Curve sketching involves using the first and second derivatives of a function to understand its shape. The first derivative tells us where the function is increasing or decreasing and helps us find local maxima and minima. The second derivative tells us about the concavity of the function—whether it's curving upwards or downwards—and helps us find inflection points. By combining this information, we can create an accurate sketch of the function's graph.
Let's practice applying these concepts to solve some problems!
Match the terms with their definitions:
| Term | Definition |
|---|---|
| 1. Inflection Point | A. A point where the function changes from increasing to decreasing or vice versa. |
| 2. Concave Up | B. The interval where the second derivative is positive. |
| 3. Local Maximum | C. A point where the second derivative changes sign. |
| 4. Critical Point | D. A point where the first derivative is zero or undefined. |
| 5. Concave Down | E. The interval where the second derivative is negative. |
Complete the following paragraph with the correct terms:
To sketch a curve, we first find the __________ points by setting the first derivative equal to zero. Then, we use the second derivative to determine the __________ of the function. A point where the concavity changes is called an __________ point. If the first derivative changes sign at a critical point, we have a local __________ or minimum. Finally, we plot these key points and connect them to create the sketch.
Explain how knowing the first and second derivatives of a function helps you accurately sketch its graph. Provide an example to illustrate your explanation.
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