Main content
Course: Calculus 1 > Unit 5
Lesson 10: Connecting f, f', and f''- 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
© 2024 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 gives us all sorts of interesting information about the original function . Let's take a look.
How tells us where is increasing and decreasing
Recall that a function is increasing when, as the -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 , but we do have the graph of its derivative, .
We can still tell when increases or decreases, based on the sign of the derivative :
- The intervals where the derivative
is (i.e. above the -axis) are the intervals where the function is . - The intervals where
is (i.e. below the -axis) are the intervals where is .
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:
at a certain point or interval.- The graph of
is below the -axis at that point/interval.
(The same goes for and being above the -axis.)
How tells us where has a relative minimum or maximum
In order for a function 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 , this means that the graph crosses the -axis at the point, so the graph is above the -axis before the point and below the -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 -value doesn't mean the function must have a relative maximum or minimum at that -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 increasing is that is positive, or above the -axis. One of the students' justifications was "It's above the -axis." The justification didn't specify what is above the -axis: the graph of ? The graph of ? 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?(3 votes)
- To past the AP EXAM😇(47 votes)
- What does it mean when f' of a certain value doesn't exist?(2 votes)
- If f’ does not exist at a certain point, then the function is not differentiable at that point. This could mean that there is a discontinuity at that point or maybe there is a “cusp” or sharp turn in the graph.(1 vote)
- 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)
- That other answer is incorrect. As per the quoted statement, there is no direct causation between the derivative increasing and either the function, itself, being positive OR increasing. Recall that the derivative is the slope at any given instant/point (ie, where the function is going): essentially that quote is stating that where the function is going is a separate matter from where the function is currently at. Say a continuous and differentiable function is approaching a local minimum between outputs f(x) = -1 and f(x) = -3; somewhere therein the function's derivative would be increasing (as the function as about to turn around -- ie: attain a derivative of 0,) while the function itself is both negative (below the x-axis,) AND decreasing (approaching the local minimum.)(3 votes)