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Chapter 3: Problem 2

Explain why \(f(x)\) could be positive or negative at a point where \(f^{\prime}(x )< 0\).

Short Answer

Expert verified
Question: Explain why a function f(x) could be positive or negative at a point where its derivative f'(x) is less than 0. Answer: A function f(x) could be positive or negative at a point where its derivative f'(x) is less than 0 because the derivative only indicates the direction (increasing or decreasing) of the function around that point, not the actual positive or negative value of the function. If f'(x) is negative, it means the function is decreasing at that point. However, the value of f(x) could still be positive or negative depending on the specific value of the function at that point.

Step by step solution

01

Understanding the function and its derivative

A function f(x) is a mathematical relationship between the input (x) and the output (f(x)). The derivative of a function, denoted f'(x), represents the rate of change of the function with respect to the input variable x. In other words, f'(x) measures how fast the function f(x) is increasing or decreasing at a given point x.
02

Interpreting positive and negative values of the function

A function f(x) is positive at a point x if its output f(x) is greater than 0, and negative if its output f(x) is less than 0. This means that when we plug in x into the function f(x), the resulting value is either greater than or less than 0, depending on whether the function is positive or negative.
03

Interpreting a negative derivative

If the derivative f'(x) is less than 0 at a point x, it means that the function f(x) is decreasing at that point. When the rate of change of the function (i.e., the derivative f'(x)) is negative, it indicates that as x increases, the function f(x) decreases.
04

Positive or negative function values with a decreasing function

Now that we understand what it means for a function to be positive or negative and how a negative derivative signifies a decreasing function, let's consider the value of f(x) at a point where f'(x) is negative. Since the function is decreasing at that point, it could either be positive (f(x) > 0) or negative (f(x) < 0) depending on the exact value of the function at that x. The derivative only informs us about the direction (increasing or decreasing) of the function around the point x, not the actual positive or negative value of the function. For example, consider the function f(x) = -x^2 + 5x. The derivative of this function is f'(x) = -2x + 5. Let's find a point x where f'(x) < 0. If we choose x = 3, then f'(3) = -2(3) + 5 = -1, which is negative. So, the function is decreasing at x = 3. However, the value of the function at this point is f(3) = -3^2 + 5(3) = -9 + 15 = 6, which is positive. The function f(x) is positive at x = 3 despite the negative derivative. Thus, it is possible for a function to be either positive or negative at a point where its derivative is less than 0.

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Key Concepts

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

Derivative
In calculus, the derivative is a fundamental concept that allows us to understand how a function behaves at various points. Essentially, the derivative of a function, denoted as \( f'(x) \), is a measure of how the function's value changes as its input, \( x \), changes. It gives us the slope of the function's graph at any given point.
  • If \( f'(x) > 0 \), the function is said to be increasing at that point.
  • If \( f'(x) < 0 \), the function is decreasing.
  • If \( f'(x) = 0 \), the function may have a local minimum or maximum, or it could be constant.
The derivative helps to determine not just the speed of change but also the "direction" of change in the function. This direction can be upwards (increasing) or downwards (decreasing). Hence, knowing the derivative is crucial for sketching the behavior of functions and solving many practical problems involving rates of change.
Function Behavior
The behavior of a function at a specific point or over an interval can be described by looking at its values as well as its derivative. A function \( f(x) \) can tell us whether its values are high (positive) or low (negative) at certain points.
  • If \( f(x) > 0 \), it's above the x-axis, meaning the function's output is positive at that input \( x \).
  • If \( f(x) < 0 \), it lies below the x-axis, indicating a negative output.
Even if the derivative \( f'(x) < 0 \), which indicates that the function is decreasing at that point, \( f(x) \) could still be either positive or negative. The derivative provides insight into whether the function's path is sloping downwards, but the specific location of \( f(x) \) on the y-axis鈥攚hether it's above or below鈥攄epends on its numerical value at that point.
Rate of Change
The rate of change of a function provides insight into how fast a function is changing as the input changes. It is quantified by the derivative \( f'(x) \). The concept of the rate of change is tied deeply into derivatives because the derivative represents the instantaneous rate of change.Consider these interpretations:
  • A positive rate of change (positive derivative) indicates the function is increasing, or moving upwards, as \( x \) increases.
  • A negative rate of change (negative derivative) tells us the function is decreasing, or moving downwards, as \( x \) increases.
Understanding the rate of change is essential, as it allows us to analyze how quickly the output of a function responds to a change in input. This is especially useful for making predictions in real-life applications, such as calculating velocity in physics or trends in business.

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Most popular questions from this chapter

a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. b. Graph the tangent and normal lines on the given graph. \(\left(x^{2}+y^{2}\right)^{2}=\frac{25}{3}\left(x^{2}-y^{2}\right)\) \(\left(x_{0}, y_{0}\right)=(2,-1)\) (lemniscate of Bernoulli)

Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. $$x^{2}\left(3 y^{2}-2 y^{3}\right)=4$$

Let \(f\) and \(g\) be differentiable functions with \(h(x)=f(g(x)) .\) For a given constant \(a,\) let \(u=g(a)\) and \(v=g(x),\) and define $$H(v)=\left\\{\begin{array}{ll} \frac{f(v)-f(u)}{v-u}-f^{\prime}(u) & \text { if } v \neq u \\ 0 & \text { if } v=u. \end{array}\right.$$ a. Show that \(\lim _{x \rightarrow u} H(v)=0\) b. For any value of \(u\) show that $$f(v)-f(u)=\left(H(v)+f^{\prime}(u)\right)(v-u).$$ c. Show that. $$h^{\prime}(a)=\lim _{x \rightarrow a}\left(\left(H(g(x))+f^{\prime}(g(a))\right) \cdot \frac{g(x)-g(a)}{x-a}\right).$$ d. Show that \(h^{\prime}(a)=f^{\prime}(g(a)) g^{\prime}(a)\).

The bottom of a large theater screen is \(3 \mathrm{ft}\) above your eye level and the top of the screen is \(10 \mathrm{ft}\) above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of \(3 \mathrm{ft} / \mathrm{s}\) while looking at the screen. What is the rate of change of the viewing angle \(\theta\) when you are \(30 \mathrm{ft}\) from the wall on which the screen hangs, assuming the floor is horizontal (see figure)?

Let \(b\) represent the base diameter of a conifer tree and let \(h\) represent the height of the tree, where \(b\) is measured in centimeters and \(h\) is measured in meters. Assume the height is related to the base diameter by the function \(h=5.67+0.70 b+0.0067 b^{2}\). a. Graph the height function. b. Plot and interpret the meaning of \(\frac{d h}{d b}\).
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