AP Calculus BC Study group

Schedule, problems, and resources

Quick resources

• Paul’s online math notes
• Stewart calculus (online pdf, great for comprehensive self-study and challenge problems)
• Practice exams
• FRQ practice by topic
• Study guide for differential calc
• Study guide for integral calc
• Overall study guide
• 100 integrals youtube video (bprp) good for computational practice
• Quizlet 1
• Quizlet 2 (derivatives)
• Quizlet 3
• Wikipedia’s comprehensive trig identity guide
• Month long study plan

Study plan

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• DM me for 1 on 1 time for help with anything (schedule in advance!!!)
• Look at quick resources. The most important are 4, 7, 9, 10, and 11
• Make some flashcards for stuff that you need help on
• Try the optional assessment on the homepage of the schoolhouse series. Once you finish all (or some if you can’t finish all of them), send it to Dylan Perazzo as a pdf for grading and additional study resources.
• Look over this presentation for useful tips or information.
• Other quick resources (if you think it will help)

Orientation

• Introductions
• Take survey
• How to study for the exam
• Practicing habits
• Overview
• Kahoot
• Start with limit/continuity
• Feel free to share this slide with your friends if they want study resources.

Limits and continuity

• Take survey if you didn’t before
• Review limit properties and strategy and continuity
• Squeeze theorem
• Algebraic limits
• Challenge problem
• Graphical limits
• One sided limits
• Epsilon-delta definition if there is time

Not gone over in class:

• Extra problems
• One sided limits
• Misconceptions

Review: Limit properties

Review: Limit strategy

• Direct substitution
• If asymptote (i.e. equivalent to a/0 where a is not zero), limit DNE
• If it evaluates to a real number, this is the limit (it may not be discontinuous)
• If that doesn’t work, it may be in an indeterminate form (0/0, inf/inf, etc.), try
• Factoring
• Using conjugates
• Using trig identities
• L’Hopital’s rule
• Otherwise, try
• Squeeze theorem
• Logarithm (if raised to a power)
• If none of the above work, try approximating

Squeeze theorem

Squeeze theorem

Squeeze theorem

Review: What is continuity

Challenge problem:

Epsilon-delta definition of a limit

Infinite limit and limit at infinity

What are one-sided limits?

Limit from the right

Limit from the left

Note: the epsilon-delta definition of a one-sided limit is analogous to the regular definition, except for the fact that the limit of x can only approach from a particular side. Try defining this precisely as an exercise.

Misconceptions

• Continuous functions are differentiable
• This is not true always because the function has to be smooth for it to be differentiable. For example, in the function |x| the derivative from the left and right at zero are different. (-1 and 1)
• A positive derivative implies the function is increasing at a point
• The explanation for this one is more complicated (recommend watching this) This is not true because increasing is a property that can only apply to a neighborhood around a point, which can have different behaviour from the point itself
• A vertical asymptote is continuous because the function is infinity and the limit is infinity
• The definition of continuity is that f(c) is the same as the limit as x->c of f(x). There are actually three conditions here. The limit and the function both need to exist. In the case of a vertical asymptote, neither do, so it is discontinuous.
• 1/x is discontinuous.
• It is discontinuous at 0, but the definition of continuity requires that the limit be the same as a function for every point in the domain. 0 is not in the domain of 1/x.

Misconceptions

• A limit is the same as the boundary of a function
• These two are very different. A boundary is something a function can not go past. More specifically, an upper bound is a number that is greater than all y values of the function, and the lower bound a number lower than all y values.
• A limit is unreachable. That is, the function at a point cannot equal the limit.
• This is not necessarily true. The reasoning for this misconception is most probably because people often see examples of limits in the context of discontinuities where the function does not exist at a point. The limit still exists and is reachable even if the limit is just the same as the function at that point. In fact, this is the definition of continuity.
• A limit is an approximation of a function which can always increase in accuracy.
• A limit can be a precise value of a function; it is not necessarily an approximation. This likely comes from the abundant table problems which have students guess the limit based on particular values close to the point. An approximation is a last resort.
• Limits must exist
• Sometimes, limits don’t exist, as in vertical asymptotes when they are infinite.

