AP Calculus Notes:
AB:
Pre-calc Stuff:
● Know all of this (it’s vital): (Quizlet Flashcards)
○ Video to help memorization
Limits:
● If the limit of a function as x approaches a constant (c)
exists then the function is interrupted.
● Types of Discontinuities:
○ Hole (removable)
○ Jump Discontinuity (non-removable)
○ Infinite Discontinuity (non-removable)
● If the limit from the right isn’t the same as the limit
from the left then the limit does not exist (DNE).
● Limit of a constant is the constant itself.
● Plug in the value for x (c) to solve for basic limits
(duh).
● To solve limit questions where you get the
indeterminate form of 0/0 or ∞/∞ you can:
○ SUBSTITUTE (ALWAYS CONSIDER THIS
FIRST! But since you got the indeterminate form
move on to the next steps)
○ Factor
○ Rationalize
○ Simplify/Use Identities
○ Common denominator (usually for numerator
when there are fractions).
● Keep simplifying until something can be plugged in.
● When trying to simplify a limit sometimes your can
multiply individual terms that are multiplied together
by “1” in a sneaky way. You can multiply one term’s
denominator by “x” and the other term’s numerator by
“x”. x/x=1, so this is legal.
● See “Properties of limits” section on identities sheet.
Take the limit of the function and then do whatever
else (square root, raise to power, add, subtract,
multiply, divide, etc.
●
●
(This is very important!)
○ Remember this:
■ https://photos.app.goo.gl/8rN3dVxkSdMX3Z
St9
●
● Remember for the future that
and
that a^4-b^4
.
● Check out
● When given the limit as h approaches 0 with the
definition of the derivative as the function then just
take the derivative of the stuff after the “-” in the
numerator (That’s the function).
● Use BOB0, BOTNA, and EATSDC to find horizontal
asymptotes.
● When finding horizontal asymptotes check the limit
from both sides (+ and -) when the infinities are close
(tied).
● If Bigger On Top by 1 then divide the functions to find
the oblique asymptote.
● Don’t forget your identities when solving limit
problems!!! They are incredibly useful!
● When using the definition of the derivative plug in
(x+h) in for every x in the function.
● For the alternative form, use C in the bottom and F(C)
in the top!!!
● Only cancel things out when they are a product of
something/not being subtracted.
● Always distribute the negative!!! Always check for this!
● Check your factoring!
● Check the limit from both sides! Especially when
given a piecewise function!
Continuity:
● Three conditions must exist to determine whether the
graph of the function is continuous at x as it
approaches c:
○ The function is defined at x = c.
○ The limit of the function exists at x = c from both
sides.
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○ The limit of the function exists at x = c and equals
f(c).
Reminder of Types of Discontinuities:
○ Hole (removable)
○ Jump Discontinuity (non-removable)
○ Infinite Discontinuity (non-removable)
■ For example the volcano graph, y=1/x^2
If the limit does not exist (from both sides) there’s a
jump discontinuity and your work is done.
If the limit exists, continue to see if it equals the
function at that x-value (c). If it does the function is
continuous but if it doesn’t then the function has a
jump discontinuity.
When dealing with absolute value functions plug in xvalues to figure out the shape of the graph and then
you can find the limit.
When checking functions with a radicand, the
radicand should always be positive or equal to 0.
○ Solve the radicand function
○ Should get an answer with + or -.
○ Check number within each region to find which
areas are defined.
○ Establish the domain and then proceed to answer
the question.
When checking continuity on a closed interval:
○ Check the continuity from the right of the x-value
(in the ordered pair).
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●
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○ And the the left of the y-value (in the ordered
pair)
○ If both limits are the same then the function is
continuous on the interval.
If “b” is a real number and “f” and “g” are continuous
at x = c, then...
○ Bf (number times a function)
○ Fg (multiply functions)
○ F+or-g (adding or subtracting functions)
○ f/g (g can’t equal 0) (dividing functions)
○ F of g(x) (composite functions)
○ are also continuous.
The Intermediate Value Theorem (IVT):
○ A mysterious y-value that appears between xvalues.
○ The IVT is sometimes used to find a function’s
zeros.
○ To find/use the IVT…:
■ Factor the function and set it to 0.
■ Find the roots and only choose the ones that
zare in the interval specified.
When finding Vertical Asymptotes (V.A.s) set the
denominator to 0.
○ If you canceled out a term in the numerator and
the denominator then that can’t be a V.A.
Extreme Value Theorem:
○ There are always maximums and minimums on a
continuous function on a closed interval.
