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Chapter 1: Problem 5

Table 1.12 gives values of a function \(w=f(t) .\) Is this function increasing or decreasing? Is the graph of this function concave up or concave down? $$\begin{array}{c|c|c|c|c|c|c|c} \hline t & 0 & 4 & 8 & 12 & 16 & 20 & 24 \\ \hline w & 100 & 58 & 32 & 24 & 20 & 18 & 17 \\ \hline \end{array}$$

Short Answer

Expert verified
The function is decreasing and concave up.

Step by step solution

01

Analyze Function Values for Increase or Decrease

Look at the values of the function \(w=f(t)\). For each time \(t\), observe if the subsequent value increases or decreases. Here, \(w\) values are \(100, 58, 32, 24, 20, 18, 17\) as \(t\) increases from \(0\) to \(24\). Since every subsequent \(w\) value is less than the previous, the function is decreasing.
02

Determine Concavity by Checking Rate of Change

Calculate the difference in \(w\) for each interval of \(t\): \(100-58 = 42\), \(58-32 = 26\), \(32-24 = 8\), \(24-20 = 4\), \(20-18 = 2\), \(18-17 = 1\). As \(t\) increases, these differences decrease, indicating the rate of decrease is getting smaller.
03

Analyze Concave Up or Down

If the rate of decrease of the function values is getting smaller, the function has an increasing slope, which means the graph is becoming flatter. This suggests that the graph of the function is concave up.

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Key Concepts

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

Decreasing Function
A function is considered decreasing when its output values get smaller as the input values increase. In the exercise provided, we see that the function \( w = f(t) \) shows a pattern of consistently decreasing values: 100, 58, 32, 24, 20, 18, 17. As \( t \) increases from 0 to 24, the function \( w(t) \) continues to drop, indicating it is a decreasing function.
Understanding decreasing functions is important because they often indicate situations where, in real-world contexts, something is lessening or declining over time. Whether it's a decreasing profit, temperature, or population, interpreting these patterns can be vital for decision making.
  • If each value is less than the previous, the function is decreasing.
  • In a graph, this is represented by a downward slope from left to right.
  • Decreasing functions are foundational in analyzing downward trends or reductions in various contexts.
Concavity
Concavity refers to the curvature of a graph. It tells us how the slope of the tangent line changes as you move along the graph. In the context of the exercise, we calculate the rate of change between consecutive \( w \) values to understand this curvature better. Observing the differences between consecutive \( w \) values (42, 26, 8, 4, 2, 1), we notice that the differences themselves are decreasing.
This pattern showing diminishing differences indicates a decreasing rate of change, suggesting that the function's graph is slowly flattening. When a graph's slope decreases at a decreasing rate, it implies the graph is concave up.
  • Concave up suggests the graph has a U-shaped curve, bending upwards.
  • For concave up sections, the second derivative is positive.
  • Understanding concavity helps in predicting the behavior of the function over intervals.
Rate of Change
The rate of change measures how much a function's output value changes for each unit increase in the input value. In the case of the given function \( w = f(t) \), we determined the rate of change by calculating the difference between consecutive \( w \) values.
The calculated differences (42, 26, 8, 4, 2, 1) show a consistent decrease. This means that the rate at which \( w(t) \) decreases is becoming less steep, which is a critical insight. A positive or negative rate of change tells us whether a function is increasing or decreasing, whereas the consistency of these changes lends insight into the function鈥檚 overall behavior.
  • Rate of change is a fundamental concept for understanding how a variable changes in relation to another.
  • It helps in identifying trends and predicting future behaviors.
  • In calculus, it is closely linked to the derivative of a function.

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Most popular questions from this chapter

Find the future value in 15 years of a \(\$ 20,000\) payment today, if the interest rate is \(3.8 \%\) per year compounded continuously.

The functions \(r=f(t)\) and \(V=g(r)\) give the radius and the volume of a commercial hot air balloon being inflated for testing. The variable \(t\) is in minutes, \(r\) is in feet, and \(V\) is in cubic feet. The inflation begins at \(t=0 .\) In each case, give a mathematical expression that represents the given statement. The volume of the balloon if its radius were twice as big.

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