Misconceptions

• If the limit exists everywhere, the function is continuous.
• The limit must be exactly equal to the function to be continuous. Otherwise, the limit evaluates what is called a point discontinuity.
• A limit can always be found by substitution.
• Substituting the x value for the limit will just give you the function value. This may not exist or may be different from the limit. Either way, substitution only works if the result is a real number.
• Dividing by zero in a limit means the limit DNE.
• This is only if the numerator (what you’re dividing from) is not zero. If both the numerator and denominator are zero, it is in indeterminate form, which may mean it does exist.
• More misconceptions about indeterminate forms.
• If the limit can be put (by substitution) into the form infinity/infinity or 0/0, then it is indeterminate and you can use any of the processes on slide 5.

Differentiation: definition and rules

• Contextual discussion: how should we define derivatives qualitatively?
• Definition review
• Cheat sheet
• Power rule derivation
• Basic derivative rules
• Quick practice
• Graphical derivatives
• Graph sketching

If time:

• Misc problems
• Challenge problem

How many definitions can you think of?

How many definitions can you think of?

• Instantaneous rate of change
• Slope of a line
• Rise over run of the changes dy and dx
• Velocity
• Scaling factor of the real line
• Wikipedia says: sensitivity of a change of output with respect to input
• An approximation of how much y will change with a unit change in x

Deriving the power rule (for integer powers) in 2 ways

Basic derivative rules

Basic derivative rules

Speedy practice

Speedy practice

Speedy practice

Speedy practice

Speedy practice

Speedy practice

Speedy practice

Speedy practice

Speedy practice

Speedy practice

Graphical derivatives

• Find the family of graphs of tangent lines to the function x^3
• Which tangent lines intersect with the origin?
• Find the second derivative of x^3 and interpret its graphical meaning

Graphical derivatives: sketching review process

• Domain/discontinuities:
• Find the particular points where the function is undefined or the function is discontinuous. The domain is all values for which the function is defined, not necessarily when it is continuous. There are four types of discontinuities you may encounter (you can search online)
• Intercepts
• Set y=0 to find x values that intercept the y axis and set x=0 to find the y value at x=0 (possible multiple values if it is not a function)
• For example: on the unit circle setting y=0 gets x=+/-1 and x=0 gives y=+/-1, these are important points on the graph

Graphical derivatives: sketching review process

3) Symmetries

• If it is an even function then f(-x)=f(x) so the graph is reflected across the x axis. If it is odd then f(-x)=-f(x) so the graph is rotated 180 degrees around the origin. If it is periodic than f(x+c)=f(x) so the function is repeated multiple times. These are very important properties of the graphs of certain functions.

4) Asymptotes

• If any of the discontinuities were asymptotes, graph those asymptotes using limits. Also find the end behaviour by letting x go to infinity or -infinity to find horizontal asymptotes

Graphical derivatives: sketching review process

5) increasing/decreasing intervals

• The function is increasing when the derivative is positive and decreasing when the derivative is negative. So find the derivative and incorporate this into the graph

6) Extrema

• When the derivative is zero, it will be either a local maximum, a local minimum, or a saddle point. If necessary, use the second derivative test to find the concavity of the point. If the second derivative is positive, it is a local minimum, it the second derivative is negative, it is a local maximum, and if the second derivative is zero it is inconclusive.

Graphical derivatives: sketching review process

7) Concavity

• Compute the second derivative and find the intervals when it is positive or negative. Then the function will be concave up or down respectively. When the second derivative is zero, it is an inflection point.

Graphical derivatives

Graph this function, including

• Domain/discontinuities
• Intercepts
• Symmetries
• Asymptotes
• Intervals of increasing/decreasing
• Extrema
• Concavity

Find the approximate values and characterization of the extrema and inflection points using a computer algebra system

Misc problems

Misc problems

Challenge problem

Differentiation: advanced rules

• Define implicit differentiation
• Finding derivatives using implicit differentiation
• Using implicit differentiation in a differential equation
• FRQ practice
• Graphical interpretation of inverse derivative

If there is time:

• Optimization process and problem
• Optimization frq

Implicit differentiation process

• Differentiate both sides of an equation with respect to x
• When there is a function of y, use the chain rule, that is, find the derivative with respect to y and multiply by dy/dx
• Rearrange to find the value you are looking for.
• This means getting e.g. dy/dx on one side of the equation, factoring it out, and dividing
• If you are finding dy/dx, there will probably be a function of x and y.
• If you are finding a higher derivative, you may need to plug in dy/dx etc. according to the original equation (example of this in slide 60)

What is implicit differentiation?

