Sketching and connecting a function with its first and second derivatives

Welcome to the Great Connector: Functions and Derivatives

In previous chapters of Unit 5, you learned how to find critical points and determine concavity. Now, it is time to put those pieces of the puzzle together! This chapter is all about the visual and analytical "handshake" between a function \(f(x)\), its slope-maker \(f'(x)\), and its bend-maker \(f''(x)\). Think of this as learning to translate between three different languages that all describe the same story.

Whether you are given a graph of the derivative and asked to sketch the original function, or given an equation and asked to describe its behavior, these connections are the "secret sauce" for the AP Calculus BC exam.

1. The Relationship Between \(f(x)\) and \(f'(x)\)

The most important thing to remember is the Slope-Height Connection: The slope of the function \(f(x)\) at any point is exactly equal to the y-value (height) of the graph of \(f'(x)\).

Key Connections:

1. Increasing vs. Positive: When \(f(x)\) is increasing, its slope is positive. Therefore, the graph of \(f'(x)\) must be above the x-axis (\(f'(x) > 0\)).
2. Decreasing vs. Negative: When \(f(x)\) is decreasing, its slope is negative. Therefore, the graph of \(f'(x)\) must be below the x-axis (\(f'(x) < 0\)).
3. Relative Extrema: If \(f(x)\) has a local maximum or minimum at \(x = c\), then \(f'(c)\) must be \(0\) (an x-intercept) or undefined. Crucial Tip: For an actual extremum to exist, the graph of \(f'(x)\) must actually cross the x-axis, changing from positive to negative or vice versa.

Quick Review: We already covered the details of the First Derivative Test in Chapter 5.4, but remember that if \(f'(x)\) changes from positive to negative, \(f(x)\) has a relative maximum.

Key Takeaway: Don't look at where \(f'(x)\) is going up or down; look at whether \(f'(x)\) is above or below the x-axis to tell what \(f(x)\) is doing.

2. The Relationship Between \(f'(x)\) and \(f''(x)\)

The relationship between the first derivative and the second derivative is exactly the same as the relationship between the function and the first derivative! It is just one level "lower" in the hierarchy.

Key Connections:

1. Slope of the Derivative: The slope of the graph of \(f'(x)\) is the y-value of \(f''(x)\).
2. Derivative Increasing vs. Second Derivative Positive: When \(f'(x)\) is increasing, its derivative \(f''(x)\) is positive (\(f''(x) > 0\)).
3. Derivative Decreasing vs. Second Derivative Negative: When \(f'(x)\) is decreasing, its derivative \(f''(x)\) is negative (\(f''(x) < 0\)).

Did you know? In physics terms (Unit 4), this is the connection between velocity and acceleration. If velocity is increasing, acceleration is positive!

3. Connecting \(f(x)\) to \(f''(x)\) (Concavity)

This is where we "skip" a level to see how the second derivative affects the shape of the original function.

Key Connections:

1. Concave Up: When \(f''(x) > 0\), the function \(f(x)\) is concave up (it looks like a cup \(\cup\)). At the same time, the first derivative \(f'(x)\) is increasing.
2. Concave Down: When \(f''(x) < 0\), the function \(f(x)\) is concave down (it looks like a frown \(\cap\)). At the same time, the first derivative \(f'(x)\) is decreasing.
3. Points of Inflection: A point of inflection occurs on \(f(x)\) where the concavity changes. On the graph of \(f'(x)\), this corresponds to a relative extremum (a peak or a valley). On the graph of \(f''(x)\), this is where the graph crosses the x-axis.

Key Takeaway: Concavity of \(f(x)\) is linked to the sign of \(f''(x)\) and the direction (inc/dec) of \(f'(x)\).

4. Master Summary Table

If you are feeling overwhelmed, use this table as your "cheat sheet" for translations:

If \(f(x)\) is... \(\rightarrow\) Then \(f'(x)\) is... \(\rightarrow\) Then \(f''(x)\) is...
1. Increasing \(\rightarrow\) Positive (above x-axis) \(\rightarrow\) (Not enough info)
2. Decreasing \(\rightarrow\) Negative (below x-axis) \(\rightarrow\) (Not enough info)
3. Concave Up \(\rightarrow\) Increasing \(\rightarrow\) Positive (above x-axis)
4. Concave Down \(\rightarrow\) Decreasing \(\rightarrow\) Negative (below x-axis)
5. Relative Extremum \(\rightarrow\) X-intercept (crossing x-axis) \(\rightarrow\) (Sign tells max/min)
6. Inflection Point \(\rightarrow\) Relative Extremum \(\rightarrow\) X-intercept (crossing x-axis)

5. Strategy for Sketching Graphs

When the AP exam asks you to sketch a graph or identify a correct graph, follow these steps:

How to sketch \(f(x)\) given \(f'(x)\):

Step 1: Identify X-intercepts of \(f'(x)\). These are the potential hills and valleys (critical points) of your function \(f(x)\). Mark them on your x-axis.
Step 2: Check Signs. Where \(f'(x)\) is above the x-axis, draw your function \(f(x)\) going up. Where it is below, draw \(f(x)\) going down.
Step 3: Identify Max/Min of \(f'(x)\). Where \(f'(x)\) has a "peak" or "valley," your function \(f(x)\) must have an inflection point (change in bend).
Step 4: Connect the dots smoothly. Note: Without a starting point (like \(f(0) = 2\)), you can shift your graph of \(f(x)\) up or down anywhere—the shape is what matters!

How to sketch \(f'(x)\) given \(f(x)\):

Step 1: Find the flat spots. Wherever \(f(x)\) has a local max or min, put an x-intercept on your \(f'(x)\) graph.
Step 2: Check the slope. If \(f(x)\) is getting steeper, your \(f'(x)\) graph should be moving away from the x-axis. If it is flattening out, \(f'(x)\) should be moving toward the x-axis.

6. Common Mistakes to Avoid

1. "The Peak Trap": Students often see a "peak" on the \(f'(x)\) graph and think it's a maximum on the \(f(x)\) graph. NO! A peak on \(f'(x)\) is an inflection point on \(f(x)\). A maximum on \(f(x)\) only happens when \(f'(x)\) crosses the x-axis from above to below.

2. Forgetting "Undefined": Critical points and inflection points can also happen where the derivative doesn't exist (sharp turns or vertical tangents). Always check for "holes" or "cusps" in the derivative graph.

3. Confusing Height and Slope: When looking at a graph, always ask yourself: "Am I looking at the function, the velocity, or the acceleration?" Height on one graph is slope on the graph above it.

Key Takeaway: Always justify your sketches using the phrases: "Since \(f'(x)\) changes from positive to negative at \(x = c\), \(f(x)\) has a relative maximum at \(x = c\)." or "Since \(f'(x)\) is increasing, \(f(x)\) is concave up." Using these logical links will guarantee you points on the Free Response Questions (FRQs)!