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Magazine Chapter 6: Problem 6
Find an antiderivative. $$h(z)=\frac{1}{z}$$
Short Answer
Expert verified
The antiderivative is \( \ln |z| + C \).
Step by step solution
01
Identify the Function
We're given the function \( h(z) = \frac{1}{z} \). We need to find an antiderivative of this function, which is a function whose derivative is \( h(z) \).
02
Recall the Antiderivative Rule for \( \frac{1}{z} \)
The antiderivative of \( \frac{1}{z} \) is \( \ln |z| + C \), where \( C \) is a constant. This result comes from the fact that the derivative of \( \ln |z| \) is \( \frac{1}{z} \).
03
Write Down the Antiderivative
Since we know the form of the antiderivative, we can directly write: \( abla(z) = \ln |z| + C \). Here, \( abla(z) \) is an antiderivative of \( h(z) \), and \( C \) is any constant.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Calculus
Calculus is a branch of mathematics that focuses on the study of change. It deals with a variety of concepts that involve rates of change and accumulations. Calculus is mainly divided into two parts:
- **Differential Calculus:** Concerned with the concept of the derivative and how functions change.
- **Integral Calculus:** Focused on finding the total accumulation of a quantity, which is where antiderivatives come into play.
Logarithmic Function
A logarithmic function is a mathematical function related to the concept of an exponent. It is the inverse of an exponential function. In simpler terms, while an exponential function raises a number to a power, a logarithmic function represents the power to which a base number is raised to produce a given number.
- **Form:** The general form is \( y = \log_b (x) \), where \( b \) is the base.
- **Natural Logarithm:** When the base \( b \) is \( e \) (Euler's number, approximately 2.718), it's referred to as the natural logarithm, noted as \( \ln x \).
- **Properties:** The derivative of \( \ln |z| \) is \( \frac{1}{z} \), which is crucial in finding antiderivatives of certain functions.
Differentiation
Differentiation is a process in calculus that determines the rate of change of a function. It involves calculating the derivative of a function and is widely used to solve problems involving motion, optimization, and rates of change.
- **Derivative:** This is a measure of how a function changes as its input changes, often symbolized as \( f'(x) \) or \( \frac{df}{dx} \).
- **Key Rules:** Rules like the power rule, product rule, and chain rule help in finding the derivative of various functions. The reverse of this process helps find antiderivatives.
- **Application:** In the exercise, differentiation helps verify the correctness of an antiderivative. If the derivative of \( \ln |z| + C \) is \( \frac{1}{z} \), the solution is confirmed.
Constant of Integration
The constant of integration is a concept that arises when we determine an antiderivative of a function. In indefinite integrals, this constant represents an unknown constant that could potentially shift the function vertically.
- **Role:** When finding an antiderivative for a function \( f(x) \), we add a constant, usually noted as \( C \), to signify that multiple functions can have the same derivative.
- **Importance:** The presence of \( C \) highlights the family of antiderivatives that are parallel due to the vertical shift, but are identical in terms of their derivative.
- **In Practice:** In our specific exercise, after finding that the antiderivative of \( \frac{1}{z} \) is \( \ln |z| \), we include \( C \) to express that any vertically shifted version of this function will have the same rate of change \( \frac{1}{z} \).
