New Format Online AP Calculus Study Guide (AB+BC) – 2020 Changes

Published on Mar 26, 2020
👋🏼 Hi, I’m Jamil Siddiqui from East Bridgewater, MA! I’m an AP Calculus teacher and streamer at Fiveable. This year’s exam is different than we expected, but I’m here to help. I’ve put together this study guide to help keep you on track while you are studying from home. You can follow this guide on your own with a free Fiveable account! I’ll also be joining a group of students live on Thursdays @ 7pm ET during cram sessions. Pick up your cram pass to join us.

Also, make sure to study with the Fiveable Official AP Calc Formula Sheet created by Sumi Vora!

Format of the New Exam Calc AB Exam

This year, the AP Calculus AB exam will look different than you were expecting. As we’re all on quarantine 😷 due to COVID-19, the College Board has decided to update the format and content of the test to fit an online testing format.

You’ll have 45-minutes to take the exam online and it will only cover units 1-7. If you have already studied content from unit 8, don’t stress! It’s all worth knowing. 

These units are on the AB exam. 

1 – 👑 Limits & Continuity
2 – 🤓 Differentiation: Definition and Basic Derivative Rules
3 – 🤙🏽 Differentiation: Composite, Implicit & Inverse Functions
4 – 👀 Contextual Applications of Differentiation
5 – ✨ Analytical Applications of Differentiation
6 – 🔥 Integration and Accumulation of Change
7 – 💎 Differential Equations

Not on the exam:

8 – 🐶 Applications of Integration

What will be on the test?

  • Q1: multi-focus free response
    • 60% of your score, 25 minutes to answer
    • Will cover 2 or more of the eligible units
  • Q2: multi-focus free response
    • 40% of your score, 15 minutes to answer
  • Other info
    • “Keyboarding tip sheet” will be provided with suggestions on how to type mathematical symbols
    • Calculators are allowed, but not required.
    • You’re advised to submit “simplified” answers

When is the exam and how do I take it?

May 12 @ 2p Eastern! Wherever you are in the world, this is the time you’ll take the test. Unless you have been approved for the make-up date in June, but only your school can request that. You’ll take the test online. There will be a practice simulation posted by College Board within the next few weeks.

Format of the New Exam Calc BC Exam

You’ll have 45-minutes to take the exam online and it will only cover units 1-8 and 5 topics in unit 10 (10.2, 10.5, 10.7, 10.8, and 10.11). If you have already studied content from units 9-10, don’t stress! It’s all worth knowing. 

What will be on the test?

  • Q1: multi-focus free response
    • 60% of your score, 25 minutes to answer
    • Will cover 2 or more of the eligible units
  • Q2: multi-focus free response
    • 40% of your score, 15 minutes to answer
  • Other info
    • “Keyboarding tip sheet” will be provided with suggestions on how to type mathematical symbols
    • Calculators are allowed, but not required.
    • You’re advised to submit “simplified” answers

When is the exam and how do I take it?

May 12 @ 2p Eastern! Wherever you are in the world, this is the time you’ll take the test. Unless you have been approved for the make-up date in June, but only your school can request that. You’ll take the test online. There will be a practice simulation posted by College Board within the next few weeks.

How do I prepare for the exam?

With so many school closures and the stress of a global pandemic, this review season will be different than usual. If this is your first AP exam, welcome! Don’t worry, it’s not usually this chaotic. 

We’ve put together this plan for you to follow between now and May. This will cover all of the units and leave you time to practice questions before test day. Some classes may have done units out of chronological order throughout the year, which is ok. The units don’t have to be taught in order. If you are learning new material on your own and need some help, use the chat bubble on http://hi.fiveable.me. We’ll answer any questions you may have. 

What resources does this study plan use?

All of the required resources are free. You’ll need to create a free Fiveable account to jump in.  We’ve also linked a few other websites, articles, and YouTube videos that you can access for free. Some of the suggested resources include paid products. There are some documentaries that you can find on streaming sites with a paid membership and we’ll also list streams and practice questions that require a paid cram pass on Fiveable.

PRE-WORK: SET-UP YOUR STUDY ENVIRONMENT

Before we begin, take some time to get organized. Remote learning can be great, but it also means you’ll need to hold yourself accountable more than usual. 