● Infinite Limits:
○ The limit of 1/x as x approaches 0 from the right,
increases without bound (+∞).
○ The limit of 1/x as x approaches 0 from the left ,
decreases without bound (-∞).
○ When the limit of a function as x approaches a
from the left or the right equals positive or
negative infinity there’s a V.A.
■ x=a is a vertical asymptote of y=f(x).
○ To do these types of problems:
■ Set denominator to 0.
■ Take the limit of those values.
■ If you get a positive number/0 that’s
undefined and going towards positive infinity.
■ If you get a negative number/0 that's
undefined and going towards negative
infinity.
■ If the limits don’t match then the limit does
not exist (DNE) and there’s possible a V.A.
and there’s a discontinuity (if you canceled
out terms then there’s a hole).
○ 1/+or-∞ = 0
Derivatives:
● Concept (2-1):
○ Secant line touches at two points (average rate of
change).
○ Tangent line touches at one point (instantaneous
rate of change).
○ Definition of the derivative (slope of tangent line):
■
■ When using the definition of the derivative
plug in (x+h) in for every x in the function.
■ Don’t forget to distribute the negative!
○ The alternative form of the derivative (finding the
average rate of change):
■
■ For the alternative form, use A in the bottom
and F(A) in the top!!!
○ Once you simplify the equations for the definition
of the derivative or the alternate form you must
plug in an x-value (c) where you want to find the
instantaneous slope.
○ When asked for the equation of the tangent line
use point slope form (y-y1)=m(x-x1) to easily
write the equation.
○
○
○
○
■ You may have to change this to standard
form by subtracting y1 and then distributing
m.
f’(x) is first derivative
f”(x) is second derivative or the derivative of the
derivative.
■ This can also be written as:
● d^2y/dx^2
● F^(2)(x)
Differentiability = Whether Derivative Exists
Sharp turns or cusps cause problems for the
derivative.
■ Justification for sharp turn can be like this:
●
○ Continuity does not guarantee differentiability
(but it can imply it, don't rely on this!).
○ Differentiability (as long as it’s not a piecewise
function) guarantees differentiability.
■ If you run into a piecewise function confirm
continuity then prove differentiability.
■ When checking differentiability for piecewise
functions the derivatives form both sides
must match for it to exist and the function to
be differentiable at that point.
○ Graphs of the Derivatives of Functions:
■ Look at graphs and assign tangent lines.
■ Look at what those tangent lines are doing
and label positive or negative slope.
■ Label if slope is zero
■ Concentrate on what slope is at 0.
■ These slopes will translate to values on the
graph of the derivative.
● Positive slope=positive values on
derivative graph.
● Negative slope=negative values on
derivative graph.
● Zero slope=crosses x-axis on derivative
graph.
● If slope is positive at 0 then the graph of
the derivative should be above the xaxis and visa versa.
● Derivative Rules (2-2):
○ Linearization (Not in this section but on getafive):
■ Linearization=linear approximation
■ If you can come up with a tangent line
equation, you can use it to approximate the
values of f(x)≈L(x).
● If you’re not given a derivative and/or a
point use the equation/x-values given to
find it.
● Once you have the derivative and a
point, write an equation using pointslope form and change it to standard
form.
● Replace y with L(x) and plug in what
they want you to find (ex: f(6.2) ) for “x”.
● Solve
○ Derivative of a constant is 0 (2-2).
■
○ The power rule is the fastest way to find common
derivatives (2-2):
■
○ The constant multiple rule (2-2):
■
■ Derivative of constant times function =
constant times derivative of function.
○ Sum and difference rule allows you to take
individual derivatives of functions added or
subtracted together (2-2).
■
○ THE PRODUCT RULE:
■
■ Use whenever you’re trying to take the
derivative of two functions being multiplied
together.
■ Or you see two variables in a derivative
problem. PUT PARENTHESIS AROUND
THE VARIABLES BEFORE YOU EVEN
START T O INDICATE YOU’RE USING THE
PRODUCT RULE.
○ THE QUOTIENT RULE:
■
● Use whenever you see two functions
being divided in a derivative problem.
○ Finding tangent lines:
■ Find the derivative of the equation given at
the x-value given. This is your slope
■ Use the x-value given to find the respective
y-value by plugging it into the original
function. This will give you a point if you
weren’t originally given one.
■ Use point slope form (y-y1)=m(x-x1) to make
an equation.
■ Change it to slope intercept form if
necessary.
● If asked for the normal line just take the
opposite reciprocal of the slope you
found.
○ Finding the tangent lines of an ellipse:
■ There are four.
