Problem 72 - 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. $$ f(x)=2 x^{4}-17 x^{3}+46 x^{2}-43 x+12,\left[0, \frac{9}{2}\right] $$

Step-by-Step Solution

Answer

Critical points of \(F\) are where \(f = 0\); \(F\) increases where \(f > 0\); \(F\) shows curvature change at zeros of \(f'\).

Step 1: Plot Functions

Use a computer algebra system (CAS) to plot the two functions, \(f(x) = 2x^4 - 17x^3 + 46x^2 - 43x + 12\) and \(F(x) = \int_{0}^{x} (2t^4 - 17t^3 + 46t^2 - 43t + 12) \, dt\), over the interval \([0, \frac{9}{2}]\). This visualization helps understand the relationship between \(f\) and \(F\).

Step 2: Solve F'(x)=0

Calculate \(F'(x)\), which by the Fundamental Theorem of Calculus, is equal to \(f(x)\). Then solve \(f(x)=0\) to find critical points where \(F(x)\) has potential extrema. Where \(F'(x)=0\), the graph of \(F\) has horizontal tangents, indicating local maxima or minima, depending on the second derivative test.

Step 3: Analyze Graphs at Critical Points

Note that at points where \(F'(x)=0\) (which is where \(f(x)=0\)), the graph of \(f\) crosses the x-axis, which corresponds to local extrema in \(F\). This is consistent with the Fundamental Theorem, which states that \(F'(x) = f(x)\).

Step 4: Determine Intervals of Increase and Decrease

Check where \(f(x) > 0\) to see where \(F(x)\) is increasing, and where \(f(x) < 0\) to see where \(F(x)\) is decreasing over \([0, \frac{9}{2}]\). This will provide intervals where the function \(F\) changes its monotonicity.

Step 5: Calculate and Plot f'(x)

Calculate the derivative \(f'(x)\), giving: \(f'(x) = 8x^3 - 51x^2 + 92x - 43\). Plot \(f'(x)\) alongside \(F(x)\). The zeros of \(f'(x)\) can indicate points of inflection in \(F(x)\), where the curvature changes.

Step 6: Connect Derivatives to Graphs

Notice that at zeros of \(f'(x)\), the graph of \(F(x)\) tends to change its curvature. This supports the observation that where the rate of change of \(f(x)\) shifts (increasing to decreasing or vice versa), the graph of \(F(x)\) undergoes a concavity change. This aligns with Part 1 of the Fundamental Theorem as \(F\) reflects cumulative behavior of \(f\).

Key Concepts

Integral Calculus

Integral calculus is one of the main branches of calculus that deals with the concept of integrals and the accumulation of quantities. In this exercise, we are looking at a specific type of integral called the definite integral, which calculates the net area under a curve defined by function \( f(x) \) over a specific interval \([a, b]\).

The integral is represented by the notation \( F(x) = \int_{a}^{x} f(t) \, dt \). This creates a new function \( F(x) \), the antiderivative or the accumulated area function. While \( f(x) \) represents the rate of change at any given point \( x \), \( F(x) \) represents the total accumulation up to that point.

The Fundamental Theorem of Calculus (FTC) establishes a connection between derivatives and integrals. It states that if \( F(x) \) is the antiderivative of \( f(x) \), then the derivative of \( F(x) \) is \( f(x) \). This links the "rate of change" to the "total change" and allows us to calculate areas and solve a multitude of problems in different contexts.

Derivatives

Derivatives measure how a function changes as its input changes. In simplest terms, the derivative \( F'(x) \) of a function gives the slope of the tangent line to the function's graph at the point \( x \). Finding derivatives is a crucial part of understanding the behavior of functions and is essential for solving many problems in calculus.

In the exercise, the derivative \( F'(x) \) is calculated from the integral of \( f(x) \). By the Fundamental Theorem of Calculus, \( F'(x) = f(x) \), which means \( F(x) \) differentiates back to give \( f(x) \). This is because the derivative tells us how \( F(x) \), the accumulated area, changes at any point \( x \).

When solving \( F'(x) = 0 \), we're finding points where the slope of the tangent (and thus \( f(x) \)) is zero. These points are candidates for local maxima, minima, or saddle points in \( F(x) \), where the graph flattens out (tangential line has a slope of zero). These concepts help in understanding the nature and behavior of the integral function \( F(x) \).

Function Analysis

Analyzing functions helps us understand their behavior across different intervals. We look at aspects like increasing and decreasing behavior, identified by finding where the derivative \( f(x) \) is positive or negative.

In our exercise, it's established that where \( f(x) > 0 \), the function \( F(x) \) is increasing, and where \( f(x) < 0 \), \( F(x) \) is decreasing. This is because the sign of \( f(x) \) (derivative of \( F \)) indicates the direction of change of \( F(x) \).

If \( f(x) \) crosses the x-axis, it highlights critical changes in the behavior of \( F(x) \). Graphically, this is seen as crossing the x-axis, which is associated with turning or flat points (local maximum or minimum) when analyzing the overall trend of \( F(x) \). Additionally, the second derivative \( f'(x) \) assists in identifying inflection points within \( F(x) \), where the concavity of \( F(x) \) changes.

Critical Points

Critical points of a function are points where its derivative is zero or undefined. These are essential for identifying potential maxima, minima, or points of inflection.

In the context of this exercise, solving \( F'(x) = f(x) = 0 \) gives us the x-values where the graph of \( F(x) \) might have a local extremum. These points are found by setting the derivative equal to zero and solving for \( x \).

After identifying these critical points, further analysis using the second derivative tells whether each point is a local maximum, minimum, or an inflection point. If \( f'(x) = 0 \), the curvature of \( F(x) \) may change, indicating points of inflection. Examining these points provides insights into the function's graph and the overall trend of the cumulative changes represented by \( F(x) \). This detailed perspective on critical points is pivotal in calculus, enabling deeper understanding of function graphs and behaviors.