🖥 Create a study space.
Make sure you have a designated place at home to study. Somewhere you can keep all of your materials, where you can focus on learning, and where you are comfortable. Spend some time prepping the space with everything you need and you can even let others in the family know that this is your study space. 

📚 Organize your study materials.
Get your notebook, textbook, prep books, or whatever other physical materials you have. Also create a space for you to keep track of review. Start a new section in your notebook to take notes or start a Google Doc to keep track of your notes. Get your self set up!

📅 Plan designated times for studying.
The hardest part about studying from home is sticking to a routine. Decide on one hour every day that you can dedicate to studying. This can be any time of the day, whatever works best for you. Set a timer on your phone for that time and really try to stick to it. The routine will help you stay on track.

🏆 Decide on an accountability plan.
How will you hold yourself accountable to this study plan? You may or may not have a teacher or rules set up to help you stay on track, so you need to set some for yourself. First set your goal. This could be studying for x number of hours or getting through a unit. Then, create a reward for yourself. If you reach your goal, then x. This will help stay focused!

 


 

UNIT 1: Limits and Continuity

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Big takeaways:

Unit 1 is the basic idea of all of Calculus. The limit is the concept that makes everything click. You ask someone what they learned in calculus and they will most likely answer “derivatives and integrals”. Limits help us to understand what is happening to a function as we approach a specific point. Limits can be one or two sided, but the sides have to match in order for the limit from both directions to exist! Well without the limit we wouldn’t have either. A major concept used throughout curriculum and within theorems is continuity. We prove continuity using limits and learn how to do that within this unit. We also learn about the Intermediate Value Theorem and the Squeeze Theorem, although this topic most likely won’t be directly tested on this year’s exam, it’s the bread and butter of what’s to come. 

Content to Focus On:

  • Methods of Finding Limits
    • From a table
      • Be able to estimate values not given in a table by using values that were given in a table.
    • From a graph
      • Be able to use a graph to interpret a limit at an x value.
    • Algebraically
      • Limit Properties
      • Rationalization
      • Some limits look unsolvable, (00 or ), but using algebraic manipulation, you can solve them! In unit 4 we learn about another helpful tool for this, L’Hospital’s rule!
    • Infinite Limits
  • Squeeze Theorem
    • The squeeze theorem helps us to find limits of functions that we do not know, by using the limits of a function that is greater than or equal to and a function that is less than or equal to. If we can find the limits of those two functions and they are equal, then our function should have that limit too!
  • Continuity and Discontinuity
    • Students should be able to prove if a function is continuous as a point, and know the different types of discontinuity! Removable, jump, infinite.
  • First existence theorem:
    • Intermediate Value Theorem
      • This theorem helps us to prove points exist given certain information. Don’t forget to always state or prove the conditions! In this case, it would be that the function is continuous on [a,b].

Resources to use:

 

AP Calculus Unit 1 Formulas & Theorems


 

UNIT 2: Differentiation: Definition and Basic Derivative Rules

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Big takeaways:

Unit 2 introduces the first of the 2 major halves of calculus: differentiation, or the instantaneous rate of change of a function. We start off with defining the derivative and applying it to our old friend, the limit. This unit also sets up the rules so that you can figure out the derivative of most simple functions you will find on the FRQ section! Derivatives help us determine instantaneous rates of change. We will discuss the difference between average and instantaneous rates of change and how they appear differently. One thing is for sure, you should still follow order of operations!

Content to Focus On:

  • Average Rate of Change vs Instantaneous Rate of Change
    • If we apply a limit to an average rate of change, we are looking at an instantaneous rate of change!
  • Definition of the Derivative
    • A long time ago, before we learned the power rule, etc, for the derivative, we learned how to solve instantaneous rate of change using the limit definition of a derivative.
  • Estimating The Derivative
  • Connecting differentiability and continuity. Remember, differentiability implies continuity, but not the other way around! Make sure you know how to tell if a function is differentiable. No cusps or corners, polynomials are always differentiable!
  • Finding Simple Derivatives
    • Power Rule
    • Derivative Properties
    • Trigonometric Derivatives
    • Exponential and Logarithmic Derivatives
      • **While this year it may not be necessary to memorize certain derivatives like natural log or tangent, using time out of your exam time to look them up can be costly! Make sure you are still on your game when it comes to the derivatives you need to know.
  • Product and Quotient Rules

Resources to use:

 

AP Calculus Unit 2 Formulas & Theorems


 

UNIT 3: Differentiation: Composite, Implicit & Inverse Functions

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Big takeaways:

In Unit 3, we expand on the methods of finding derivatives from Unit 2 in order to evaluate any derivative that Collegeboard can throw at you! This section includes the very important chain rule. Implicit differentiation is a big take away from this unit. It allows us to find derivatives of any variable when we may be finding the derivative with respect to something else.

