Curve Sketching Examples with First and Second Derivatives.

📚 Quick Study Guide

• 📈 First Derivative: The first derivative, $f'(x)$, tells us about the function's increasing and decreasing intervals. If $f'(x) > 0$, the function is increasing; if $f'(x) < 0$, the function is decreasing.
• 📍 Critical Points: Critical points occur where $f'(x) = 0$ or $f'(x)$ is undefined. These points are potential locations for local maxima or minima.
• 📉 Second Derivative: The second derivative, $f''(x)$, tells us about the concavity of the function. If $f''(x) > 0$, the function is concave up; if $f''(x) < 0$, the function is concave down.
• 🔄 Inflection Points: Inflection points occur where the concavity changes. These occur where $f''(x) = 0$ or $f''(x)$ is undefined.
• 📝 Curve Sketching Steps:
  1. Find $f'(x)$ and $f''(x)$.
  2. Find critical points and possible inflection points.
  3. Create sign charts for $f'(x)$ and $f''(x)$.
  4. Identify increasing/decreasing intervals and concavity.
  5. Sketch the curve.

🧪 Practice Quiz

1. If $f'(x) > 0$ on an interval, what does this tell us about the function $f(x)$ on that interval?

   1. It is decreasing.
   2. It is increasing.
   3. It is concave down.
   4. It has a local maximum.

2. What is a critical point?

   1. A point where $f''(x) = 0$.
   2. A point where $f(x) = 0$.
   3. A point where $f'(x) = 0$ or $f'(x)$ is undefined.
   4. A point where the function changes concavity.

3. If $f''(x) < 0$ on an interval, what does this tell us about the function $f(x)$ on that interval?

   1. It is increasing.
   2. It is decreasing.
   3. It is concave down.
   4. It is concave up.

4. What is an inflection point?

   1. A point where $f'(x) = 0$.
   2. A point where $f''(x) = 0$ and the concavity changes.
   3. A point where $f(x) = 0$.
   4. A point where $f'(x)$ is undefined.

5. Consider the function $f(x) = x^3 - 6x^2 + 5$. Find the critical points.

   1. $x = 0, 4$
   2. $x = 2$
   3. $x = -2, 2$
   4. $x = 0$

6. For the function $f(x) = x^3$, where is the inflection point located?

   1. $x = 1$
   2. $x = -1$
   3. $x = 0$
   4. There is no inflection point.

7. Given $f'(x) = (x-1)(x-3)$, on what interval is $f(x)$ increasing?

   1. $(1, 3)$
   2. $(-\infty, 1)$
   3. $(1, \infty)$
   4. $(-\infty, 1) \cup (3, \infty)$