) d t\) for the specified function \(f\) and interval \([a, b] .\) Use a CAS to perform the following steps and answer the questions posed. a. Plot the functions \(f\) and \(F\) together over \([a, b]\) b. Solve the equation \(F^{\prime}(x)=0 .\) What can you see to be true about the graphs of \(f\) and \(F\) at points where \(F^{\prime}(x)=0 ?\) Is your observation borne out by Part 1 of the Fundamental Theorem coupled with information provided by the first derivative? Explain your answer. c. Over what intervals (approximately) is the function \(F\) increasing and decreasing? What is true about \(f\) over those intervals? d. Calculate the derivative \(f^{\prime}\) and plot it together with \(F .\) What can you see to be true about the graph of \(F\) at points where \(f^{\prime}(x)=0 ?\) Is your observation borne out by Part 1 of the Fundamental Theorem? Explain your answer. $$ f(x)=x^{3}-4 x^{2}+3 x, \quad[0,4] $$

Step-by-Step Solution

Answer

At points where \(F'(x) = 0\), \(f(x) = 0\). Thus, \(F\) is flat, aligning with the Fundamental Theorem. \(F\) increases where \(f > 0\) and decreases where \(f < 0\), consistent with calculus principles.

Step 1: Define the Integral Function F(x)

The given function is \(f(x) = x^3 - 4x^2 + 3x\) and the interval is \([0, 4]\). We define \(F(x)\) as \(F(x) = \int_{0}^{x} (t^3 - 4t^2 + 3t) \, dt\). This will be used to plot and analyze \(F(x)\) along with \(f(x)\).

Step 2: Plot Functions f(x) and F(x) Over [0, 4]

Use a computer algebra system (CAS) to plot \(f(x)\) and \(F(x)\). You will see \(f(x)\) forming a cubic curve, while \(F(x)\) should represent the area accumulated under \(f(x)\) from \(0\) to \(x\).

Step 3: Solve F'(x) = 0

For \(F'(x)\), based on the Fundamental Theorem of Calculus, \(F'(x) = f(x)\). Therefore, solve \(x^3 - 4x^2 + 3x = 0\) to find critical points where \(F'(x) = 0\). Factor the equation to get \(x(x - 3)(x - 1) = 0\), giving critical points at \(x = 0, 1, 3\). At these points, \(f(x) = 0\) as well, suggesting that when \(F'(x) = 0\), \(f\) also intersects the x-axis.

Step 4: Analyze Increasing and Decreasing Intervals for F(x)

Determine the intervals where \(F(x)\) is increasing or decreasing. Using the sign of \(f(x) = F'(x)\), note that \(F(x)\) is increasing where \(f(x) > 0\) and decreasing where \(f(x) < 0\). Compute by testing intervals: \([0, 1], [1, 3], [3, 4]\). You will find \(F\) is increasing in \([0, 1)\) and \((3, 4]\), and decreasing in \((1, 3)\).

Step 5: Derive and Plot f'(x)

Calculate \(f'(x) = \frac{d}{dx}(x^3 - 4x^2 + 3x) = 3x^2 - 8x + 3\). Plot \(f'(x)\) together with \(F(x)\). Evaluate where \(f'(x) = 0\) to find \(x\) values where the slope of \(f(x)\) is zero. Solving \(3x^2 - 8x + 3 = 0\) using the quadratic formula gives \(x = \frac{8 \pm \sqrt{28}}{6}\). These are turning points for \(f\), but for \(F(x)\), they contribute to curvature change rather than slope zero.

Step 6: Final Observations

From the plots and calculations: \(F'(x) = 0\) implies \(f(x) = 0\) (points where \(f\) is zero are where \(F\) has horizontal tangents). This observation aligns with Part 1 of the Fundamental Theorem: \(F'(x) = f(x)\). The derivative \(f'(x)\) affects the concavity and inflection points in \(F\), confirming that changes in \(f'(x) = 0\) can result in these features but not critical points without considering \(f\).

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus provides a crucial link between differentiation and integration. Specifically, it consists of two parts. Part 1 states that, if you have a continuous function \( f \) defined on an interval \([a, b]\), then the function \( F(x) = \int_{a}^{x} f(t) \, dt \) is an antiderivative of \( f \). This means that \( F'(x) = f(x) \). In simpler terms, the derivative of the integral of a function returns the original function.
Understanding this part helps us with problems involving integral functions, such as finding \( F'(x) \) for the integral \( F(x) = \int_0^x (t^3 - 4t^2 + 3t) \, dt \). Here, the fundamental theorem tells us that \( F'(x) = f(x) = x^3 - 4x^2 + 3x \).
By recognizing this result, we can understand that any computation involving \( F'(x) \) will directly lead us back to the functions we integrated.

Integral Function

An integral function \( F(x) \) is created by accumulating the area under a curve \( f(t) \) from a starting point to \( x \). Formally, \( F(x) = \int_{a}^{x} f(t) \, dt \) represents this process. By graphing \( F(x) \) alongside \( f(x) \), we visually interpret how the accumulated area up to any point \( x \) changes.
In our example with \( f(x) = x^3 - 4x^2 + 3x \) over the interval \([0, 4]\), understanding \( F(x) \) helps us explore questions about the function's behavior like where \( F(x) \) has reached a peak, or turned flat (critical points).
Plotting \( F(x) \) provides insight into how the integral behaves. The graph tends to illustrate increases or decreases in \( F(x) \), corresponding to variations in \( f(x) \). For example, if \( f(x) \) is positive over a specific interval, \( F(x) \) will increase over that interval.

Increasing and Decreasing Intervals

To determine where a function \( F(x) \) is increasing or decreasing, we look at the derivative \( F'(x) = f(x) \). If \( f(x) > 0 \), \( F(x) \) is increasing, and if \( f(x) < 0 \), \( F(x) \) is decreasing.
For the function \( f(x) = x^3 - 4x^2 + 3x \), we need to solve \( F'(x) = 0 \) to identify the points critical for increasing or decreasing behavior. In this case, the critical points found at \( x = 0, 1, \) and \( 3 \) help us evaluate the intervals:

• \([0, 1)\): where \( f(x) \) is positive, \( F(x) \) is increasing.
• \((1, 3)\): where \( f(x) \) is negative, \( F(x) \) is decreasing.
• \((3, 4]\): where \( f(x) \) returns positive, \( F(x) \) increases again.

This analysis shows how the behavior of \( f(x) \) directly translates to the behavior of the integral function \( F(x) \).

Critical Points

Critical points of a function occur where its derivative is zero or undefined. They are key because they denote potential maxima, minima, or points of inflection. In the context of integral functions like \( F(x) \) from our exercise, solving \( F'(x) = 0 \) pinpoints those critical points by focusing where the original function \( f(x) \) hits zero.
Here, \( F(x) \) achieves critical points at \( x = 0, 1, \) and \( 3 \) because \( f(x) = x^3 - 4x^2 + 3x \) equals zero at these values. These points often signify changes in increasing or decreasing tendencies for \( F(x) \), such as turning from an upward trend to a downward or vice versa.
To further investigate, plotting the derivative \( f'(x) = 3x^2 - 8x + 3 \) reveals points where the curvature of \( f \) alters, indicating how these affect the shape of \( F(x) \) although the turning points in \( f'(x) \) might not always affect \( F(x) \) directly unless combined with additional context.