Content to Focus On:

  • Chain Rule
    • Chain rule allows us to differentiate composite functions. But make sure you see every function! Sometimes chain rule can be more than one chain! For example: (sin(x2))2. We have 3 functions to consider here!
  • Implicit Differentiation
    • Implicit differentiation gives us the tools to derive any variable with respect to any variable. The main concept is that if the variable you are deriving is not the same as the variable you are taking the derivative of, chain rule applies and you must also multiply by the derivative of that variable. For example, the derivative of y2with respect to x is 2ydydx
  • Derivatives of Inverse Functions (Including Trigonometric Functions)
    • Remember how to find an inverse? Switch x and y and solve for y? Well then take the derivative of that! or follow the formula we have! The important part to remember on problems like this is the x and y values. If you are supposed to be using the x value of an inverse function, that means it was the y value of your original function!

Resources to use:

 

AP Calculus Unit 3 Formulas & Theorems


 

UNIT 4: Contextual Applications of Differentiation

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Big takeaways:

Unit 4 allows you to apply the derivative in different contexts, most of which have to do with different rates of change. You will also learn how to solve problems containing multiple rates, how to estimate values of functions, and how to find some limits you may not have known how to solve before! Related rates is a very popular topic in the FRQ section of the exam. It is very important you label every variable you use! Limits you may not have been able to know how to solve before you can now solve using L’Hospital’s rule, but make sure you know the requirements!

Content to Focus On:

  • Rates of Change
    • Position, Velocity, Acceleration
      • Velocity is the derivative of position, and acceleration is the derivative of velocity. Velocity has a direction. If the velocity and acceleration match signs, the particle is speeding up. If their signs are opposite, the particle is slowing down. Speed is the absolute value of velocity and has not direction. On the AP test, you can be asked to map out the path of a particle, which direction it is going and if it is speeding up or slowing down.
    • Related Rates
      • Related Rates have to do with nothing other than relating rates! There are a set of steps to follow when solving a related rate: 
      • 1. Assign letters to quantities and label their derivatives with respect to time. If you have A= area, then dA/dt would be the rate of change of the area.
  1. Identify the rates that are known and the rate that needs to be found.
  2. Find an equation that relates the variables whose rates of change you need in step 2. Draw a picture to help you find this equation!
  3. Differentiate the equation with respect to time. Remember it is necessary to think about this as implicit differentiation.
  4.  Substitute in all known values and variables, and solve for the unknown rate of change.
      • One special type of related rate is that of a cone. You should practice a related rate problem with a cone, because it is necessary to write r, the radius, in terms of h before differentiating.
  • Linearization and Tangent Line Approximations
    • In these types of problems, we use a tangent line approximation for one value that we know, and use it to approximate another very close value. Let’s day, x0 is the one we know about, and x1 is the one we need. Then we would use: f(x1)-f(x0)=f'(x0)(x1x0) to find f(x1).
  • L’Hopital’s Rule
    • When using L’Hospital’s rule, we have a few things we need to remember! To apply L’Hospital’s rule, your limit needs to be in one of two forms: 00 or . To show this in a free response question, you need to show the limits in the numerator and the denominator go to either 0 or infinity separate from each other, then you can use L’Hospital’s rule, which says: xaf(x)g(x)=xaf'(x)g'(x)

Resources to use:

  • 🎥 Watch these videos:
  • 📖 Read these study guides (Emphasize 4.2, 4.5-4.7)

 

AP Calculus Unit 4 Formulas & Theorems


 

UNIT 5: Analytical Applications of Differentiation

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Big takeaways:

Unit 5 continues our discussion on the applications of derivatives, this time looking at graphs and how the value of the first and second derivatives of a graph influences its behavior. We will review two of the three existence theorems, the mean and extreme value theorems. We’ll also learn how to solve another type of problem commonly seen in the real world (and also on FRQ problems): optimization problems.

