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Chapter 4: Problem 14

For Activities 7 through \(18,\) write the first and second derivatives of the function. \(b(t)=7-3\left(0.02^{2}\right)\)

Short Answer

Expert verified
Both derivatives are 0.

Step by step solution

01

Identify the function

The given function is \(b(t)=7-3\left(0.02^{2}\right)\). We first note that this is a constant function because the term \(0.02^{2}\) is merely a number, and there is no variable \(t\) present in the function.
02

Calculate the first derivative

Since the given function is a constant function (as detailed in Step 1), its derivative with respect to \(t\) is zero. Therefore, the first derivative, \(b'(t)\), is \(0\).
03

Calculate the second derivative

The second derivative of a constant (or zero function) is also zero. Therefore, \(b''(t) = 0\).

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Key Concepts

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

Understanding Derivatives
In calculus, a derivative represents how a function changes as its input changes. It is the mathematical way to express the rate of change or the slope of a function's graph at any given point. The derivative is found by taking the limit as the change in the input approaches zero. Derivatives help us understand behavior such as:
  • The speed of a moving object
  • The change in population over time
  • The slope of a curve

To find a derivative, we typically look for patterns and use rules such as the power rule, the product rule, and the chain rule. These rules simplify the process, especially for polynomial functions and combinations thereof.
Exploring Constant Functions
A constant function is one of the simplest types of functions in mathematics. It is a function that always returns the same value, no matter the input. Its formula looks like this: f(x) = c, where c is a constant.
In the context of derivatives, the derivative of a constant function is always zero. This is because there is no change in the function's value as the input changes, indicating a slope of zero.
For example, with the function given as b(t) = 7 - 3(0.02^2), it simplifies to a constant because it doesn't contain any variable terms. Thus, both its first and second derivatives are zero.
Calculating First and Second Derivatives
Derivatives can be applied multiple times to analyze a function's changing behavior more deeply.
  • First Derivative (\(f'(x)\)): This represents the rate of change or the slope of the function. It tells us whether the function is increasing or decreasing at specific points.
  • Second Derivative (\(f''(x)\)): This measures the rate of change of the rate of change, or the concavity of the function. It helps determine if the slope is increasing or decreasing, giving insights into the function's curvature.

When dealing with constant functions like b(t) = 7 - 3(0.02^2), the first derivative is zero, signifying no rate of change. Consequently, the second derivative is also zero, indicating constant slope and no curvature.

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Most popular questions from this chapter

A leaking container of salt is sitting on a shelf in a kitchen cupboard. As salt leaks out of a hole in the side of the container, it forms a conical pile on the counter below. As the salt falls onto the pile, it slides down the sides of the pile so that the pile's radius is always equal to its height. The height of the pile is increasing at a rate of 0.2 inch per day. a. How quickly is the salt leaking out of the container when the pile is 2 inches tall? b. How much salt has leaked out of the container by this time?

Write the indicated related-rates equation. \(s=\pi r \sqrt{r^{2}+b^{2}} ;\) relate \(\frac{d b}{d t}\) and \(\frac{d r}{d t}\), assuming that \(s\) is constant.

A set of 24 duplexes ( 48 units) was sold to a new owner. The previous owner charged 700 per month rent and had a 100 % occupancy rate. The new owner, aware that this rental amount was well below other rental prices in the area, immediately raised the rent to 750 Over the next 6 months, an average of 47 units were occupied. The owner raised the rent to 800 and found that 2 of the units were unoccupied during the next 6 months. a. What was the monthly rental income i. for the previous owner? ii. after the first rent increase? iii. after the second rent increase? b. Construct an equation for the monthly rental income as a function of the number of \(\$ 50\) increases in the rent. c. If the occupancy rate continues to decrease as the rent increases in the same manner as it did for the first two increases, what rent amount will maximize the owner's rental income from these duplexes? How much will the owner collect in rent each month at this rental amount? d. What other considerations besides rental income should the owner take into account when determining an optimal rental amount?

For Activities 7 through \(18,\) write the first and second derivatives of the function. \(L(t)=\frac{100}{1+99.6 e^{-0.02 t}}\)

During the summer, an art student makes and sells necklaces at the beach. Last summer she sold the necklaces for 10 each and her sales averaged 20 necklaces per day. When she increased the price by 1, she found that she lost 2 sales per day. Assume that this pattern would continue if she kept increasing the price; that is, for every dollar she raised the price, she would sell 2 fewer necklaces per day. a. Write a model for the student's daily revenue from the sale of x necklaces. b. The material used in each necklace costs 6 . Write the student's daily cost in terms x. c. What price will maximize the student's profit from sales at the beach? What will be the profit be if this price is charged?
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