Derivative of square root of x.

(using implicit differentiation)

Derivative of arcsin(x)

(using implicit differentiation)

Graphical interpretation of the derivative of the inverse

<ignore this

Optimization and critical points

• Find the critical points of cos(x)+x and cos(x) on the interval (-2pi,2pi)
• Classify them
• Find absolute maximum and minimum

Differentiation applications

• Review of optimization
• Notable theorems
• Intermediate value theorem (IVT)
• Mean Value theorem (MVT)
• Rolle’s theorem
• L'Hospital's rule
• Newton’s method
• Pros and cons
• Related rates
• Likely most difficult of differentiation units-here is a helpful guide
• Sketching functions

Critical point processes

• 1st derivative test: when the derivative of a function is zero, it is a critical point
• Find the values at each point and create a sign chart for the derivative
• When the derivative goes from positive to positive or negative to negative after the critical point, it is a saddle point.
• When it goes from positive to negative (i.e. the derivative is decreasing) then it is a local maximum
• When it goes from negative to positive (i.e. the derivative is increasing) then it is a local minimum
• Second derivative test (rarely used but still useful in some instances). Instead of finding the sign chart, you can find the second derivative at a critical point to classify it
• If the second derivative is positive, it is a local minimum, if negative, it is a local maximum
• If the second derivative is zero, it is a saddle point which is a special type of inflection point when the first and second derivative are both zero
• In a closed interval, the endpoints are also critical points and may be maxima or minima of the function in that interval
• In an open interval, if the value at the endpoints approaches the maximum or minimum, there is no actual maximum or minimum respectively
• Points where it is not differentiable are also critical points
• Evaluate all critical points to find the global maximum and minimum

Notable theorems

Intermediate value theorem:

A continuous function on [a,b] attains all values between f(a) and f(b) between a and b

a

b

Notable theorems

Rolle’s theorem:

If a function is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), than there is a point c in [a,b] such that f’(c)=0

Notable theorems

Mean Value theorem:

If a function is continuous on [a,b], differentiable on (a,b), than there is a point c such that

Notable theorems

L’Hospital’s rule:

If a limit (when evaluated) can be put in the form 0/0, or alternatively, it is of the form

and f(a)=g(a)=0, then the limit is the same as

How would you formulate this for the other indeterminate form? (infinity/infinity) Why are they basically the same?

Newton’s method

A method for finding roots of a function:

• Think of a value that is close to the root you want to find (say x)
• Find the derivative of the function at that point (f’(x))
• Find the value of the function at that point (f(x))
• Compute the value x-(f’(x)/f(x))
• Repeat from 2 with this new x value from 4

1

3

2, (find the slope of this pink line)

4

It may be helpful to

• Draw a diagram like the one on the right (sketch the curve)
• Figure out how many roots there are to see if you may be finding one you don’t want
• Write down the formula

Pros: the perfect situation

First value x

So much closer after 1 repetition!

Root we are trying to find

Cons: it doesn’t always get closer

First value x

Root we are trying to find

It’s going farther away!

Note: it can only find one root

Differentiation review

• Related rates
• Likely most difficult of differentiation units-here is a helpful guide
• Some frq practice
• Misconceptions

Related rates (Source: link)

• Read the problem slowly and carefully.
• Draw an appropriate sketch.
• Introduce and define appropriate variables. Use variables if quantities are changing. Use constants if quantities are not changing.
• Read the problem again.
• Clearly label the sketch using your variables.
• State what information is given in the problem.
• State what information is to be determined or found.
• Use a given equation or create an appropriate equation relating the given variables.
• Differentiate this equation with respect to the time variable t
• Plug in the given rates and numbers to the differentiated equation.
• Solve for the unknown rate.
• Put proper units on your final answer.

Related rates

Suppose you have a box with changing side lengths. The width is increasing at 6 in/min. The length is decreasing at 5 in/min. The depth is increasing at 2 in/min. What is the rate of change of the surface area when the box is a cube of side length 10 inches?

