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Chapter 1: Problem 17

Find the first and second derivatives. \(f(r)=\pi r^{2}\)

Short Answer

Expert verified
The first derivative is \ f'(r) = 2 \pi r \ and the second derivative is \ f''(r) = 2 \pi.\

Step by step solution

01

Find the first derivative

To find the first derivative of the function, use the power rule for differentiation. The power rule states that if you have a term of the form \(ar^n\), the derivative will be \((n \cdot a)r^{n-1}\). For the given function \(f(r) = \pi r^2\), apply the power rule. The first derivative is computed as follows: \[f'(r) = \frac{d}{dr}(\pi r^2) = 2 \pi r.\]
02

Find the second derivative

To find the second derivative, take the derivative of the first derivative. Again, use the power rule for differentiation. The first derivative is \(f'(r) = 2 \pi r\). Apply the power rule: \[f''(r) = \frac{d}{dr}(2 \pi r) = 2 \pi.\]

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Key Concepts

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

first derivative
The first derivative of a function gives us the rate at which the function's value is changing. For a function such as \( f(r) = \pi r^2 \), we use differentiation to find how it changes with respect to \( r \). By applying the power rule, we take the exponent, multiply it by the coefficient, and then subtract one from the exponent. In this case, the function \( \pi r^2 \) becomes:
\[ f'(r) = 2 \pi r \] This tells us the rate of change of the function \( f(r) \) at any point \( r \). If \( r \) increases, the value of \( f'(r) \) will also change accordingly.
second derivative
The second derivative provides information about the curvature or concavity of a function. It tells us how the rate of change of the rate of change is behaving. To find the second derivative, we differentiate the first derivative. Starting with \( f'(r) = 2 \pi r \) and applying the power rule once more, we get:
\[ f''(r) = 2 \pi \] This is a constant, showing that the rate at which \( f'(r) \) changes is consistent, indicating that the original function \( f(r) \) has a constant rate of curvature.
power rule
The power rule is a basic rule in calculus used for differentiating functions of the form \( ar^n \). According to the power rule, the derivative of \( ar^n \) is given by \( (n \cdot a) r^{n-1} \).
This means:
- Multiply the exponent (\( n \)) by the constant (\( a \)).
- Subtract one from the exponent.
For example, to find the first derivative of \( \pi r^2 \), using the power rule, we do the following steps:
- Identify the exponent \( n = 2 \) and the coefficient \( a = \pi \)
- The derivative becomes \( 2 \cdot \pi r^{2-1} = 2 \pi r \).
Similarly, apply the power rule to find the second derivative.
differentiation
Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function, which tells us how the function changes as its input changes. Differentiation can be thought of as finding the slope of a function at any given point.
For polynomial functions, the process of differentiation primarily involves applying rules like the power rule. This process helps us understand the behavior and properties of functions, such as their increasing/decreasing nature and points of inflection. It's important to grasp this concept, as it forms the basis for more advanced calculus topics.

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Most popular questions from this chapter

In Exercises \(33-36\), refer to a line of slope \(m\). If you begin at a point on the line and move \(h\) units in the \(x\) -direction, how many units must you move in the \(y\) -direction to return to the line? \(m=2, h=\frac{1}{2}\)

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