Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

2. Intro to Derivatives

Basic Graphing of the Derivative

Calculus

2. Intro to Derivatives

Basic Graphing of the Derivative

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2:42 minutes

Problem 3.91

Textbook Question

Analyzing Graphs

Each of the figures in Exercises 91 and 92 shows two graphs, the graph of a function 𝔂 = ƒ(x) together with the graph of its derivative ƒ'(x). Which graph is which? How do you know?

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Examine the behavior of each graph: Identify key features such as increasing or decreasing intervals, local maxima and minima, and points of inflection.

Understand the relationship between a function and its derivative: The derivative ƒ'(x) represents the slope of the tangent line to the graph of ƒ(x). Where ƒ(x) is increasing, ƒ'(x) is positive; where ƒ(x) is decreasing, ƒ'(x) is negative.

Identify critical points: Look for points where the derivative graph crosses the x-axis (ƒ'(x) = 0). These correspond to potential local maxima or minima on the graph of ƒ(x).

Analyze concavity: Determine where the graph of ƒ(x) is concave up or concave down. This can be inferred from the sign of the second derivative, but since we only have ƒ(x) and ƒ'(x), look for changes in the slope of ƒ'(x).

Match the graphs: Based on the analysis, match the graph with the characteristics of ƒ(x) and ƒ'(x). The graph with the zero crossings and sign changes corresponding to the behavior of the function is the derivative graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function and Derivative Relationship

The relationship between a function and its derivative is fundamental in calculus. The derivative of a function, denoted as ƒ'(x), represents the rate of change of the function ƒ(x) at any given point. When analyzing graphs, the behavior of the derivative graph can indicate where the original function is increasing or decreasing, as well as where it has local maxima or minima.

Critical Points and Inflection Points

Critical points occur where the derivative ƒ'(x) is zero or undefined, indicating potential local maxima, minima, or points of inflection in the function ƒ(x). Inflection points are where the concavity of the function changes, which can be identified by changes in the sign of the derivative. Understanding these points helps in distinguishing between the graphs of a function and its derivative.

Graphical Interpretation of Derivatives

Graphically, the derivative of a function can be interpreted as the slope of the tangent line to the function's graph at any point. If the graph of ƒ(x) is increasing, ƒ'(x) will be positive, and if it is decreasing, ƒ'(x) will be negative. Additionally, where the derivative graph crosses the x-axis indicates where the original function has horizontal tangents, aiding in identifying the correct graphs.

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Related Practice

Textbook Question

Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>

b. Estimate the value of L'(a) for a ≥ 4 . What does this tell you about the talon lengths on these birds? (Source: ZooKeys, 132, 2011)

Textbook Question

Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>

Textbook Question

A cost function of the form C(x) = 1/2x² reflects diminishing returns to scale. Find and graph the cost, average cost, and marginal cost functions. Interpret the graphs and explain the idea of diminishing returns.

Textbook Question

Match the graphs of the functions in a–d with the graphs of their derivatives in A–D. <MATCH A-D IMAGE>

Textbook Question

Recovering a function from its derivative

b. Repeat part (a), assuming that the graph starts at (−2, 0) instead of (−2, 3).

Textbook Question

Slopes on the graph of the tangent function Graph y = tan x and its derivative together on (−π/2, π/2). Does the graph of the tangent function appear to have a smallest slope? A largest slope? Is the slope ever negative? Give reasons for your answers.

Multiple Choice

Based on the graph of $f\left(x\right)$ , describe the graph of the derivative $f^{\prime}\left(x\right)$ on the interval $\left(0,\infty\right)$ .

Multiple Choice

Based on the graph of $f\left(x\right)$ , describe the graph of the derivative $f^{\prime}\left(x\right)$ at the point $x=-1$ .

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Analyzing GraphsEach of the figures in Exercises 91 and 92 shows two graphs, the graph of a function 𝔂 = ƒ(x) together with the graph of its derivative ƒ'(x). Which graph is which? How do you know?<IMAGE>

Key Concepts

Function and Derivative Relationship

Critical Points and Inflection Points

Graphical Interpretation of Derivatives

Watch next

Analyzing Graphs

Each of the figures in Exercises 91 and 92 shows two graphs, the graph of a function 𝔂 = ƒ(x) together with the graph of its derivative ƒ'(x). Which graph is which? How do you know?

<IMAGE>