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Chapter 2: Problem 16

If \(d f / d x\) is negative, is \(f(x)\) always negative? Is \(f(x)\) negative for large \(x ?\) If you think otherwise, give examples.

Short Answer

Expert verified
A negative derivative means \(f(x)\) is decreasing; it can be positive or negative. \(f(x)\) often becomes negative for large \(x\).

Step by step solution

01

Understanding the Meaning of a Negative Derivative

If \( \frac{df}{dx} \) is negative, it means that the function \( f(x) \) is decreasing. The slope of the tangent to the function at any given point is negative, indicating a downward trend. However, this does not directly imply that \( f(x) \) itself is negative.
02

Example of Decreasing Functions

Consider a function that is decreasing but not necessarily negative, such as \( f(x) = -x + 10 \). The derivative \( \frac{df}{dx} = -1 \) is negative, showing that the function is decreasing. However, \( f(x) \) will be positive as long as \( x < 10 \). Thus, a negative derivative can result in a positive \( f(x) \).
03

Assessing Large Values of \(x\)

To determine if \( f(x) \) is negative for large \( x \), we need to analyze the behavior of the function as \( x \) approaches infinity. Taking our example \( f(x) = -x + 10 \), as \( x \) gets larger, \( f(x) \) becomes more negative. Hence, \( f(x) \) is indeed negative for large \( x \).
04

Conclusion

While a negative derivative implies a decreasing function, it does not mean that \( f(x) \) must be negative. However, for many functions like linear functions with a negative slope, \( f(x) \) can become negative when \( x \) is sufficiently large.

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Key Concepts

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

Negative Derivative and its Implications
When the derivative of a function, denoted as \( \frac{df}{dx} \), is negative, it tells us something crucial about the function's behavior. Specifically, it indicates that the function \( f(x) \) is decreasing. But what does this mean?
  • A negative derivative implies that the slope of the function is moving downwards. Imagine a hill that slopes downwards, this hill visually represents a function with a negative derivative.
  • Even though the function is decreasing, it doesn't automatically mean that the actual values of the function \( f(x) \) are negative. These are two distinct concepts. A decreasing trend doesn't dictate the sign of the function's values.
  • For example, consider the function \( f(x) = -x + 10 \). Here, \( \frac{df}{dx} = -1 \), indicating that the slope is negative, so the function is indeed decreasing. However, depending on the value of \( x \), the output \( f(x) \) can still be positive.
By understanding that a negative derivative simply reflects a downwards trend, we can prevent the common mistake of assuming negativity in the function's output.
Decreasing Function Explained
A function is termed decreasing when it consistently goes downwards as you move along the X-axis. This concept closely ties in with our previous discussion on negative derivatives.
  • When \( \frac{df}{dx} \) is negative, it indicates a consistent downward movement of the function. Imagine skating down a slope, where each movement takes you a little lower.
  • This doesn't mean the values of the function \( f(x) \) are themselves negative. Instead, it means the function consistently outputs smaller or lesser values."
  • Taking our previous example, \( f(x) = -x + 10 \), shows a decreasing pattern as the situation consistently declines with increasing \( x \).
The key tack to understanding decreasing functions is focusing on the trend, not just the function's current position on the axis.
Understanding Function Behavior for Large \( x \)
Have you ever wondered what happens to a function when its variable, \( x \), gets very large? This is often crucial for understanding the function's broader behavior.
  • As \( x \) grows towards infinity, it's important to observe whether \( f(x) \) trends towards positive or negative values, or stabilizes at some specific level.
  • In many cases, like our example function \( f(x) = -x + 10 \), as \( x \) increases beyond certain boundaries (here it's 10), \( f(x) \) tends to become negative because \( -x \) dominates the positive constant. This makes the function trend towards negative values for large \( x \).
Understanding behavior at extremes helps in grasping the full picture of the function, allowing us to predict its long-term patterns effortlessly.

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Most popular questions from this chapter

In Problems \(1-20\), find the numbers \(c\) that make \(f(x)\) into (A) a continuous function and (B) a differentiable function. In one case \(f(x) \rightarrow f(a)\) at every point, in the other case \(\Delta f / \Delta x\) has a limit at every point. $$ f(x)=\left\\{\begin{array}{cc} (\sin x) / x^{2} & x \neq 0 \\ c & x=0 \end{array}\right. $$

In Problems \(1-20\), find the numbers \(c\) that make \(f(x)\) into (A) a continuous function and (B) a differentiable function. In one case \(f(x) \rightarrow f(a)\) at every point, in the other case \(\Delta f / \Delta x\) has a limit at every point. $$ f(x)=\left\\{\begin{array}{cc} (\sin x-x) / x^{e} & x \neq 0 \\ 0 & x=0 \end{array}\right. $$

True or false, with a good reason: (a) The derivative of \(x^{2 n}\) is \(2 n x^{2 n-1}\). (b) By linearity the derivative of \(a(x) u(x)+b(x) v(x)\) is \(a(x) d u / d x+b(x) d v / d x\) (c) The derivative of \(|x|^{3}\) is \(3|x|^{2}\). (d) \(\tan ^{2} x\) and \(\sec ^{2} x\) have the same derivative. (e) \((u v)^{\prime}=u^{\prime} v^{\prime}\) is true when \(u(x)=1\).

Show by example that these statements are false: (a) If \(a_{n} \rightarrow L\) and \(b_{n} \rightarrow L\) then \(a_{n} / b_{n} \rightarrow 1\) (b) \(a_{n} \rightarrow L\) if and only if \(a_{n}^{2} \rightarrow L^{2}\) (c) If \(a_{n}<0\) and \(a_{n} \rightarrow L\) then \(L<0\) (d) If infinitely many \(a_{n}\) 's are inside every strip around zero then \(a_{n} \rightarrow 0\).

Find the limits if they exist. An \(\varepsilon-\delta\) test is not required. $$ \lim _{t \rightarrow 2} \frac{t^{2}+3}{t-2} $$
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