Problem 58 - Essential Calculus

Question

Find an equation for the function \(f\) that has the indicated derivative and whose graph passes through the given point. $$ f^{\prime}(x)=\pi \sec \pi x \tan \pi x, \quad\left(\frac{1}{3}, 1\right) $$

Step-by-Step Solution

Answer

The function \(f(x)\) that has the derivative \(f'(x) = \pi \sec(\pi x) \tan(\pi x)\) and passes through the point \((1/3, 1)\) is \(f(x)= \pi \sec(\pi x) + (1-2\pi)\).

Step 1: Find the Integral

First calculate the indefinite integral of \(f'(x)\) to find \(f(x)\). It's noted that the integral of \(\sec(x) \tan(x)\) with respect to \(x\) is \(\sec(x)\). Therefore: \[ \int f'(x) \, dx = \int \pi \sec(\pi x) \tan(\pi x) \, dx = \pi \int \sec(\pi x) \tan(\pi x) \, dx = \pi\sec(\pi x) + C \] where \(C\) is the constant of integration.

Step 2: Find the Constant

Plug the given point \((1/3, 1)\) into the function \(f(x)\) to solve for \(C\). This means that when \(x = 1/3\), \(f(x) = 1\), therefore: \[ 1 = \pi \sec\left(\pi \cdot \frac{1}{3}\right) + C\] Solve this equation for \(C\) to get the final function.

Step 3: Solve for the Constant

The secant function can be calculated and rearranged to find an expression for \(C\). Solving for \(C\) gives: \[ C = 1 - \pi \sec(\pi/3) = 1 - \pi \cdot 2 \] Therefore, \(C = 1 - 2\pi\).

Step 4: Write the Final Function

Substitute the value of \(C\) back into \(f(x)= \pi \sec(\pi x) + C\) to get the final function: \[ f(x)= \pi \sec(\pi x) + (1-2\pi) \]

Key Concepts

Indefinite Integral

An indefinite integral represents the antiderivative of a function, which is essentially a reverse operation of taking the derivative. In the context of our calculus problem, the indefinite integral is used to recover the original function, denoted by f(x), given its derivative f'(x). Intuitively, this process is akin to retracing your steps to get back to your starting point after walking a certain path.

When computing the indefinite integral, we look for a function that, when differentiated, would give us the provided function. This requires knowledge of basic integration rules and often involves recognizing patterns or using substitution methods. The integral of the product of the secant and tangent functions, as seen in the step-by-step solution, is a standard integral that yields the secant function, further modified by the constant multiplier π.

Secant Function

The secant function, denoted as sec(x), is one of the trigonometric functions and the reciprocal of the cosine function. More formally, sec(x)=1/cos(x). In trigonometry, it represents the ratio of the length of the hypotenuse to the adjacent side in a right-angled triangle.

Understanding the secant function is crucial for solving integrals involving trigonometric identities, like in our example where the secant is paired with the tangent function. The behavior of the secant function, especially its values at key angles, affects the computation of the integral and the constant of integration, as it did when evaluating sec(π/3) during the problem-solving process.

Constant of Integration

The constant of integration is denoted by C and it emerges naturally when dealing with indefinite integrals. Since the derivative of a constant is zero, when we take the antiderivative, the original constant term is unknown, thus we include C to account for all possible constants.

In the exercise solution, after integrating the secant and tangent product, the constant of integration was determined by using the given point on the graph. The value of C ensures that our function goes through that specific point, fulfilling the conditions of the problem. Without C, we would have an incomplete answer, as it constitutes the particular solution to the differential equation represented by the integral.

Calculus Problem Solving

Calculus problem-solving often entails integrating or differentiating functions to find quantities like areas, tangents, or antiderivatives—key abilities for any calculus student. The example problem demonstrates this by walking through the process of solving an integration problem that is connected to a specific point on the graph of the function.

In our showcased problem, critical thinking and analytical skills come into play, first in recognizing that the integral of sec(x)tan(x) is sec(x), and then in determining the constant of integration by applying the given point. The methodology emphasizes the importance of breaking down complex problems into manageable steps and knowing how certain functions behave to navigate through calculus problems effectively.