/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Draw the graph of a function \(y... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 22

Draw the graph of a function \(y=f(x)\) with the stated properties. Both the function and the slope decrease as \(x\) increases. [Note: The slope is negative and becomes more negative.]

Short Answer

Expert verified
Draw a concave down curve that becomes steeper as x increases.

Step by step solution

01

Identify the properties of the function

The function has two key properties: both the function itself and its slope decrease as x increases. This means the function must be decreasing, and its slope is negative and becomes more negative as x increases.
02

Understanding the slope behavior

Since the slope is negative and becomes more negative, the graph should become steeper as x increases. This indicates a function that is concave down.
03

Sketch the graph

Start from a point on the y-axis and draw a curve that moves downward as x increases. Make sure the slope (or steepness) of the curve increases in magnitude, meaning the curve becomes steeper as it progresses.

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Key Concepts

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

negative slope
A negative slope means that as you move from left to right along the x-axis, the value of the function decreases. Imagine you are walking on a hill; a negative slope means you are walking downhill. The more negative the slope, the steeper the descent.
Think of the slope as the 鈥榮teepness鈥� of the line or curve. In mathematical terms, slope is the ratio of the change in the y-value to the change in the x-value, denoted as \( \frac{dy}{dx} \).
Because the function in this problem has a negative slope that becomes more negative as x increases, the graph gets steeper in the downward direction as you move to the right. This behavior is crucial when sketching the graph.
concave down
Concave down indicates that the slope decreases as x increases. In simple terms, the smile on the graph would be turned upside down, similar to a frown.
To be more precise, in a concave down function, the second derivative \( \frac{d^2y}{dx^2} \) is negative. This mathematical property ensures that the slope becomes more negative (steeper downwards) as x increases.
For our example function, both the function and its slope are decreasing. This aligns perfectly with a concave down graph where the slope keeps getting smaller and smaller as we move along the x-axis.
graph sketching
When sketching the graph of a function like this, follow these steps:
  • Start at a point on the y-axis (initial value).
  • Ensure the graph moves downward as x increases, indicating that the function itself is decreasing.
  • Draw the curve such that it becomes steeper. This matches the slope becoming more negative.
Remember that making the slope steeper as you move right is key here. This ensures the graph is both decreasing and concave down.
Graph sketching is more visual and intuitive. Always think about how the slope and curvature change. In this problem, because of the negative and increasingly negative slope, your sketch will look like a curve that starts off gentle and becomes steeper as you move right.

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Most popular questions from this chapter

Consider a smooth curve with no undefined points. (a) If it has two relative maximum points, must it have a relative minimum point? (b) If it has two relative extreme points, must it have an inflection point?

Some years ago, it was estimated that the demand for steel approximately satisfied the equation \(p=256-50 x\) and the total cost of producing \(x\) units of steel was \(a(x)=182+56 x\). (The quantity \(x\) was measured in millions of tons and the price and total cost were measured in millions of dollars.) Determine the level of production and the corresponding price that maximize the profits.

The first and second derivatives of the function \(f(x)\) have the values given in Table 1. (a) Find the \(x\) -coordinates of all relative extreme points. (b) Find the \(x\) -coordinates of all inflection points. $$\text { Table 1 Values of the First Two Derivatives of a Function }$$ $$\begin{array}{ccc} \hline x & f^{\prime}(x) & f^{\prime \prime}(x) \\ \hline 0 \leq x < 2 & \text { Positive } & \text { Negative } \\ 2 & 0 & \text { Negative } \\ 2 < x < 3 & \text { Negative } & \text { Negative } \\ 3 & \text { Negative } & 0 \\ 3 < x < 4 & \text { Negative } & \text { Positive } \\ 4 & 0 & 0 \\ 4 < x \leq 6 & \text { Negative } & \text { Negative } \\ \hline \end{array}$$

The graph of each function has one relative extreme point. Find it (giving both \(x\) - and \(y\) -coordinates) and determine if it is a relative maximum or a relative minimum point. Do not include a sketch of the graph of the function. $$f(x)=\frac{1}{4} x^{2}-2 x+7$$

Each of the graphs of the functions has one relative maximum and one relative minimum point. Find these points using the first-derivative test. Use a variation chart as in Example 1. $$f(x)=-x^{3}-12 x^{2}-2$$
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