Need extra practice for your AP Calculus Unit 5 review? Outlined are the topics and derivative application practice problems aligned with College Board’s curriculum to study for a Unit 5 test.
As College Board outlines, the topics to review for Unit 5 are:
· Using the Mean Value Theorem
Mean Value Theorem (MVT) says that if a function f(x) is continuous over [a,b] and differentiable over (a,b), there there exists a point c, a<b<c, where
It is crucial for MVT free response questions on the AP Calculus exam to indicate that the hypotheses are met. That is, you must explicitly state the given function is continuous over [a,b] and differentiable over (a,b) before jumping in and showing the instantaneous rate of change, f’(c), equals the average rate of change, (f(b)-f(a))/(b-a). Most of the free response questions for MVT are given a table, like the 2018 problem below. Notice the problem stem says “twice-differentiable function H” so you can use that to show the hypotheses are satisfied:
· Extreme Value Theorem, Global vs. Local Extrema, and Critical Points
Extreme Value Theorem (EVT) says any continuous function on a closed interval will have an absolute or global maximum and minimum. Some functions also have relative or local extrema. Draw any curve with a relative max or min and you’ll notice the slope (derivative) at that point is 0 or undefined. These are called critical points. For more details or explanations on these theorems and Unit 5 topics, check out this study guide.
A critical point for f(x) occurs when f’(x) is zero or undefined. These are locations of possible relative maximums and minimums. We’ll determine if a critical point is a relative maximum or minimum in a later section.
2. If f’(x)=xcosx, what are the critical values in the interval [0, 2π)?
· Determining Intervals of Increasing and Decreasing
A function f(x) is increasing when f’(x)>0 and decreasing when f’(x)<0. See an example problem with a graph of f’ in the next section.
· Using the First Derivative Test to Determine Relative (Local) Extrema
A relative (local) minimum for f(x) occurs when f’(x) changes from negative to positive. A relative (local) maximum for f(x) occurs when f’(x) changes from positive to negative.
Since increasing, decreasing, and relative extrema are all based on f’(x), it may be helpful to make a sign chart for f’(x), especially when given different contexts like the graph below.
· Using the Candidates' Test to Determine Absolute (Global) Extrema
Repeating EVT, a continuous function will have an absolute (global) min and max on a closed interval. Sometimes the absolute extrema occur at a critical point and other times it occurs at the endpoint. I recommend the Candidates' Test to determine absolute extrema. The process of using the Candidates' Test is to take the derivative and find the critical points, where f’ is zero or undefined. Then make a table, with the critical points and endpoints (the candidates). Plug those x-values into f(x) to see the highest and lowest y-values, absolute max and min, respectively. I like the Candidates' Test because it is the only justification needed for an absolute extrema, unlike for relative extrema when you must explain in words how f’(x) is changing signs.
Something important to note: be careful to notice if the question asks where the absolute max/min occurs or what the absolute max/min value is. Where the absolute max/min occurs is looking for the x-value, typically, while the absolute max/min value is the y-value.
· Determining Concavity of Functions (and Points of Inflection)
Concavity and points of inflection are based on f’’(x). A function f(x) is concave up when f’’(x)>0 and concave down when f’’(x)<0. A point of inflection occurs when f’’(x) changes signs.
It may be helpful to make a sign chart for f’’(x) to answer questions about concavity and inflection points.
5. Given f’’(x)=x(x+4)(x-1)^2, then the graph of f(x) has inflection points when x=
(A) 0 only
(B) -4 only
(C) 0 and 1
(D) -4 and 0
(E) -4, 0, and 1
· Using the Second Derivative Test to Determine Extrema
Students often ask me if they need to know the second derivative test or if they can just use the first derivative test. Check out that post for the answer to that question, along with examples.
For the Second Derivative Test, the beginning process is the same as the First Derivative Test: find the critical points. The difference is, from there, you’ll want to find the second derivative f’’(x) for the Second Derivative Test. Plugging the critical points into f’’(x) will determine concavity. If f(x) is concave up at a critical point, that critical point is a relative minimum (draw a picture to visualize this). If f(x) is concave down at a critical point, that critical point is a relative maximum.
A sample justification for using the Second Derivative test needs to include both that the x-value is a critical point and its concavity: “f has a relative minimum at x=2 because f’(2)=0 and f’’(2)>0.”
· Sketching Graphs of Functions and their Derivatives
Some teachers over-emphasize this section, in my opinion. Based on the AP Calculus exam, this is not commonly assessed.
7. Sketch a graph of f’ given the graph of f(x) below:
· Connecting a Function, its First Derivative, and Second Derivative
From the released AP Calculus exams I've seen, opimization isn't on the AP exam. The questions are time consuming and if you can’t come up with the equations, it’s impossible to showcase your Calculus knowledge of applying the First Derivative Test. On your test, I would skip these problems until the end to ensure you have enough time for other questions. Even though you won't likely see optimization on the AP exam, these are relevant application problems if you're going into a STEM field!
9. What is the radius of the cylinder with the maximum value, given that the sum of the height and diameter is 18 inches?
· Exploring Behaviors of Implicit Relations
Implicit relationships are similar to functions, they just use dy/dx instead of f’(x). Take a look at the AP Calculus AB 2021 question below:
AP Calculus Unit 5 Review
For your AP Calculus Unit 5 review, make sure you are practicing derivative application problems from multiple perspectives: given functions, tables, graphs, and word problems. If you need extra practice, check out the 7 best resources to study for AP Calculus tests. These resources have questions that match the rigor of your tests.
If you need further explanation on how to approach some of these difficult Unit 5 review questions, especially when all of the topics are mixed together, consider joining Calculus Crew. I lead group sessions in Calculus Crew where we focus on solving my past test questions and incorporate previous AP exam questions. I also answer any questions students have. You’ll get more practice with problems given graphs, tables, and word problems to be prepared for in class tests and the AP exam.
To learn more about Calculus Crew and to join our thriving group, visit the Calculus Crew website today!
Solutions to the above problems: