Draw the derivative of these graphs. The diagram above depicts the graph of a function $ f(x) $.**Graph Description:**- **Axes:** The graph plotted is in a Cartesian plane with the horizontal axis labeled as $ x $ and the vertical axis labeled as $ f(x) $.- **XY Coordinates:** The x-axis ranges from -3 to 3, while the y-axis spans from -4 to 4. Both axes are marked at regular intervals of 1 unit.- **Function Behavior:** - The function appears to be a polynomial with multiple local extrema. - The graph intersects the y-axis at $ f(0) = 4 $. - The graph has local minima and maxima. Specifically, it demonstrates a local minimum at approximately $ x = -2 $ and $ x = 2 $, where $ f(x) \approx -4 $. There is also a local maximum around $ x = 0 $ with $ f(x) \approx 4 $.The detailed behavior of the graph suggests features typical of a higher-order polynomial function (likely a cubic or quartic), given the changes in concavity and the presence of both local minima and maxima. Understanding and interpreting such graphs is crucial for analyzing the roots, critical points, and general behavior of polynomial functions in calculus and algebra. ### Graph of $ f(x) $The graph illustrates the function $ f(x) $ plotted against $ x $.#### Description:- The x-axis is labeled as $ x $, with marked points at -1, 0, 1, 2, and 3.- The y-axis aligns with $ x = 0 $.#### Behavior of $ f(x) $:1. **For $ x < 0 $**: - The function $ f(x) $ rises steeply as $ x $ increases from -1 to 0.2. **For $ x \geq 0 $**: - The function $ f(x) $ continues to increase from $ x = 0 $ to $ x = 1 $, though the rate of increase slows down. - From $ x = 1 $ to $ x = 3 $, the function $ f(x) $ slightly decreases or plateaus.#### Key Observations:- At $ x = 0 $, the function $ f(x) $ has a distinct change in concavity, moving from a steep incline to a more gradual slope.- After reaching a peak around $ x = 1 $, the function maintains a nearly constant value or experiences a slight decline as $ x $ increases further.This graph can help in understanding the general behavior and trends of the function $ f(x) $, providing insights into its nature and critical points.

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Answered: Draw the derivative of these graphs.

Math

Calculus

Draw the derivative of these graphs.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

Question

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Draw the derivative of these graphs.

The diagram above depicts the graph of a function $ f(x) $.

**Graph Description:**
- **Axes:** The graph plotted is in a Cartesian plane with the horizontal axis labeled as $ x $ and the vertical axis labeled as $ f(x) $.
- **XY Coordinates:** The x-axis ranges from -3 to 3, while the y-axis spans from -4 to 4. Both axes are marked at regular intervals of 1 unit.
- **Function Behavior:**
- The function appears to be a polynomial with multiple local extrema.
- The graph intersects the y-axis at $ f(0) = 4 $.
- The graph has local minima and maxima. Specifically, it demonstrates a local minimum at approximately $ x = -2 $ and $ x = 2 $, where $ f(x) \approx -4 $. There is also a local maximum around $ x = 0 $ with $ f(x) \approx 4 $.

The detailed behavior of the graph suggests features typical of a higher-order polynomial function (likely a cubic or quartic), given the changes in concavity and the presence of both local minima and maxima. Understanding and interpreting such graphs is crucial for analyzing the roots, critical points, and general behavior of polynomial functions in calculus and algebra.

### Graph of $ f(x) $

The graph illustrates the function $ f(x) $ plotted against $ x $.

#### Description:
- The x-axis is labeled as $ x $, with marked points at -1, 0, 1, 2, and 3.
- The y-axis aligns with $ x = 0 $.

#### Behavior of $ f(x) $:
1. **For $ x < 0 $**:
- The function $ f(x) $ rises steeply as $ x $ increases from -1 to 0.

2. **For $ x \geq 0 $**:
- The function $ f(x) $ continues to increase from $ x = 0 $ to $ x = 1 $, though the rate of increase slows down.
- From $ x = 1 $ to $ x = 3 $, the function $ f(x) $ slightly decreases or plateaus.

#### Key Observations:
- At $ x = 0 $, the function $ f(x) $ has a distinct change in concavity, moving from a steep incline to a more gradual slope.
- After reaching a peak around $ x = 1 $, the function maintains a nearly constant value or experiences a slight decline as $ x $ increases further.

This graph can help in understanding the general behavior and trends of the function $ f(x) $, providing insights into its nature and critical points.

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