● y^2+x^2=1 is the equation for a circle
with radius 1(Just an FYI).
● When trying to find multiple horizontal
tangent lines:
○ Set derivative of function to 0
○ Plug what you get into ORIGINAL
function to get a point.
○ Slope is 0 because the tangent
lines are “horizontal”
○ Use point-slope form to write the
equations.
○ Derivative of position, s(t), is velocity, v(t), and
derivative of velocity is acceleration, a(t).
■ Direction change occurs when graph crosses
x-axis (ex: going from positive to negative)
and v(t)=0
■ Thing is slowing down when slope is
negative (mostly) but say when v(t) and a(t)
have different signs. Or look at graph and
find where this is true.
■ Thing is speeding up when slope is positive
(mostly) but say when v(t) and a(t) have the
same sign. Or look at graph and find where
this is true.
■ Thing is moving right when v(t) >0
■ Thing is moving left when v(t)<0
■ Thing stops when v(t)=0
○ I v(t) I = speed
○ Trig. Rules for Derivatives:
■
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■
■
■
■
■ I will add these to the quizlet of things to
know for calculus.
● More Rules When Differentiating:
○ When differentiating, you can rewrite square
roots or cube roots etc. as to the ½ or ⅓ etc.
power. Remember exponent over root for this.
○ Find common denominator when you have
fractions in the numerator (most typical scenario).
○ If you have a trig function to a power (ex: cos^2x)
rewrite it to (cosx)^2.
○ If you are taking the derivative of -sinx, leave the
negative! It will be -cosx.
○ If there’s is product rule that needs to be done,
take out the constant if there is one, and put the
product rule work in brackets.
○ When you rewrite radicands to a power you
should put them back ESPECIALLY IF YOU’RE
ADDING OR SUBTRACTING THEM because
you’ll have to find a common denominator.
● When solving for variables in a differentiation
problem:
○ Find two limits (one from both sides) and use
what they equal to make an equation with the two
variables. You will have to plug in x-value and
simplify at times for this to work.
○ Find the derivatives of the original functions.
○ Set derivatives equal to each other to solve for
one variable.
○ Use equations from limits to solve for the other
variable.
● Chain Rule:
○
■ Take the derivative of the outside (using
power rule) and leave the inside.
■ Then multiply by the derivative of the inside
(the base).
■ You may have to take the derivative again in
which you must chain rule again.
● This is typical when you have an angle
of a trig function and you have to do
something like:
○ f’(x) = -sin(x) * x’
○ If you use the product rule or quotient rule, do
that before the chain rule.
○ You know those problems where you’re given
f(x)=a number and g(x)=a number and
f’(x)=another number and g’(x)=yet another
number. Yeah those problems! Simplify inside
parenthesis before you solve!!!
● Implicit Differentiation:
○ We’ve been doing explicit differentiation where
we differentiate y with respect to x.
○ Implicit differentiation is where you can
differentiate two variables, like x and y, on the
side side on the equation.
○ Use product rule a lot.
○ When you take the derivative of “y” it is dy/dx.
■ Ex: y^2 The derivative would be the
derivative of y^2 = 2y times dy/dx
■ Final Answer = 2y*dy/dx
○ When doing implicit differentiation problems:
■ Find the derivatives (both sides, RIGHT
SIDE TOO!)
■ Isolate the terms with dy/dx.
■ Factor out dy/dx.
■ Divide by what's left and simplify.
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■ When finding vertical tangent lines set the
rise to 0 and when finding horizontal tangent
lines set the run to 0.
● Don’t forget to get rid of complex
fractions.
● When taking out a negative you can
take it out of just the numerator or the
denominator but typically just one.
● This will typically open up answers since
-x-2 will typically not be in the numerator
of a final answer on the MC.
READ THE PROBLEM!
READ THE PROBLEM CAREFULLY!
READ IT TWICE!
WRITE IT DOWN AGAIN/WRITE DOWN KEY INFO!
ASK “WHAT IS THIS PROBLEM ASKING OF ME?”
Related Rates:
● You must know formulas from geometry to find areas
of shapes, circumferences, volume, etc.
○ Circle:
■ Area = πr2
■ Circumference = 2πr
○ For more check this out: https://goo.gl/1vWGXw
● How to Solve:
○ Sketch and label a model of the problem with the
given info.
○ List known values in proper notation.
○ List value to be found.
○ Find an equation (remember the formulas) that
can give you the answer to what you’re looking
for with your given info. If the problem looks
especially complex (cone problems) try and see if
you can eliminate variables.
■ In cone problems you can use congruence to
find the height of the inner cone along with
the height to radius ratio.
