AP Calculus AB/BC

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The Riemann Sum is a way of approximating the area under a curve on a certain interval [a, b] developed by Bernhard Riemann. The way a Riemann sum works is that it approximates the area by summing up the area of rectangles and then finding the area as the number of rectangles increases to infinity with an infinitely thin width. The following gif may give you a better visual example:

Riemann sums are useful because if you're dealing with a complex function, the area underneath it can be approximated with just a few rectangles. As the number of rectangles increases, you can see that the accuracy of the approximation increases. A perfect Riemann sum has infinitely many rectangles with an infinitely thin width, so of course, an infinite Riemann sum cannot be summed up by hand (unless you want to sit there for a VERY long time, and by that I mean literally forever). The formal definition of a Riemann sum is:

Woah! That looks really really scary, doesn't it? Well, it's actually quite simple, so let's break it down. The large E looking symbol, if you are unfamiliar, is the Greek letter sigma and is used for summing together a bunch of elements or numbers. 

The use of sigma here makes sense since we're summing together a bunch of rectangles. Now, let's tackle the inside of the sum, the f(xk)*Δx. Remember, we're summing together the 

, so this must be the area of some rectangle. Recall that A = bh (or alternatively A = lw) for rectangles. So one of these must be the height of our rectangles, the other must be our width.

You can see in the image that Δx = (b-a)/n. What does this mean? Our area is being taken on the interval [a, b], so b-a is just the length of our interval. For example, if we were finding the area on the interval [3, 8], b - a = 5. Similarly, n is the number of rectangles we have (in a perfect Riemann sum, n approaches infinity). Therefore, (b-a)/n is the equivalent of saying we're taking our interval and chopping it up into n sized pieces on that interval. Here's a visual:

In this example, we are finding the sum from 0 to 8, so b - a = 8. We have n = 4 rectangles, so Δx = (b - a)/n = 8/4 = 2. This makes sense, we took a range of 8 and cut it into 4 

 rectangles (The rectangles in a Riemann sum will have the same width if you are just given a number of rectangles and an interval).

If Δx is the width of our rectangles, f(xk) must be the height! This makes pretty intuitive sense since, for some x value, f(x) tells you essentially the height of the graph at that x. What is k, though? k tells us essentially what rectangle we're on. So for example, at k = 1, we are looking at the first rectangle. Well, how do we find f(xk)? First, we must find xk itself. You'll see in the formula above that xk = a + kΔx. Let's think about this for a second. a is our initial value, our starting point on our interval. k is what rectangle we're on and Δx is the width of each rectangle. xk, therefore, is a way of calculating the x value of each rectangle. In this instance, we're looking at the right x value of each. If we start at k = 0, we have a 

So imagine you are given this equation: f(x) = x^2. Your interval is [0,5] and n = 5. For the purposes of simplicity, I am going to demonstrate a Left Riemann Sum.

1. Understand the information the question gives you

Equation: f(x) = x^2
Interval: [a,b] and in the case of this problem [1,5] - these are the parameters for the section under the curve that you are estimating
N: n=4 - this number determines how many rectangles you are splitting the section under the curve into. (Note: More rectangles = more precision, but also means more work)

2. Find Δx - the width of each of your rectangles

The top left (for a Left Riemann Sum) corner of each rectangle meets the curve (If you were to take a Right Riemann Sum you would use the top right corner of each rectangle and if you were to use a Midpoint Riemann Sum the height would be where the middle of each rectangle hit the curve)

Our heights for this specific problem would be 1, 4, 9, and 16 

4. Find the area of each rectangle and add them together

^In simplest terms, this equation will help you solve any Riemann Sum. Note that all the steps are the same for Right Riemann Sums except for #3. Just remember to use the top left corner of your rectangles for each Left Riemann Sum and the top right corner for each Right Riemann Sum.

For some quick background, when you use the areas of rectangles to estimate the area under a curve, that estimate is a Riemann Sum. When using this method you have two options: a Left Riemann Sum or a Right Riemann Sum. But what's the difference and when should you use each one?

Right Riemann Sum: The right corner of each rectangle touches the curve.

Left Riemann Sum: The left corner of each rectangle touches the curve.