Problem solving skills:

• Use the definition
• Draw a diagram

Problem solving skills:

• Think graphically
• Simplify the givens first

Problem solving skills:

• Try rearranging equations/inequalities
• Substitution

Problem solving skills:

• Think graphically
• Simplify the problem (Think: what are the critical points? Having a plan will always help)

Problem solving skills:

• Manipulate what you are given by plugging in
• Interpret functional relationships
• Use the definition

Problem solving skills:

• Relating derivatives
• Using theorems

Problem solving skills:

• Think both algebraically and graphically
• Know when to use certain tools (implicit differentiation, tangent line)

Problem solving skills:

• Simulate relationships algebraically and geometrically

Problem solving skills:

• Read the problem well
• Interpret graphically to intuit the procedure

Misconceptions

• The derivative of a product is the product of the derivatives
• Use the product rule instead! Notice that the derivative of x*x is not 1*1=1 but rather 1*x+x*1 using the product rule to get 2x or just multiplying and using the power rule to get the derivative of x^2=2x.
• The derivative of sin(x^2) is cos(x^2) (and similar problems)
• The problem with this is you are not taking into account the chain rule. For more complicated functions this is even harder to catch. Make sure when you are taking the derivative of a function OF a function that you take the derivative of the outside function with respect to the inside multiplied by the derivative of the inside function with respect to x.
• The derivative of f(g(x)) is g’(x)*f’(g(x))
• If you know what you are talking about it is OK to use this notation, but just be careful with it! Notice that the factor f’(g(x)) is not taking the derivative of f with respect to x, but rather with respect to g. For example, the derivative of sin(x^2) is not 2x*cos(x), but rather 2x*cos(x^2). Note here the student making the mistake also took away the x^2, which is not the correct answer.

Misconceptions

• If the derivative is zero at a point, the point must be a maximum or minimum
• There are many other scenarios in which it is neither.
• 1. if it is a local maximum or minimum, but there is another point in the function where it is the global maximum or minimum. Local mins and maxes are not necessarily global, you must test all of the critical points to find the global minimum or maximum
• 2. It is a saddle point. In this case, it is not even a local maximum or minimum. This called a saddle point because it looks like a saddle on a graph (see critical point classification slide)
• 3. Points in which the function is not continuous can be global maxima or minima. These include points of discontinuity, endpoints of an interval, non-differentiable points and more

Saddle point

See types of critical points on next slide

Corner: non-differentiable, continuous, is global minimum in this case

Cusp: derivative is infinite at the point, derivative goes from negative to positive

Saddle point: derivative is zero and the derivative changes from positive to positive or negative to negative

Local maximum or minimum with zero derivative: derivative is zero and it is a mountain or a valley (positive to negative or negative to positive derivative respectively)

Endpoints: Check the endpoints as well! On an open interval still check, but the endpoints do not exist in the domain.

Jump discontinuity: The function goes from one value to another instantly, basically jumping to a different value

Removable discontinuity OR hole: a point that jumps to one value than jumps back immediately after

Infinite derivative: not a cusp, like a saddle point but sideways. Derivative stays positive or negative.

Asymptote: If the function is asymptotic the critical point behavior is a little more weird, there probably is no global maximum or minimum corresponding to asymptote

Misconceptions

• If the point is a maximum or minimum, the derivative is zero
• Note: this is the converse of the previous statement in slide 89 which is also untrue.
• Every single one of the critical points in the previous slide have the possibility to express global maximum or minimum behavior except for the saddle point. (I say behavior instead of is because of the next misconception.) The only one that always has derivative equal to zero is the local maximum or minimum with zero derivative.
• Since this is probably all very confusing, to sum it up, whenever you are trying to find a global extremum, check and compare all non-differentiable points (including endpoints and discontinuities) and zero derivative points.
• Every interval of a function must have a global maximum or minimum
• This is not necessarily true. Note that in some of the graphs mentioned in the previous slide, it may be the situation that there is no way to find a global maximum or minimum because the actual global maximum or minimum is not in the domain. (See this)
• Note: For continuous functions and closed intervals this is true (extreme value theorem)
• If a function has second derivative and first derivative zero at a point, then it is a saddle point
• This is actually untrue (I fell for this myself!) because it is actually an inconclusive test. x^4 at 0 is a counterexample. The reason people think this is true is because if the second derivative is not zero and the first derivative is zero then it actually gives you information, namely the second derivative shows whether it is a local maximum or minimum

Fundamentals of Integration

• What is integration?
• Fundamental theorem of calculus: an intuitive explanation and proof
• Antidifferentiation and integration are the same!
• Finding integrals using area
• Finding indefinite integrals by working backwards

What is integration?

