Apply what you learned in this unit by creating a presentation in which you come up with your own twice-differentiable function and draw its graph without a calculator by analyzing its properties. You can create a slideshow presentation, a video, or a written report. These properties must be included in your presentation: zeros, symmetry, and first- and second-order derivatives, local and global extreme values, the concavity test, concave up, and concave down. Then, graph your function using your graphing calculators to verify your work.
Title: Exploring the Properties of a Twice-Differentiable Function: Analysis and Graphing
Slide 1: Introduction
- Start with a brief introduction to the topic of twice-differentiable functions, highlighting their importance in mathematical analysis and graphing.
- Emphasize that we will be creating our own function, analyzing its properties, and graphing it without using a calculator.
Slide 2: Function Definition
- Introduce our created function: f(x) = x^3 - 3x^2 - 2x + 2
- Explain that this function is chosen specifically to exhibit various properties we plan to analyze.
Slide 3: Zeros
- Discuss the concept of zeros as points where the function intersects the x-axis.
- Solve f(x) = 0 to find the zeros of our function.
- Show solutions: x = -1, x = 1, x = 2.
Slide 4: Symmetry
- Explain the concept of symmetry.
- Analyze the function's symmetry by evaluating f(-x) and comparing it with f(x).
- Show f(-x) = (-x)^3 - 3(-x)^2 - 2(-x) + 2 = -x^3 - 3x^2 + 2x + 2.
- Compare f(-x) with f(x) to conclude that our function is not symmetric.
Slide 5: First-Order Derivatives
- Introduce the first derivative, f'(x), and discuss its significance in analyzing a function's critical points and local extrema.
- Calculate f'(x) = 3x^2 - 6x - 2 using the power rule and simplification.
- Discuss the critical points by equating f'(x) = 0 and solving for x.
- Show solution: x = (3 ± √21)/3, approximately -0.58 and 2.25.
Slide 6: Local and Global Extreme Values
- Explain the relationship between critical points and local extrema.
- Determine the nature of the critical points using the second derivative test.
- Introduce the second derivative, f''(x), and discuss its significance in verifying the nature of critical points.
- Calculate f''(x) = 6x - 6 using the power rule and simplification.
- Analyze f''(x) at the critical points obtained in the previous slide.
Slide 7: Concavity Test
- Introduce the concept of concavity and the concavity test.
- Solve f''(x) > 0 to identify where the function is concave up.
- Solve f''(x) < 0 to identify where the function is concave down.
- Discuss the transition points where the concavity changes.
Slide 8: Final Analysis
- Summarize all the properties we have analyzed so far: zeros, symmetry, critical points, and concavity.
- Combine the information to create a sketch of the function's graph.
Slide 9: Graphing the Function
- Use graphing calculators or appropriate graphing software to plot the function.
- Verify our analysis of zeros and critical points.
- Compare the calculated graph with our previously drawn sketch.
Slide 10: Conclusion
- Recap the importance of examining the properties of a twice-differentiable function for understanding its behavior.
- Highlight the benefits of creating and analyzing our own function without relying on a calculator.
- End with a closing thought about the power of mathematical analysis in graphing functions.
Note: The presentation can be adapted according to the chosen format - slideshow, video, or written report. The objective is to include the essential properties and steps for analysis while ensuring clarity and coherence.