Problem 45 - Calculus Early Transcendentals

Question

Use the definition of a derivative to find \(f^{\prime}(x)\) and \(f^{\prime \prime}(x)\) . Then graph \(f, f^{\prime},\) and \(f^{\prime \prime}\) on a common screen and check to see if your answers are reasonable. \(f(x)=1+4 x-x^{2}\)

Step-by-Step Solution

Answer

The derivatives are \(f'(x) = 4 - 2x\) and \(f''(x) = -2\).

Step 1: Find the First Derivative using the Definition

The derivative of a function \(f(x)\) at a point \(x\) is defined as \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\). Let's find \(f'(x)\) for \(f(x) = 1 + 4x - x^2\). First, calculate \(f(x+h) = 1 + 4(x+h) - (x+h)^2 = 1 + 4x + 4h - x^2 -2xh - h^2\). The expression for the difference \(f(x+h) - f(x)\) is \(4h - 2xh - h^2\). Therefore, \(f'(x) = \lim_{h \to 0} \frac{4h - 2xh - h^2}{h} = \lim_{h \to 0} (4 - 2x - h) = 4 - 2x\).

Step 2: Find the Second Derivative

We have \(f'(x) = 4 - 2x\). The second derivative \(f''(x)\) can be found by differentiating \(f'(x)\) with respect to \(x\). \(f''(x) = \frac{d}{dx}(4 - 2x) = -2\).

Step 3: Graphing and Analysis Check

To verify your solutions, graph \(f(x) = 1 + 4x - x^2\), \(f'(x) = 4 - 2x\), and \(f''(x) = -2\) on a common screen. The graph of \(f(x)\) should be a downward opening parabola. The graph of \(f'(x)\) should be a straight line with a negative slope crossing the x-axis at \(x = 2\). \(f''(x)\) is a constant line at \(-2\), indicating a constant rate of change in the slope of \(f(x)\). If these characteristics match, the derivatives are correct.

Key Concepts

First Derivative

Understanding the first derivative is crucial for grasping how a function behaves. In this case, the function given is a polynomial function, specifically a quadratic: \( f(x) = 1 + 4x - x^2 \). To find the first derivative \( f'(x) \), we use the definition of a derivative. This is expressed as:\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]In this exercise, after substituting \(f(x+h)\) and simplifying, the first derivative is found to be \( f'(x) = 4 - 2x \). This derivative represents the slope of the tangent line to the curve at any point \(x\). It tells us how the function \( f(x) \) is changing at any particular point. For polynomial functions, knowing the first derivative helps you understand the behavior and motion of the function, such as where it increases or decreases.

Second Derivative

The second derivative \( f''(x) \) provides more insight into the nature of the function. For polynomial functions like our quadratic \( f(x) = 1 + 4x - x^2 \), the second derivative is found by differentiating the first derivative again. Given our first derivative \( f'(x) = 4 - 2x \), the second derivative turns out to be a constant \( f''(x) = -2 \). This second derivative tells us about the concavity of the function: whether it curves upwards or downwards. When the second derivative is negative like \(-2\), it indicates that the function \( f(x) \) is concave down. Understanding concavity helps in identifying points of inflection and understanding the broader shape of the graph.

Graphing Derivatives

Graphing derivatives provides a visual insight into the behavior and characteristics of a function. For the function \( f(x) = 1 + 4x - x^2 \), we graph not only the function itself but also its first and second derivatives. The graph of the function \( f(x) \) is a parabola opening downwards, due to the negative \( x^2 \) term.

• The first derivative \( f'(x) = 4 - 2x \) is a straight line. It crosses the x-axis at \( x = 2 \), indicating a change in the direction or slope of the original function. Before this point, the function increases and after this, it decreases.
• The second derivative \( f''(x) = -2 \) is constant and below the x-axis. This horizontal line confirms the constant concave down nature of the original function.

Using graphs helps students to visually confirm their calculations, making it clear how the original function behaves and changes.

Polynomial Functions

Polynomial functions, such as \( f(x) = 1 + 4x - x^2 \), play a significant role in calculus because of their simple structure and predictable behavior. They are easy to differentiate and integrate, making them ideal for teaching the foundational concepts of calculus.These functions are composed of powers of \( x \) with coefficients, and their degree is determined by the highest power of \( x \). In this case, it's a quadratic function, meaning it has a degree of two. This gives it the characteristic parabola shape when graphed.Understanding polynomial functions allows students to explore concepts like zeroes, turning points, and extremas. These functions serve as an entry-point into more complex calculus concepts and help in grasping how functions can be manipulated through differentiation and integration.