What is integration?

• Anti-differentiation
• Area under curve
• Accumulation of change
• Riemann sum

Fundamental theorem of calculus

2nd Fundamental theorem of calculus

Integrals using area

Integrals using area

Integrals using area

Integration rules

Practice applying integration formulas

Integration techniques

• U-substitution
• Trig substitution
• Long division
• Partial fractions
• Integration by parts
• Overall guide to integration
• Improper integrals

U-substitution

• You may be able to use u-sub when you see a function of a function
• Set a new variable u equal to a function of x
• Find the derivative of u wrt. x
• Change all instances of the function u(x) to u.
• Change dx to f(u)du using (2)
• Simplify and eliminate all remaining instances of x.
• You should now be just in the u world, and the integral should be easier

Long division (and) trig sub

• LONG DIVISION:
• Use long division whenever you have a rational function of polynomials f(x)/g(x) where the degree of f > degree of g

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• TRIG SUB:
• Pretty difficult to spot and use! The best way to learn trig sub is to memorize the formulas for when you should use trig sub and be able to recognize when trig sub will be useful.

(Practice)

Partial fraction decomposition

• Use whenever you have a rational function f(x)/g(x) and deg(f) ≤ deg(g)
• Factor the denominator
• Try to rewrite the rational function as a sum of other rational functions using the following rule:
• Solve for the constants

by multiplying out the

denominator.

(Practice)

Integration techniques cont. + applications of integration

• Integration by parts
• Overall guide to integration
• Improper integrals

If time (applications/formulas):

• Area between curves
• Volume by cross-sections
• Volumes of revolution
• Average value of a function
• Arc length
• Surface area

Integration by parts (via DI/tabular method)

D I

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f(x) g(x)

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f’(x) I(g(x))

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f’’(x) I(I(g(x)))

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.

.

+

-

+

Integrate successive terms

(Practice)

Guide to integration

Note: it is more important to practice by for example watching this youtube video and get used to when you should use different techniques. That being said - some guidance is useful for studying. see stewart’s calculus pg. 526-530

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Improper integrals

• Whenever you have infinity involved (either vertically or horizontally) in a definite integral, you have to use improper integration by evaluating the integral as a limit. Nothing changes with the procedure except for the fact that you have to use limits at the end.

Applications of integration: formulas and their derivations

Area between two curves

Applications of integration: formulas and their derivations

Volume by cross-sections

Where A is cross-sectional area

Applications of integration: formulas and their derivations

Disk method

Washer method

Shell method

Applications of integration: formulas and their derivations

Average value of a function

Applications of integration: formulas and their derivations

Arc length

Applications of integration: formulas and their derivations

Surface area

Applications of integration

• A prototypical example
• Disk or shell?
• Frq problems

A prototypical example function for applications

Consider the function y=x on the interval [0,3]

• Find the total area under the curve
• Find the area between this function and y=3-x
• If the area between this function, the x-axis, and x=3 is the base of a volume with cross-sections perpendicular to the x-axis equilateral triangles, find volume
• Find the average value of the function on the interval
• If this function is revolved around the x-axis, find the volume of the resulting solid
• Find the total arc length

When to use disk/washer or shell technique

Disk/washer

Shell

When to use disk/washer or shell technique

Disk/washer

Shell

• Usually when revolving around x-axis, use disk/washer method
• When the function is difficult or nigh impossible to integrate, it may take more time than necessary

• Usually when revolving around y-axis, use shell method
• When the function is difficult or nigh impossible to integrate, it may take more time than necessary

Frq practice

Frq practice

Frq practice

Frq practice

Differential equations

• Parabola exercise
• What types of slope fields are there?
• What are integral curves?
• Euler’s method
• Solving DE’s
• Logistic model

Earth’s gravity makes a parabola

• If acceleration is constant (i.e. gravity), what differential equation does this make?
• Solve this DE using integration.
• Find an exact solution if the initial position is +5m and the initial velocity is +5m/s.

You can model almost anything with DE’s

• Kinematics (and much of physics)
• Population growth
• Economics/finance
• Sociology

Slope fields

Euler’s method

Parametrics + polar

• What are parametrics in Cartesian 2-space?
• First and second derivative
• Arc length
• Vector-valued functions, kinematics in 2d space

Parametric equations

Plotting parametrics + polar

Parametric differentiation

• There are three important derivatives that are important in a parametric equation in 2d. These are dx/dt, dy/dt, and dy/dx. By the chain rule we know dy/dx=dy/dt*dt/dx, and that dt/dx=(dx/dt)^-1:

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• To find the second derivative (d^2)y/(dx^2), this is equal to

Arc length

Vector-valued functions

How are they different from parametric curves?

