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Chapter 1: Problem 14

Find the first and second derivatives. \(y=100\)

Short Answer

Expert verified
The first derivative is 0, and the second derivative is also 0.

Step by step solution

01

Identify the function

The given function is a constant function: \[ y = 100 \].
02

Find the first derivative

For a constant function, the first derivative is always zero. Therefore, \[ \frac{dy}{dx} = 0 \].
03

Find the second derivative

For the first derivative which is also a constant (0), the second derivative is also zero. Therefore, \[ \frac{d^2y}{dx^2} = 0 \].

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Key Concepts

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

constant function
A constant function is a type of function where the value of the function remains the same, no matter what the input is. In simpler terms, it is a flat line on a graph. For example, if we have \(y = 100\), it means that the output is always 100, regardless of the value of \(x\).

Because the output never changes, the function is 'constant' over its entire domain. In mathematical notation, you can write a constant function as \(y = c\), where \(c\) is a constant value.

Key characteristics of a constant function include:
  • The graph is a horizontal line.
  • It intersects the y-axis at \(c\).
  • The slope of the function is zero.
first derivative
The first derivative of a function gives us the rate at which the function's value changes with respect to changes in its input.

In other words, it tells us the slope of the function at any given point. For a constant function like \(y = 100\), the slope is zero everywhere.

This is because there is no change in the function鈥檚 value as \(x\) changes.
In mathematical terms, if \(y = c\), where \(c\) is a constant, the first derivative is given by:

\(\frac{dy}{dx} = 0\)


The process of differentiation reduces the function's constant value to zero, showing that a constant function has no rate of change.
second derivative
The second derivative provides information about the curvature of the function. It indicates how the rate of change of the function is changing.

For a given function \(y\), the second derivative is the derivative of the first derivative. When the first derivative itself is a constant value, the second derivative will be zero.

For our constant function \(y=100\), we have:

\(\frac{dy}{dx} = 0\)

Since the first derivative is zero, taking the derivative of zero gives us:

\(\frac{d^2y}{dx^2} = 0\)

This result tells us that the constant function has no curvature, confirming it is a perfectly flat, horizontal line.

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Most popular questions from this chapter

Use limits to compute \(f^{\prime}(x)\). [Hint: In Exercises \(45-48\), use the rationalization trick of Example \(8 .]\) \(f(x)=\sqrt{x+2}\)

Find the indicated derivative. \(\frac{d y}{d x}\) if \(y=\frac{x-1}{3}\)

Let \(f(t)\) be the temperature of a cup of coffee \(t\) minutes after it has been poured. Interpret \(f(4)=120\) and \(f^{\prime}(4)=-5 .\) Estimate the temperature of the coffee after 4 minutes and 6 seconds, that is, after \(4.1\) minutes.

Compute the following limits. \(\lim _{x \rightarrow-\infty} \frac{1}{x^{2}}\)

Use limits to compute \(f^{\prime}(x)\). [Hint: In Exercises \(45-48\), use the rationalization trick of Example \(8 .]\) \(f(x)=\frac{x}{x+1}\)
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