In all honesty, there are no set restrictions that determine when to use each one. Neither method is strictly better than the other.  It is up to you to look at how the rectangles fit within the curve to determine whether a Right or Left Riemann Sum will provide you with a more accurate estimate, but for the most part, the AP exam will tell you which approximation to use.

👉 2007 AP Calculus AB Right Reimann Sum FRQ

2011 AP Calculus AB Trapezoidal Reimann Sum FRQ

2012 AP Calculus BC Midpoint Reimann Sum FRQ

You may have noticed earlier that we discussed whether or not a Riemann sum is an overapproximation or an underapproximation. This depends on two things: if the function is increasing or decreasing, and if we are using a righthand Riemann sum or a lefthand Riemann sum. If a function is increasing, lefthand is an underapproximation and righthand is an overapproximation and vice versa.

Riemann sums are an incredibly powerful tool when approximating areas, and they can even be used, when infinitesimal, to calculate exact areas. Approximating areas is useful, especially when you're dealing with a function where finding an anti-derivative is difficult or would take too long. Using Riemann sums in calculus opens up a whole new world of integration

With Reimann Sums, it is important to master all Applications of Integration.

AP Calculus: Riemann Sums

I’ll be honest with you - AP Calculus AB is one of the most difficult courses out there. Lucky for you, I'm here to share my tips and tricks on how to succeed on the AP exam - from solving complex, multi-step problems to understanding the wording of problems on the test.

Before you walk into your first class, you should know what the AP Calculus AB exam looks like. It helps to review 

 to look at the time breakdown of each section.

Go through the questions and answers of different styles of FRQs. Although you may not know how to solve them yet, it helps to see what types of responses graders are looking for so you can start working on your FRQ skills from day 1.

The wording of FRQs is often very specific (ex: "increasing" v. "positive" slope), so it's important to pay close attention to the way problems are written from the beginning.

Remember: on the calculator section, your calculator can work out long and complicated derivatives and integrals for you in a matter of seconds, so don't waste your time solving them by hand!

Of course, it also helps to know some of the actual content of the exam before the first day of class.

It helps to already know the graphs of some basic parent functions (ex: y=x, y=logx, etc.) as well as the trig functions (y=sinx, y=cosx, y=tanx, and their inverses and reciprocals). Already having these memorized will help in understanding(and visualizing, if you're more of a visual learner) their derivatives and integrals later on in the course.

Know your basic log rules and exponent rules by heart. It saves time on the exam by not having to work everything out the long way.

The rules and theorems of algebra are employed in almost every topic, so it's important to memorize them early on so you can implement them more easily. Being fluent in algebra is almost a requirement for succeeding in calculus

Make sure to understand the "why" behind each step in solving a problem. Simply memorizing is easy but NOT beneficial in the long run.

Live Stream Replay: Interpreting the Meaning of Derivative/Integral

In general, you should be working out AP-style problems throughout the entire year. As soon as you've covered a topic, find some AP FRQ practice problems and answers either on the College Board website (linked above) or in a prep book (below). This will help you get used to the wording of the test and focus your attention on what concepts you need more help in.

(Greenhall Publishing) are excellent resources. Both have step-by-step problems and solutions for all the concepts that are covered in the AP curriculum. The Princeton Review releases a new edition every year, but I've found that you can buy older editions for a much lower price than the current year's (after all, the structure of the exam hasn't been changed in quite some time).

*Remember: more students are seeing L'Hospital's rule than ever before, so make sure to look at previous FRQ's to see what College Board is and isn't testing.*

Khan Academy online has a great database of step-by-step problems that you'll almost definitely see on the test

 on YouTube. He offers reviews and helpful tricks for both Algebra and Calculus concepts, plus videos of him working out the steps of some trickier derivatives and integrals.

Most importantly, ask questions if you're confused about anything. If you don't understand the basics of a certain topic, it will be much more difficult for you to grasp the more complicated manifestations of that basic topic. Ask your teacher for help after class, find a friend to study with you at lunch, or come chat with us here at Fiveable!

