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Chapter 4: Problem 32

In Exercises \(29-34,\) find all possible functions \(f\) with the given derivative. $$f^{\prime}(x)=\sin x$$

Short Answer

Expert verified
The function \(f(x)\) with the derivative \(f^{\prime}(x)=\sin x\) is \(f(x)=-\cos x + C\), where \(C\) is the constant of integration.

Step by step solution

01

Identify the Given Function

The derivative of the function \(f(x)\) is given as \(f^{\prime}(x)=\sin x\). The task is to find the original function \(f(x)\).
02

Integrate the Function

The original function can be found by integrating its derivative. We know that the integral of \(\sin x\) with respect to \(x\) is \(-\cos x\). So, \(f(x)\) will be the integral of \(f^{\prime}(x)\), which is \(-\cos x\).
03

Add the Constant of Integration

When we find the integral of a function, we always add the 'constant of integration', usually denoted by \(C\), because the derivative of a constant is zero. So to the antiderivative \(-\cos x\), we add the constant of integration \(C\). Therefore, the general solution for \(f(x)\) is \(f(x)=-\cos x + C\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration
Integration is essentially the reverse process of differentiation. It allows us to find an original function given its derivative. In this exercise, we used integration to retrieve a function whose derivative is known. Specifically, integrating \(\sin x\) gives us \(-\cos x\).
  • The integral of a function provides a family of functions.
  • This includes the original function and other functions differing by a constant.
  • Integration in mathematics helps solve many real-world problems, such as finding areas under curves.
Understanding integration makes it possible to transition between rates of change and accumulation of quantities. This core concept is fundamental in courses like calculus.
Constant of Integration
A constant of integration is added to an indefinite integral to represent all possible solutions. This concept arises because many different functions can have the same derivative. For example, \(f(x) = -\cos x + C\) is a solution to \(f'(x) = \sin x\) because:
  • The derivative of a constant is zero.
  • Therefore, any constant included in the function will disappear in differentiation.
  • To account for this, we add \(C\) during integration.
The constant of integration is important because it ensures that we represent all possible antiderivatives. Remember that each value of \(C\) corresponds to a different parallel shift of the curve.
Original Function
In this problem, the original function \(f(x)\) is equal to \(-\cos x + C\). Finding the original function means identifying from where the derivative \(\sin x\) originated.
  • The original function is \(-\cos x\), which gives a derivative of \(\sin x\).
  • This means that finding an antiderivative effectively reverses the process of differentiation.
  • Knowing the original function is crucial for solving practical problems involving motion, fluid dynamics, or any area where change is occurring.
Understanding how to find the original function helps connect the derivative back to its originating expression and validates how integration serves as an inverse operation to differentiation.
Trigonometric Functions
Trigonometric functions like \(\sin x\) and \(\cos x\) are fundamental in mathematics. They play a crucial role because they model periodic phenomena such as light, sound waves, and tides.
  • The integral of \(\sin x\) is \(-\cos x\), a typical relationship between sine and cosine.
  • Knowing these relationships helps in solving differential equations or predicting systems' behavior over time.
  • Recognizing these functions both in their derivative and integral forms strengthens problem-solving skills involving oscillating systems.
Grasping trigonometric functions and their integrals solidifies foundational mathematical understanding and opens up numerous applications in science and engineering fields.

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