Content to Focus On:

  • Two of the three Existence Theorems
      • It is important when using theorems to make sure their conditions are met before you use them!
    • Mean Value Theorem
      • In order to use the Mean Value Theorem, the functions must be continuous on a closed interval and differentiable on that same interval, but open. Once you can say for sure that these things are true, the mean value theorem tells us that there must be a point in that interval where the average rate of change equals the instantaneous rate of change, or the derivative, at a point within the interval. f'(c)=f(b)-f(a)b-a.
    • Extreme Value Theorem
      • For this existence theorem, the condition is that the function continuous on a finite closed interval. If it is, that means there must be an absolute maximum and an absolute minimum.
  • Increasing/Decreasing Functions and The First Derivative
    • If the derivative of a function is positive, then the function is increasing! If the derivative is negative, then the function is decreasing!
  • Local and Global Extrema
    • First and Second Derivative Tests for Local Extrema
    • Candidates Test for Global Extrema
  • Concavity, Inflection Points, and The Second Derivatives
    • If the second derivative is positive, the function will be concave up. If the second derivative is negative, the function will be concave down. Whenever the second derivative changes sign, there will be an inflection point, a change in concavity, on the original function.
  • Curve/Derivative Sketching
    • All of the information you learned from how the first and second derivative relate to the original function can help you to sketch graphs based on the information you have about their derivatives!
  • Optimization Problems
    • These are also known as applied maximum and minimum problems. It is problems about finding an absolute maximum and an absolute minimum in an applied situation. When trying to find an absolute max or min, you must find all critical points (f'(x)=0 or undefined). Then plug those and the endpoints into the original function to find out which has the highest or lowest value, depending on what you are looking for.

Resources to use:

 

AP Calculus Unit 5 Formulas & Theorems


 

UNIT 6: Integration and Accumulation of Change

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Big takeaways:

Unit 6 introduces us to the integral! We will learn about the integral first as terms of an area and Reimann sums, then working into the fundamental theorem of calculus and the integrals relationship with the derivative. We will learn about the definite integral and the indefinite integral. We learned about methods of integration from basic rules to substitution. In BC, there is also the methods of integration by parts, partial fractions, and improper integrals. In some cases the integral is called the antiderivative, and if f is the function, then F would be the antiderivative of that function.

Content to Focus On:

  • Approximating an integral as an area
    • When given a graph, we can find the area under the curve and between the x axis to find the integral. However, if an area is underneath the x-axis, that would be considered negative.
  • Riemann Sums
    • A way of evaluating an area under a curve by making rectangles to find the area of an adding them up. This is useful with a table of values or with a curve that we do not know the exact area of its shape. There are four kinds: Left, Right, Midpoint, and Trapezoidal sum. Depending on the function, these may be over or under estimates of the actual area. 
    • Students should also be able to write a Reimann sum in summation notation and integral notation.
  • The Fundamental Theorem of Calculus
    • This theorem tells us two very important things: 
      • abf(x)dx=F(b)-F(a)
      • ddx(axf(t)dt)=f(x), where a is a constant.
    • We use this theorem very often when working with integrals. 
  • Definite vs. Indefinite Integrals
    • Integrals must always include a dx (if not x, whichever variable you are using) at the end. 
    • If an integral is indefinite, it has no bounds. This means it cannot be evaluated using the fundamental theorem of calculus. Instead, we add on a +C, to make it know that there could have been a constant at the end of the function, and that there are lots of possibilities for what that constant would be. 
    • Definite integrals have bounds and we use the fundamental theorem of calculus often with them. The bounds let us know the endpoints of the integral, or in terms of a graph, what two points we are finding the area under the curve between. 
  • Integral Rules
    • Make sure you remember how to integrate and its connection to deriving!
    • You can check an integral by taking its derivative to see if you get back to where you started. 
  • U-Substitution
    • When you have a composition of functions in an integral, it is necessary to use u-substitution to make sure you are integrating each part. It is usually the inside function which is chosen. Sometimes it can be hard to tell. You will know once you try! Idf you find yourself going in circles and needing to substitute more, I would try using a different piece of the function as u. 

