Problem 37 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The antiderivative of \\(h(t) = ext{cos }t\\) is \\(H(t) = ext{sin }t + C\\).

Step 1: Identify the Antiderivative Rule

The antiderivative of a function, also known as the indefinite integral, is a function whose derivative is the original function. The antiderivative of \( ext{cos }t\) is \( ext{sin }t\) because \(rac{d}{dt} [ ext{sin }t ] = ext{cos }t\).

Step 2: Apply the Antiderivative Rule

Apply the rule identified in the previous step. The general antiderivative of \(h(t) = ext{cos }t \) is \(H(t) = ext{sin }t + C\), where \(C\) is the constant of integration, accounting for any vertical shift in the function.

Step 3: Include the Constant of Integration

In indefinite integrals, always add the constant of integration \(C\). Thus, the antiderivative of \(h(t) = ext{cos }t \) is \(H(t) = ext{sin }t + C\).

Key Concepts

Integration

Integration is a fundamental concept in calculus, acting as the reverse operation of differentiation. To integrate a function, we aim to find an antiderivative or a function whose derivative matches the given function. This process uncovers the 'accumulated' quantity described by the function itself, such as area under a curve. When integrating the function \( h(t) = \cos t \), we look for a new function that differentiates back to \( h(t) \). In this scenario, \( \sin t \) becomes the antiderivative because its derivative is \( \cos t \).

• Integration finds a function which differentiates into the original.
• Antiderivatives help reveal accumulated quantities mathematically.
• Involves reversing differentiation, linking directly to function derivations.

This integral calculation forms what is called an indefinite integral, and it’s different from a definite integral which calculates a numerical value, generally representing area.

Function Derivatives

Function derivatives indicate how a function changes at any given point and is a core notion in calculus. Derivatives offer the rate of change and can outline the slope of a curve at a particular point. The derivative of \( \sin t \) is \( \cos t \), which enables finding the antiderivative of \( h(t) = \cos t \). This connection is pivotal because it anchors the computation of an antiderivative.

• Derivatives showcase a function’s alteration rate.
• They reveal the slope at a specific point on a curve.
• Link with antiderivatives helps in reversing derivatives to understand original functions.

Understanding derivatives is essential in properly comprehending how functions evolve and their accumulated result, forming a deep link with integration.

Indefinite Integrals

Indefinite integrals are centered around finding antiderivatives. Unlike definite integrals with set bounds, indefinite integrals don’t evaluate to a number. Instead, they represent families of functions due to the additive constant \( C \). The result always involves \( + C \) because integration accounts for any constant differential shift undefined in indefinite integrals, offering flexible equations that suit varied starting points in a function’s behavior.

• Represent families of functions due to the constant \( C \).
• Highlight obtaining a general form instead of specific numerical values.
• Illustrate the full scope of related functions shifting along the vertical axis.

Integrating \( \cos t \) results in \( \sin t + C \), signifying any vertical shift possible within the solution family, thereby explaining numerous real-world phenomena where starting levels aren't standardized.