Problem 24 - Applied Calculus

Question

Find an antiderivative. $$ h(t)=\cos t $$

Step-by-Step Solution

Answer

An antiderivative of \( h(t) = \cos t \) is \( H(t) = \sin t + C \).

Step 1: Understand the Problem

We need to find an antiderivative of the function \( h(t) = \cos t \). This means we are looking for a function \( H(t) \) such that \( H'(t) = h(t) \).

Step 2: Recall the Antiderivative Rule for Cosine

The antiderivative of \( \cos t \) is well-known. The derivative of \( \sin t \) is \( \cos t \). Therefore, an antiderivative of \( \cos t \) is \( \sin t \).

Step 3: Add the Constant of Integration

An antiderivative is not unique; it includes a constant of integration \( C \). Thus, the general antiderivative of \( h(t) = \cos t \) is \( H(t) = \sin t + C \), where \( C \) is any real number.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is often encountered in various branches of mathematics, including calculus. When looking at the cosine function, it is designated as \( \cos(t) \), where \( t \) is either an angle in the context of trigonometry or a real number in calculus.
Some key characteristics of the cosine function include:

• It is periodic with a period of \( 2\pi \).
• The cosine of an angle or number can be understood as the horizontal coordinate of a point on the unit circle.
• The cosine function is even, meaning \( \cos(-t) = \cos(t) \).

Understanding these properties is essential when finding derivatives and antiderivatives, as seen in calculus exercises. For the cosine function, knowing that its derivative is \( -\sin(t) \) helps establish that its antiderivative is \( \sin(t) \). This interplay between trigonometric functions and their derivatives/antiderivatives is a core component of integral calculus.

Constant of Integration

When finding antiderivatives, one must not forget the constant of integration, often denoted by \( C \). This constant arises because the derivative of a constant is zero, meaning there are infinitely many functions whose derivative could be the original function plus any constant number.
Consider the integral of a function. If you take the derivative of \( \sin(t) \) to get \( \cos(t) \), there also exists a family of functions \( \sin(t) + C \), where \( C \) is any real number, that also satisfies the derivative \( \,'' H'(t) = \cos(t) \,'' \).
Key points about the constant of integration:

• Adding \( C \) ensures the result covers all possible antiderivatives.
• It reflects the idea that many functions can differ by only a constant yet have identical derivatives.
• In physical applications, boundary conditions or initial values can be used to solve for the specific value of \( C \).

Remembering to include this constant when finding antiderivatives is crucial for creating a complete solution, particularly in integral calculus.

Integral Calculus

Integral calculus is a major branch of calculus, alongside differential calculus. It focuses on finding antiderivatives, also known as indefinite integrals, as well as determining the area under curves, known as definite integrals. In integral calculus, functions are reconstructed from their rate of change, which is a vital concept in many applications.
The process of finding an antiderivative, like \( \int \cos(t) \, dt \), involves recognizing known derivative and antiderivative pairs, and understanding how different functions integrate over their domains. Some related aspects include:

• Indefinite integrals represent a family of functions differing by the constant of integration \( C \).
• Definite integrals evaluate the area under a curve over a specific interval, giving a single numeric result.
• Integral calculus is essential in physics, engineering, and other sciences for solving real-world problems involving accumulation and area.

Mastering these concepts requires practice and a solid understanding of function behavior. The mathematics of antiderivatives opens the door to solving complex problems involving rates of change and accumulation both analytically and numerically.