Now that you know what lies ahead of you in your calculus class, you need no longer fear the AP Calculus AB exam. I know it seems daunting, but don't stress yourself too much about it now - go enjoy the rest of your summer while it's still here. When school starts back up again, you can be sure that you have all the skills and resources you need to succeed.

What to Know Before Taking AP Calculus AB

Watch AP Calculus teacher Jamil Siddiqui, the Massachusetts Teacher of the Year 2019, explain graphical limits

 of a function as it approaches a certain value which is especially important in calculus. In mathematical terms, the limit is asking the question "What value is 'y' getting close to as 'x' approaches a number?" and its represented by the expression:

Out loud, this would sound something like "the limit of f(x) as x approaches c"

The 'lim' shows that we're finding the limit and not the value of the function

The 'x →c' tells us the value that x is getting closer and closer to but never actually reaching

The 'f(x)' is meant to represent the functions you're working with

Study Guide: Determining Procedures for Determining Limits

1.) The limit of the function approaching from the right exists

Graphically, this would look like following a function to a certain point on the x-axis from the right

2.) The limit of the function approaching from the left exists

Graphically, this would look like following a function to a certain point on the x-axis from the left

3.) The limit from the right equals the limit from the left ​​

Study Guide: Determining Limits using the Squeeze Theorem

A limit MUST meet all three criteria in order for it to exist, which is important to remember especially when FRQ's come into play. If the limit from the left and the limit from the right are 

 equal, then the limit does not exist. An example of this would look something like this graph:

Here we can see that as we follow the function from the right hand side, it's going towards y = 2. On the left hand side, we see that the function is approaching y = 1. Since 2 ≄ 1, the limit does not exist at x = 1.

Study Guide: Exploring Types of Discontinuities

Watch AP Calculus teacher Suzanne Ferrell-Locke explain limits @ infinity.

​This is when the degree of the polynomial on the top of the function is greater than the degree of the polynomial in the denominator In this case, the answer you are finding is all dependent on whether or not you're being asked to be approaching negative or positive infinity. If you're approaching positive infinity, then the limit of a top-heavy function will also be infinity and vice versa.

This is when the degrees of the numerator and denominator are the same. To find the limit here, you'll need to divide the leading coefficients of the numerator and denominator by each other.

For example, here we see that the degrees of the top and bottom are 3 and that the leading coefficient of the numerator is 5 and 2 for the denominator.  Therefore, the limit of f(x) as x approaches positive infinity is 5/2.

This is when the degree of the denominator is greater than that of the numerator. If a function is bottom heavy, then the limit is zero, regardless if it is approaching positive or negative infinity

📌Here are some extra resources to help you out on limits approaching infinity! 

What is a Limit?

AP Calculus AB/BC Exams

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Unit 1: Limits & Continuity﻿

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Unit 3: Differentiation: Composite, Implicit & Inverse Functions﻿

Unit 4: Contextual Applications of the Differentiation﻿

Unit 5: Analytical Applications of Differentiation﻿

Unit 6: Integration and Accumulation of Change﻿

Unit 7: Differential Equations﻿

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Unit 10: Infinite Sequences and Series (BC Only)﻿

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Live Cram Sessions 2020

Free Review 2020

Free Response Questions (FRQ)

Unit 1: Limits & Continuity

Unit 2: Differentiation: Definition & Fundamental Properties

Unit 3: Differentiation: Composite, Implicit & Inverse Functions

Unit 4: Contextual Applications of the Differentiation

Unit 5: Analytical Applications of Differentiation

Unit 6: Integration and Accumulation of Change

Unit 7: Differential Equations

Unit 8: Applications of Integration

Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)

Unit 10: Infinite Sequences and Series (BC Only)

Multiple Choice Questions (MCQ)

Blogs

Jenni MacLean

Jerry Kosoff

Jamil Siddiqui

Meghan Dwyer

Suzanne Ferrell-Locke

Jennifer Durost

Bob Amar

Amy Dunn-Ruiz

Mark Howell

Bryan Passwater

Peter Cao

Jed Quiaoit

Sander Owens

Brandon Wu

Catherine Liu

Jillian Holbrook

Jacob Jeffries

Sumi Vora

Anusha Tekumulla

John Le

Nick Molnar

Leen Amro