Problem 27 - Essential Calculus Early Transcendentals

Question

Find \(f\) $$f^{\prime}(t)=2 \cos t+\sec ^{2} t, \quad-\pi / 2

Step-by-Step Solution

Answer

\( f(t) = 2\sin t + \tan t + 4 - 2\sqrt{3} \)

Step 1: Identify the Problem

We need to find the function \( f(t) \) given its derivative \( f'(t) = 2 \cos t + \sec^2 t \) and the value \( f(\pi / 3) = 4 \). This involves integration and using initial conditions to find the constant of integration.

Step 2: Integrate the Derivative

Integrate \( f'(t) = 2 \cos t + \sec^2 t \) with respect to \( t \).\[ f(t) = \int (2 \cos t + \sec^2 t) \, dt = \int 2 \cos t \, dt + \int \sec^2 t \, dt \]Calculate each integral separately:\[ \int 2 \cos t \, dt = 2 \sin t + C_1 \]\[ \int \sec^2 t \, dt = \tan t + C_2 \]

Step 3: Combine Integrals

Combine the results of the integrals to find the general form of \( f(t) \):\[ f(t) = 2 \sin t + \tan t + C \]Here, \( C \) is the constant of integration, which includes contributions from both \( C_1 \) and \( C_2 \).

Step 4: Apply Initial Condition

Use the initial condition \( f(\pi / 3) = 4 \) to find \( C \):Substitute \( t = \pi/3 \) into \( f(t) = 2 \sin t + \tan t + C \):\[ f(\pi/3) = 2 \sin(\pi/3) + \tan(\pi/3) + C = 4 \]Since \( \sin(\pi/3) = \sqrt{3}/2 \) and \( \tan(\pi/3) = \sqrt{3} \), we substitute back:\[ 2 \left(\frac{\sqrt{3}}{2}\right) + \sqrt{3} + C = 4 \]\[ \sqrt{3} + \sqrt{3} + C = 4 \]\[ 2\sqrt{3} + C = 4 \]\[ C = 4 - 2\sqrt{3} \]

Step 5: Write the Final Function

Substitute \( C \) back into the expression for \( f(t) \):\[ f(t) = 2 \sin t + \tan t + (4 - 2\sqrt{3}) \]This gives us the function \( f(t) \) on the interval \(-\pi/2 < t < \pi/2\).

Key Concepts

Definite Integrals

Definite integrals are used to find the area under a curve between two specific values. They differ from indefinite integrals, which do not have boundary limits. In the solution to the problem, we focused on finding a function given its derivative rather than calculating a definite integral. However, understanding definite integrals is essential for comprehending calculus's broad utility, such as calculating the total accumulation of quantities.

Key points about definite integrals include:

• They are denoted as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration.
• The result is a number representing the net area between the curve and the x-axis from \(a\) to \(b\).
• In applied scenarios, definite integrals can compute total distance, area, volume, and more.

Derivative and Integration

Differentiation and integration are fundamental calculus operations that are essentially inverse processes. Differentiation finds the rate of change of a function, while integration finds the accumulated quantity. When given a derivative like in our exercise, we use integration to return to the original function (or a version thereof, including a constant).

Important aspects to remember are:

• The fundamental theorem of calculus links the two, stating that the integral of a function's derivative over an interval gives the function's net change over that interval.
• Integrating a derivative requires looking for an antiderivative and usually results in an equation with a constant, \(C\), known as the constant of integration.

In solving for \(f(t)\), we focused on integrating each term in the derivative expression separately and then applying initial conditions to solve for \(C\). This approach ensures we accurately determine the specific function that corresponds to the given original condition \(f(\pi / 3) = 4\).

Trigonometric Functions

Trigonometric functions like \( \cos(t) \), \( \sin(t) \), and \( \tan(t) \) are crucial in calculus. These functions appear frequently in problems involving periodic phenomena. For this exercise, they form part of the derivative we integrated.

Understanding basic trigonometric values is essential:

• \( \sin(\pi/3) = \sqrt{3}/2 \)
• \( \cos(\pi/3) = 1/2 \)
• \( \tan(\pi/3) = \sqrt{3} \)

When integrating such functions, you should know common integrals:

• \( \int \cos(t)\, dt = \sin(t) + C \)
• \( \int \sec^2(t)\, dt = \tan(t) + C \)

Recognizing these key values and integral forms helps greatly in solving problems involving trigonometric functions.

Integration Techniques

There are various techniques to approach integration problems, each useful in different scenarios depending on the form of the function. In this exercise, we employed basic integration rules due to the simplicity of the terms involved.

Some common integration techniques include:

• Substitution: Used when a function is a result of another function's composition, making the integral easier to evaluate by substituting variables.
• Integration by Parts: Useful when an integral involves a product of two functions. This is based on the integral product rule.
• Partial Fraction Decomposition: Applicable for rational functions, it involves breaking down complex fractions into simpler ones that can be integrated individually.
• Trigonometric Identities: Sometimes, knowing trigonometric identities can simplify the integrals involving trigonometric functions.

In our problem, each part of the derivative was easily integrated using the fundamental rules due to the presence of straightforward trigonometric forms \( \cos(t) \) and \( \sec^2(t) \). Knowing when to use different techniques is key to efficiently solving integration problems.