BC Only

  • Integrating using integration by parts
    • Integration by parts you can use when you are multiplying two functions, it looks like this:
      • udv=uv-vdu.
    • When trying to figure out which function to prioritize as u, make sure to go to this helpful order:
      • LIPET (Logs, Inverse Trig, Polynomials, Exponentials, Trig). Whichever one appears first, use that for u, then use the other for dv. Find du and v, and plug in from there!
  • Integrating using partial fraction decomposition
    • This is helpful when dealing with a rational function in your integral. The denominator should be a higher power than the numerator to use this. It helps turn terms into sums of ratios of linear non repeating decimals that are integrable. 
  • Evaluating Improper Integrals
    • An improper integral has one or both limits at infinity. In order to integrate these, we denote that infinite bound as a variable, a, and take the limit as a approaches infinity of the integral of our function. The fundamental theorem of calculus really helps us here!

Resources to use:

 

AP Calculus Unit 6 Formulas & Theorems


 

UNIT 7: Differential Equations

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Big takeaways:

Unit 7 takes us to a great connection between integrals and derivatives in differential equations. We will learn how to model and solve differential equations using initial conditions. This always requires separation of variables when it comes to free response questions! We will also learn how to identify differential equations from their slope fields and how to draw those fields as well. BC Calculus adds in Euler’s method and logistic models in this unit. 

Content to Focus On:

  • Differential Equations
    • Modeling
      • If you are given information, can you write a differential equation from it? 
    • Verifying Solutions
      • Being able to plug in given information and deduce if the solution is true. 
    • Separation of variables
      • This appears often in the FRQ section. Students are given a differential equation and need to put all of the terms for one variable on one side and for the other variable on the other, then they integrate both sides and solve for y, usually y, but really whichever the dependent variable it. 
      • NOTE: If you skip the step of separating variables on AP exams in the past, they have offered you no credit for the rest of that section of the problem. 
    • Using initial conditions
      • Once you have done your separation of variables, don’t forget to have a +C! Ths is where the initial condition comes in. You plug in the terms given from x and y in your initial condition, then you solve for C, rewriting the function at the end with C plugged in!
  • Slope fields
    • Slope fields are coordinate planes of 1 by 1 sections of slopes at each x and y value. If given a differential equation and asked to find a slope field, you plug in an x and y pair into the differential equation, the value it outputs is the slope at the point, and is the steepness you use to draw a line at that particular pair. 

BC Only

  • Euler’s Method
    • This is another method for approximating a solution to a differential equations or a point on a solution curve. 
  • Logistic Models with Differential Equations
    • This is a joint proportional differential equation. dydt=ky(a-y). You can interpret the initial condition and the differential equation without solving the differential equation. 
    • The limiting value, also known as the carrying capacity can be determined using a logistic growth model. 
    • The logistic growth model can also be used to determine the value of the dependent variable in the logistic differential equation at the point when it is changing the fastest. 

Resources to use:

 

AP Calculus Unit 7 Formulas & Theorems


 

UNIT 8: Applications of Integration (BC Only)

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Big takeaways:

Unit 8 teaches us more useful applications of the integral as we enter the 3D graph world! We learn how to find the average value of a function and the particle motion according to an integral. BC Calculus also adds in the arc length here! We also use cross sections and disks and washers to find volume of shapes as functions are revolved around either a vertical or horizontal line. We put our spatial reasoning and geometry skills to use in this section. 

Content to Focus On:

