Calculus I (Increasing/Decreasing)
Objectives:
(1) Be able to identify when a function is increasing/decreasing.
(2) Understand the relationship between when functions are increasing and decreasing and the slopes of tangent lines.
Requirements:
(1) Read through the following outline of ideas discussed in class.
(2) Consider what changes can be made or what other materials/resources can be included to better help explain the ideas present.
(3) Make changes to the lesson below that help to improve the quality of the content provided (be sure to provide appropriate citations for all external sources).
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4.5 Functions; Increasing and Decreasing
Prior to Calculus we are often introduced to the ideas of what it means for a function to be increasing or decreasing. However, these prior notions were often loosely defined and relied on a very intuitive approach to understanding these pieces of terminology.
General Idea: Identify the intervals on which the function pictured below is increasing, decreasing, and constant.
- This function is increasing on the intervals (-1,1) & (4,5).
- This function is decreasing on the interval (1,4).
- This function is constant on the interval (5,7).
Note: It would also be correct to say the function is increasing on [-1,1] and [4,5], decreasing on [1,4], and constant on [4,5]. This is dependent on how one defines these terms. These notes will always use open intervals when describing these properties.
We notice that at points where the function is decreasing the slopes of the tangent lines to those points are negative. Similarly, at points where the function is increasing the slopes of the tangent lines to those points are positive. We can also see this in the diagrams below, and this observation is summarized by the following theorem.
| Decreasing = negative tangent slopes | Increasing = positive tangent slopes |
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| Theorem |
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Let f(x) be a function that is continuous on [a,b] and differentiable on (a,b). Then,
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Example 1: Determine on what intervals f(x)=x3+3x2 is increasing and decreasing.
We calculate that f'(x)=3x2+6x. To make it easier to see when f'(x)>0 and f'(x)<0 we will find when f'(x)=0.
We clearly see that f'(x)=0 when 3x2+6x=0 → 3x(x+2)=0 → x=0 or x=-2.
We set up the following number line and using test values we find when the derivative is positive and negative.
Based on the number line we conclude that:
f(x) is increasing on (-∞,-2) and (0,∞).
f(x) is decreasing on (-2,0).
Example 2 (you try): A drug company is testing a drug that is supposed to help alleviate nausea symptoms. The drug is quite potent and the company wants to fully understand the effects experienced by the drug after it is ingested. The drug enters the blood and dissipates from it at a rate according to the function Q(t)=t/(t2+1) where Q measures the quantity of the drug in the system and t is the number of hours since the drug was ingested. Determine at what point in time at which the total amount of drug in the system begins to decrease.
We calculate that Q'(t)=[(t2+1)-t(2t)]/(t2+1)2 = (1-t2)/(t2+1)2 = (1-t)(1+t)/(t2+1)2.
We clearly see that Q'(t)=0 when t=1 or t=-1.
Q'(t) is never undefined.
We really only need to consider t=1 since t=-1 is not a valid value for time.
Based on the number line we conclude that the amount of drug in the bloodstream begins to decrease after 1 hour.