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Chapter 3: Problem 35

Find the derivative. Assume \(a, b, c, k\) are constants. $$V=\frac{4}{3} \pi r^{2} b$$

Short Answer

Expert verified
\( \frac{dV}{dr} = \frac{8}{3} \pi b r \)

Step by step solution

01

Identify the function to differentiate

We have the function \( V = \frac{4}{3} \pi r^{2} b \) and we need to find its derivative with respect to \( r \). Variables \( a, b, c, k \) are constants. Since \( b \) does not depend on \( r \), treat it as a constant.
02

Differentiate with respect to r

Use the power rule to differentiate \( r^{2} \) with respect to \( r \). The constant \( \frac{4}{3} \pi b \) can be factored out of the differentiation:\[ \frac{dV}{dr} = \frac{4}{3} \pi b \cdot \frac{d}{dr}(r^2) \] Calculate the derivative of \( r^2 \), which is \( 2r \).
03

Calculate the derivative expression

Substitute the derivative \( \frac{d}{dr}(r^2) = 2r \) back into the equation:\[ \frac{dV}{dr} = \frac{4}{3} \pi b \cdot 2r = \frac{8}{3} \pi b r \]

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Key Concepts

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

Power Rule in Differentiation
The Power Rule is one of the fundamental tools in the calculus toolbox. It helps us find the derivatives of power functions easily. If you have a function of the form \( f(x) = x^n \), the Power Rule states that its derivative is \( f'(x) = nx^{n-1} \). For example, if you have \( r^2 \), applying the Power Rule gives you \( 2r \). Here, you bring down the exponent 2, and then subtract one from the exponent to find its derivative.
  • This rule applies to all real numbers \( n \).
  • It simplifies the process by eliminating the need for complex limit calculations.
The Power Rule makes working with polynomials straightforward. Once you get the hang of it, you'll find differentiation much easier.
Understanding Derivatives
Derivatives represent the rate of change or slope of a function at any given point. Think of it like finding the speed of a moving car at a particular moment. In our example, finding the derivative of \( V = \frac{4}{3} \pi r^2 b \) with respect to \( r \) shows how the volume changes as \( r \) changes. Here's how:
  • Look at the function and decide what variable you're differentiating with respect to. In this case, it's \( r \).
  • Apply rules of differentiation to other components. Use the Power Rule for terms like \( r^2 \).
Derivatives provide indispensable insights into the behavior of functions in various fields, like physics and engineering. They're foundational for predicting future behavior and understanding trends.
Role of Constants in Differentiation
In differentiation, constants play a special role. They can influence the shape and position of a function without affecting its differentiation process directly. Constant terms remain unchanged during differentiation:
  • In our example, \( b \) and coefficients like \( \frac{4}{3}\pi \) are constants and can be factored out.
  • They simplify the differentiation process by allowing you to focus on the variable of interest, such as \( r \).
Treating constants properly is crucial for correctly differentiating functions. They often appear in formulas and equations across calculus, making them key players despite not changing during differentiation itself.

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Most popular questions from this chapter

Differentiate the functions in Problems. Assume that \(A\) \(B,\) and \(C\) are constants. $$y=e^{0.7 t}$$

If the demand curve is a line, we can write \(p=b+m q\) where \(p\) is the price of the product, \(q\) is the quantity sold at that price, and \(b\) and \(m\) are constants. (a) Write the revenue as a function of quantity sold. (b) Find the marginal revenue function.

If a someone is lost in the wilderness, the search and rescue team identifies the boundaries of the search area and then uses probabilities to help optimize the chances of finding the person, assuming the subject is immobile. The probability, \(O,\) of the person being outside the search area after the search has begun and the person has not been found, is given by $$O(E)=\frac{I}{1-(1-I) E}$$ where \(I\) is the probability of the person being outside the search area at the start of the search and \(E\) is the search effort, a measure of how well the search area has been covered by the resources in the field. (a) If there was a \(20 \%\) chance that the subject was not in the search area at the start of the search, and the search effort was \(80 \%,\) what is the current probability of the person being outside the search area? (Probabilities are between 0 and \(1,\) so \(20 \%=0.2\) and \(80 \%=0.8 .)\) (b) In practical terms, what does \(I=1\) mean? Is this realistic? (c) Evaluate \(O^{\prime}(E) .\) Is it positive or negative? What does that tell you about \(O\) as \(E\) increases?

The quantity demanded of a certain product, \(q\), is given in terms of \(p,\) the price, by $$q=1000 e^{-0.02 p}$$ (a) Write revenue, \(R,\) as a function of price. (b) Find the rate of change of revenue with respect to price. (c) Find the revenue and rate of change of revenue with respect to price when the price is \(\$ 10 .\) Interpret your answers in economic terms.

A ball is dropped from the top of the Empire State Building. The height, \(y,\) of the ball above the ground (in feet) is given as a function of time, \(t,\) (in seconds) by $$y=1250-16 t^{2}$$ (a) Find the velocity of the ball at time \(t .\) What is the sign of the velocity? Why is this to be expected? (b) When does the ball hit the ground, and how fast is it going at that time? Give your answer in feet per second and in miles per hour \((1 \mathrm{ft} / \mathrm{sec}=15 / 22 \mathrm{mph})\)
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