Sequences and series

• What is convergence and divergence?
• Important types of series
• Harmonic
• P-series
• Geometric
• Important tests for convergence/divergence
• n-th term test
• Integral test
• Comparison test
• Alternating series test
• Ratio test

What do you know about convergence and divergence of sequences and series?

• Write everything relevant in the chat.
• Possible prompts include:
• What tests are there for convergence and divergence?
• What is the formal or informal definition of convergence and divergence (sequences and series)?
• What types of convergence is there?
• What are particularly significant series?

Properties

Divergence of the harmonic series

P-series

Geometric series

N-th term test

What’s wrong with this (multiple things)? What should be the correct definition.

Integral test

Comparison test

Alternating series test

The ratio test

When to use each test

• Find the infinite limit of the terms. If it is not zero, use the n-th term test
• If it is an alternating series, try the alternating series test
• If it contains factorials or exponential function of x, try the ratio test
• If it is complicated, but you notice that changing terms to a different, comparable term, makes it simpler, try the comparison test
• You can use this in quite a variety of situations as a last resort, try comparing series to p-series or geometric series
• If you can somewhat easily integrate each term, try the integral test

Absolute convergence vs Conditional

When in doubt, write it out (telescoping series)

Exponential and Logistic modeling + DE review

• Separable equations, more solving
• Law of Exponential change
• Logistic models

Polar coordinates

• No more review - frq and mcq problems

Unit 9 - Parametric, polar, vector

• Summarize procedures for problems
• Reviewing formulas
• Exercise with vectors
• Polar review

The functions x(t) and y(t) give the x and y coordinate of a point moving through the xy plane on a parametric curve. t is the parameter (often represents time)

Question: what is the difference between parametric curves and vector-valued functions?

Distance

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Speed

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Displacement

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Velocity

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Acceleration

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• How do you find:
• Velocity from displacement
• Acceleration from velocity
• Displacement from velocity
• Velocity from acceleration
• Speed from velocity
• Speed from displacement
• Distance from velocity
• Speed from acceleration

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• When is an object:
• Speeding up
• Slowing down
• Accelerating
• Decelerating

Practice: r=1+2sin(theta)

Find:

• Yellow+green area
• Green area
• Yellow area

Unit 10 - Sequences and Series

• First I will quickly go over a series test guide and any questions.
• Use series tests in practice
• Alternating series error bound
• Find taylor series of functions

Series test summary

• Does the series look like any of the following?
• Geometric series
• P-series (including harmonic series)
• Alternating series
• Also test for absolute convergence if applicable here
• Can only test for convergence
• Do the terms converge to zero? Use the nth-term test to find out
• nth term test can only test for divergence
• Are there factorials or exponential functions involved?
• Try out the ratio test
• Is the term taken to the nth power?
• Try out the root test
• Try the limit or direct comparison test if any of the following are true:
• The terms include polynomials (not monomials)
• It looks similar to a geometric or p-series
• If it looks like it diverges, maybe limit compare it to harmonic series
• Is the term easy to integrate? Use the integral test

1

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5

6

Complete review of units

• Big activity
• I will ask everyone what they want to review

For the following functions, find

a) horizontal and vertical asymptotes

b) discontinuities

c) derivative

d) equation of the tangent line at x=1

e) concavities

f) intervals of increasing/decreasing

g) max/min points

h) indefinite integral

i) trapezoidal approximation (step size 1) of the integral from -1 to 1

j) volume if you revolve it around the x axis on [-1,1]

k) area between the function and y=x^2+1 on [-3,3]

l) arc length from -1 to 1

m) 1st order differential equation describing it

n) maclaurin series (general)

o) first four terms of maclaurin series

p) error of part (o) at x=2

q) interval of convergence of (n)

r) improper integral from -inf to 0

Draw pictures for all of the relevant parts

1.

2.

Review day

• Kahoot
• Hard problems
• Collaborate together

Review day

• Going over problems from 10 Hardest AP Calculus BC Practice Questions | CollegeVine Blog