Analytical Applications of Differentiation

Abigail Young
11 min read
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Study Guide Overview
This study guide covers applying differentiation, focusing on graphical analysis and key theorems. Topics include the Mean Value Theorem (MVT), Extreme Value Theorem (EVT), and finding local and global extrema using the First and Second Derivative Tests and the Candidates Test. Additionally, it covers determining function concavity, inflection points, sketching graphs of functions and their derivatives, and solving optimization problems.
Now that you know how to take basic derivatives, it’s time to take it up a notch and learn how to actually apply differentiation to different problems. Remember how AP Calculus is all about memorizing formulas? This still stands with these problems - the College Board will generally give you different variations of the same problem, so as long as you know how to solve these, you’ll be okay! 🙆♂️
#Graphical Analysis
Graphical analysis is a MAJOR AP TOPIC along with Calculus BC Integrals and Sequences and Series. You should expect at least one FRQ to be on Graphical Analysis along with several multiple-choice questions. 📊 This unit makes up 15-18% of the AB exam and 8-11% of the BC exam.
For the following theorems and tests, you need to know the conditions for each test as well as how to perform them.
#5.1 Using the Mean Value Theorem
The Mean Value Theorem (MVT)) is explained below.
Essentially, if the function is continuous on a closed interval (including the endpoints) and differentiable on the open interval (not necessarily including the endpoints), then there is a point within the interval where the slope of the tangent line (derivative) is equal to the average rate of change (secant line) between the endpoints of the interval.
It is important to note that the MVT applies only to functions that are both continuous and differentiable. Also, it only guarantees the existence of at least one point c satisfying the condition, but not that there is only one such point.
For example, here is a graph demonstrating the MVT with two points that satisfy the conditions.
#Image courtesy of Expii.
#5.2 Extreme Value Theorem, Global vs. Local Extrema, and Critical Points
The Extreme Value Theorem (EVT)) states that if a function f(x) is continuous on a closed interval [a, b], then the function must attain both a maximum and a minimum value on that interval.
The EVT guarantees the existence of at least one global maximum and one global minimum but does not say anything about the number of local extrema or the number of critical points. There could be more than one minimum and maximum.
Global/absolute extrema refer to the highest and lowest points of a function over the entire domain of the function, while local extrema refer to the highest and lowest points of a function over a specific subinterval of the domain. 📈
#Image courtesy of Xaktly.
A critical point of a function f(x) is a value c in the domain of the function such that either f'(c) = 0 or f'(c) does not exist. Critical points are important in finding the extrema of a function, as local extrema will always occur at critical points (where the...

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