Get Exploration Concavity And The 2nd Derivative Test For

This guide provides a clear and supportive approach to filling out the Exploration Concavity And The 2nd Derivative Test For online. Whether you are familiar with calculus concepts or new to them, this guide will walk you through each step necessary for successful completion.

Follow the steps to complete the form effectively.

1. Click ‘Get Form’ button to access the form and open it in your preferred online editor.
2. Fill in your name, period, and group in the designated fields at the top of the form. This information helps identify your submission.
3. Review the warm-up section, which includes a mnemonic for remembering concavity. Make sure to commit the phrase to memory: 'Concave up, shaped like a cup. Concave down, shaped like a frown.'
4. Follow the prompts to draw vertical lines on the provided graph that separate the regions of concavity according to the second derivative.
5. Identify critical points by noting where the first derivative equals zero or does not exist. Label these points as 'CP' on the graph provided.
6. Based on your analysis, determine if the critical points in concave regions are local maxima or minima and complete the corresponding statements.
7. For the specified function f(x) = 15x^3 - x^5, find critical points by solving f'(x) = 0 or DNE, and indicate the domain explicitly.
8. Create a sign chart (number line) to visualize the critical points and label them accordingly as 'CP'.
9. Utilize the second derivative to assess the concavity at each critical point, labeling them as maxima or minima based on your findings.
10. Perform an end behavior analysis to classify each maximum or minimum as local or absolute and make the necessary calculations.
11. Repeat similar steps for the second given function f(x) = 2x^2 - x^4 + 1, including finding critical points, creating a sign chart, and applying the second derivative test.
12. For the final function f(x) = -x^4 + 4x^3 - 4x + 1, ascertain all inflection points and assess concavity by identifying intervals of concave up and concave down behavior.
13. Conclude your filling process by comparing results with the provided graph, ensuring accuracy in your answers.

Now that you have a detailed guide, complete your document online confidently!