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Chapter 6: Problem 39

Find all functions \(f(t)\) that satisfy the given condition. $$f^{\prime}(t)=0$$

Short Answer

Expert verified
The function is a constant, \(f(t) = C\).

Step by step solution

01

Understand the Given Condition

The problem states that the derivative of the function, denoted as \(f'(t)\), is equal to 0. This means that the rate of change of the function with respect to \(t\) is zero.
02

Interpret the Condition Mathematically

When the derivative \(f'(t)\) is zero, the function \(f(t)\) must be constant. This is because a derivative represents the slope of the tangent line at any given point on the function, and a slope of zero indicates a horizontal line, which corresponds to a constant function.
03

Conclude the Form of the Function

Since \(f'(t) = 0\) indicates that the function has no change, the function \(f(t)\) must be a constant value. Therefore, \(f(t) = C\), where \(C\) is any constant.

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Key Concepts

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

derivative
In calculus, a derivative measures how a function changes as its input changes. Think of it as the function's sensitivity to change. When we talk about the derivative of a function, we're looking at the slope of its graph at any given point. This slope tells us the rate at which the function's value is rising or falling. If the derivative is zero, like in our original exercise, it means the function isn't changing at all.
rate of change
The rate of change is all about how one quantity changes in relation to another. In the context of functions, it means how the output value of a function changes as the input value changes. The derivative we talked about earlier is a precise measure of this rate of change. For example, if the rate of change (or derivative) is zero, it means the function's value remains constant regardless of changes in the input.
horizontal slope
A horizontal slope means a flat line. When we talk about the slope of a function's graph, a horizontal slope tells us that the function's value doesn't change, no matter what the input is. In mathematical terms, this horizontal slope is represented by a derivative of zero. This concept ties directly into our initial problem, where we found that any function with a zero derivative has a constant value.
constant value
A constant value means that the output of the function is the same, no matter what the input is. When a function is constant, it can be written as \(f(t) = C\), where \(C\) is a fixed number. In our original exercise, the fact that the derivative \(f'(t) = 0\) led us to conclude that \(f(t)\) must be a constant value. This constant value reflects that the function doesn't change, visually represented by a perfectly horizontal line on a graph.

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Most popular questions from this chapter

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