  • Average Value of a function
    • The average value of a function is denoted by 1b-aabf(x)dx. We can only find the average value of a function that we have a definite integral for. 
  • Position, Velocity, Acceleration
    • Position is the integral of velocity, and velocity is the integral of acceleration! If you take the absolute value of velocity, you are able to find speed. 
    • Distance, the integral of the absolute value of velocity
    • Displacement, the integral of velocity
      • To remember this, I think about a track! After one lap around, your distance is 400m, but your displacement is 0. Make sure if you are asked for distance, you remember to use absolute value!
  • Finding area between curves
    • If given two curves on a coordinate plane and endpoints, students should be able to integrate those functions, either with respect to the x or y axis, in order to determine the area between them. When talking about the x axis, all functions should be in terms of x, the bounds should be in terms of x, and it will be the integral of the upper functions minus the lower. When talking about with respect to the y axis, all functions should be in terms of y, the bounds should be in terms of y, and it will be an integral of an outer function minus the inner.
  • Finding the area between curves that intersect at more than two points. 
    • Sometimes, your function will have different parts where you may need to split the area and add your answers together, because the upper functions changes, or another function changes. 
    • When in doubt, break it up into pieces you know how to work with, and add those together. 
    • Area under the x axis is already accounted for as negative when evaluating using an integral, so don’t worry about that!
  • Volumes of cross sections
    • When using cross sections, you integrate the area of whatever shape is given to you (this could be a square, rectangle, triangle, or semicircle). 
    • You will integrate the area on a given interval, but it is important that you interpret the function for the variable in the shape that it represents. 
    • If it says perpendicular to the x-axis, all should be in terms of x.
    • Perpendicular to the y-axis, all should be in terms of y. 
  • Volumes by disks and washers
    • In this case we are integrating an area again, but this time the shape will always be a circle, so we will always be using the area formula for a circle. The variable that changes here is r, the radius, which you can interpret using the function. 
    • If the rotation is around just the x axis, all should be in terms of x (bounds and functions) and the radius will be your function. 
    • If the rotation is around just the y axis, all should be in terms of y (bounds and functions) and the radius will be your function.
    • If it is rotated around any other vertical or horizontal line, you may need to add or subtract value from your function to find the radius. 
    • A washer is when two functions are used, and you must subtract out the volume the inner function would have added into your total volume to account for the missing piece. 

BC Only

  • Arc Length
    • BC Students are also required to know how to find arc length of a smooth, planar curve and distance traveled. 
    • A definite integral can be used to calculate this. s(x)=ax1+[f'(t)]2dt

Resources to use:


 

UNIT 10: Infinite Sequences and Series (BC Only)

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Big takeaways:

Unit 10 is only BC and covers many topics under the umbrella of sequences and series. We learn how series can diverge or diverge and how to know. We learn many different types of convergence tests along with many different types of series! This year some sections have been taken out to account for what you may have learned so far in class. We outline those specific sections to focus on below. 

Content to Focus On:

  • 10.2: Working with geometric series
    • A geometric series has a constant ratio between successive terms. 
    • n=0arn=a1-r, and r>0
    • If r<1, then the series converges, but diverges otherwise. 
  • 10.5 Harmonic Series and p-series
    • This section includes the harmonic series, the alternating harmonic series, and p-series. 
    • The harmonic series is a divergent infinite series. k=11k. This form should be familiar to you. 
    • The alternating harmonic series converges conditionally. k=1(-1)k-1k because this series converges but the absolute value of its series, the harmonic series, does not. 
    • P series is any series in the form n=101np. If p is greater than 1, it converges. Otherwise it diverges!
  • 10.7: Alternating series test for convergence
    • An alternating series is one that changes sign, so it is in the form an=(-1)n+1bn where bn>0 for all n and bn is decreasing, and nbn =0, then n=1an converges.
      • This means you need to findnbn. If it is zero and all of the other conditions are met, you have a convergent series. 
  • 10.8: Ratio Test for Convergence
    • The ratio test compares a term from the series to a different value in the series by taking its limit.
    • If our series is an  then to use the ratio test, we want to find L, where L=nan+1an. If L < 1, the series is absolutely convergent. If L > 1, the series is divergent, if L=1, we have more work to do, because the series could still be divergent, conditionally convergent, or absolutely convergent. 

fNOTE: These are the only tests for convergence that are on the AP exam. If asked explicitly to use them, you should! If you are told to figure out whether a series is convergent or not and not given a specific method, feel free to use another method you learned even if it is not on the test. 

  • 10.11: Finding Taylor Polynomial Approximation of Functions
    • A taylor polynomial centered at x=a can be found using this form:

f(x)=f(a)+f'(a)(x-a)+f”(a)(x-a)22!+f”'(a)(x-a)33!+…..+f(n)(a)(x-a)nn!

    • These taylor polynomials can be used to approximate function values near x=a. 
    • The more terms you find in an nth degree taylor polynomial, the closer you are to approximating the actual function. 
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1 Comment

  1. Avatar

    The unit 4 and unit 5 formulas and theorems were on this website the last time I checked (yesterday) but now they are not. Formulas and theorems are available for all units except 4 and 5. Can you please upload them back? Thank you!

    Reply

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