Main content
AP®︎/College Calculus AB
Course: AP®︎/College Calculus AB > Unit 5
Lesson 10: Connecting a function, its first derivative, and its second derivative- Calculus-based justification for function increasing
- Justification using first derivative
- Justification using first derivative
- Justification using first derivative
- Inflection points from graphs of function & derivatives
- Justification using second derivative: inflection point
- Justification using second derivative: maximum point
- Justification using second derivative
- Justification using second derivative
- Connecting f, f', and f'' graphically
- Connecting f, f', and f'' graphically (another example)
- Connecting f, f', and f'' graphically
© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice
Justification using first derivative
Let's take a close look at how the behavior of a function is related to the behavior of its derivative. This type of reasoning is called "calculus-based reasoning." Learn how to apply it appropriately.
A derivative f, prime gives us all sorts of interesting information about the original function f. Let's take a look.
How f, prime tells us where f is increasing and decreasing
Recall that a function is increasing when, as the x-values increase, the function values also increase.
Graphically, this means that as we go to the right, the graph moves upwards. Similarly, a decreasing function moves downwards as we go to the right.
Now suppose we don't have the graph of f, but we do have the graph of its derivative, f, prime.
We can still tell when f increases or decreases, based on the sign of the derivative f, prime:
- The intervals where the derivative f, prime is start color #1fab54, start text, p, o, s, i, t, i, v, e, end text, end color #1fab54 (i.e. above the x-axis) are the intervals where the function f is start color #1fab54, start text, i, n, c, r, e, a, s, i, n, g, end text, end color #1fab54.
- The intervals where f, prime is start color #aa87ff, start text, n, e, g, a, t, i, v, e, end text, end color #aa87ff (i.e. below the x-axis) are the intervals where f is start color #aa87ff, start text, d, e, c, r, e, a, s, i, n, g, end text, end color #aa87ff.
When we justify the properties of a function based on its derivative, we are using calculus-based reasoning.
Common mistake: Not relating the graph of the derivative and its sign.
When working with the graph of the derivative, it's important to remember that these two facts are equivalent:
- f, prime, left parenthesis, x, right parenthesis, is less than, 0 at a certain point or interval.
- The graph of f, prime is below the x-axis at that point/interval.
(The same goes for f, prime, left parenthesis, x, right parenthesis, is greater than, 0 and being above the x-axis.)
How f, prime tells us where f has a relative minimum or maximum
In order for a function f to have a relative maximum at a certain point, it must increase before that point and decrease after that point.
At the maximum point itself, the function is neither increasing nor decreasing.
In the graph of the derivative f, prime, this means that the graph crosses the x-axis at the point, so the graph is above the x-axis before the point and below the x-axis after.
Common mistake: Confusing the relationship between the function and its derivative
As we saw, the sign of the derivative corresponds to the direction of the function. However, we can't make any justification based on any other kinds of behavior.
For example, the fact that the derivative is increasing doesn't mean the function is increasing (or positive). Furthermore, the fact that the derivative has a relative maximum or minimum at a certain x-value doesn't mean the function must have a relative maximum or minimum at that x-value.
Want more practice? Try this exercise.
Common mistake: Using obscure or non-specific language.
There are a lot of factors at play when we’re looking at the relationship between a function and its derivative: the function itself, that function’s derivative, the direction of the function, the sign of the derivative, etc. It's important to be extremely clear about what one is talking about at any given time.
For example, in Problem 4 above, the correct calculus-based justification for the fact that h increasing is that h, prime is positive, or above the x-axis. One of the students' justifications was "It's above the x-axis." The justification didn't specify what is above the x-axis: the graph of h? The graph of h, prime? Or maybe something else? Without being specific, such a justification cannot be accepted.
Want to join the conversation?
is there any use to knowing the justification?(2 votes)
To past the AP EXAM😇(24 votes)
I do not understand this statement "For example, the fact that the derivative is increasing doesn't mean the function is increasing (or positive)."(1 vote)
If the derivative is increasing, the function has to be increasing, but needn't be positive. For example, take the graph of x^2 - 4. In the interval [0,2], the derivative is increasing, the function is increasing, but the function is still negative.(1 vote)
