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

5. Graphical Applications of Derivatives

Curve Sketching

Calculus

5. Graphical Applications of Derivatives

Curve Sketching

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2:48 minutes

Problem 4.4.8d

Textbook Question

Sketch a graph of a function f with the following properties.

f' < 0 and f" < 0, for 8 < x < 10

Verified step by step guidance

Understand the given conditions: f'(x) < 0 means the function f is decreasing in the interval 8 < x < 10, and f''(x) < 0 means the function is concave down in the same interval.

Visualize the behavior of the function: Since f'(x) < 0, the slope of the tangent line to the graph of f is negative, indicating that the graph is sloping downward. Additionally, f''(x) < 0 implies that the graph is curving downward, resembling a 'downward bowl' shape.

Sketch the graph in the interval 8 < x < 10: Start by drawing a curve that decreases (slopes downward) and is concave down. Ensure the curve does not flatten or change concavity within this interval.

Consider the behavior outside the interval: While the problem does not specify the behavior of f for x ≤ 8 or x ≥ 10, ensure the graph transitions smoothly into the interval 8 < x < 10 without abrupt changes.

Label the graph appropriately: Mark the interval 8 < x < 10 on the x-axis and ensure the graph clearly reflects the properties f'(x) < 0 and f''(x) < 0 within this interval.

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Key Concepts

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

First Derivative Test

The first derivative of a function, denoted as f', indicates the slope of the tangent line to the graph at any point. If f' < 0, the function is decreasing in that interval. Understanding this concept is crucial for sketching the graph, as it informs us that the function is moving downward between x = 8 and x = 10.

Second Derivative Test

The second derivative, denoted as f'', provides information about the concavity of the function. If f'' < 0, the function is concave down, meaning that the slope of the tangent line is decreasing. This concept is essential for sketching the graph, as it indicates that the function not only decreases but does so at an accelerating rate in the specified interval.

Graph Behavior

Understanding the overall behavior of a function based on its derivatives is key to sketching its graph. In this case, since f' < 0 and f'' < 0 for 8 < x < 10, the graph will show a downward slope that becomes steeper as x increases. This behavior helps in accurately representing the function's characteristics in the specified interval.

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Sketch a graph of a function f with the following properties.f' < 0 and f" < 0, for 8 < x < 10

Key Concepts

First Derivative Test

Second Derivative Test

Graph Behavior

Watch next

Sketch a graph of a function f with the following properties.

f' < 0 and f" < 0, for 8 < x < 10