○ Implicitly differentiate both sides with respect to
time (t). Only the changing variables!!!
■ Chain Rule if applicable (along with anything
else learned so far such as product rule,
quotient rule, etc.)
○ You will most likely find you’re missing a variable
and have to solve for it.
○ Evaluate for the answer with the given info.
○ Don’t forget units in answer and also answers
can be negative as things do down, backwards,
etc.
Extrema:
● Extreme = extreme values of a function such as max
or min.
● Found when the slope is zero or undefined.
● Max or min must be definite (no holes) otherwise the
max DNE (can’t say approaching a number).
○ Function must be continuous and on a closed
interval to have a (relative) max and/or min.
○ If a function is continuous everywhere (all real
numbers) then it is continuous on the given
closed interval.
● To find Extrema:
○ Find the derivative of the function(s).
○ Set the derivative (y’) to zero and DO NOT
simplify.
○ Solve for x (and other vars if present) and make
sure the critical numbers are within the interval.
■ If you have a fractional function set the top
and bottom to 0.
○ Take the critical numbers and the numbers from
the end of the given interval and plug them into
the original function in a process called the
candidates test to figure out the absolute and
relative max and mins.
■ Simplifying no + or - when taking square
roots.
■ Use + or - when solving and taking a square
root.
○h
Derivatives of Logarithmic and Exponential Functions:
● Derivatives of logarithms:
○ Basics of logarithms:
■ The domain is (0, infinity) and the range is (infinity, +infinity).
■ The function is continuous, increasing, and
one-to-one.
■ The graph is concave downward.
○ Logarithmic Properties:
■ ln(1) = 0
■ ln(ab) = ln(a) + ln(b)
■ ln(a/b) = ln(a) - ln(b)
■ ln(an) = n*ln(a)
○ The derivative of the natural log function is 1/x:
○ The derivative of some quantity “u” is 1/u times
the derivative of the inside:
○ The derivative of a logarithm with base “a” (not
natural log) can be rewritten like this:
○ Use logarithmic basics and properties BEFORE
differentiating.
○ Logarithmic differentiation can occasionally be
used to differentiate non-logarithmic functions.
■ You can take the natural log of an annoying
function (such as a complex fraction) to
simplify things.
○ Due to the fact that logarithms can't be taken of
negative numbers, you will find many problems
with absolute value signs.
○ d/dx [ln(u)] = d/dx [ln(-u)]
■ d/dx I u I = (u/ I u I) * u’
■ Example: f(x) = [ln l cos(x) l]
■ f’(x) = -sin(x)/cos(x)
■ f’(x) = -tan(x)
○ You can find extrema of logarithmic functions by:
■ Taking their derivative
■ Simplifying or guessing and checking to find
critical values.
■ Using the candidates test to find out
extrema.
● Derivatives of Exponential Functions:
○ Remember the inverse relationship where ln(ex) =
x and x = eln(x).
○ eaeb = ea+b
○ ea/eb = ea-b
○ The domain of f(x) = ex is (-infinity, +infinity) and
the range is (0,+infinity).
○ The function f(x) = ex is continuous, increasing,
and one-to-one throughout its domain.
○ The graph is concave upward.
○ The limit as f(x) approaches -infinity is 0.
○ The limit as f(x) approaches +infinity is +infinity.
○ d/dx[ex] = ex (1)
○ d/dx[eu] = (eu)* (du/dx)
Rolle’s Theorem and the Mean Value Theorem (MVT):
● MVT:
○ Continuous on [a,b] and differentiable on (a,b). If
these conditions are not met than do not use the
MVT or Rolle’s Theorem.
○ Now find the derivative of the function
○ Set derivative equal to slope of secant line:
■ f’(c) = f(b) - f(a) / b - a
○ Solve and only choose answers that fit within the
interval.
● Rolle’s Theorem:
○ Special version of the MVT.
○ Continuous on [a,b] and differentiable on (a,b).
○ f(a) = f(b) = 0 (This is when/why you use Rolle’s
Theorem).
○ Set derivative to 0.
○ Solve and only choose answers that fit within the
interval.
Increasing and Decreasing Functions (3-3):
● Begin by finding the domain (any exceptions?)
● Find the derivative of the function
● Set the derivative to 0 to find critical numbers.
● Make a sign chart:
○ Plug in numbers that are greater, less than, or
between critical values (lable critical values with a
“c”).
○ If the chart goes from positive to zero (or vice
versa) than the function is increasing. (draw a
line under chart)
○ Chart is zero (draw a flat line under chart)
○ If the chart goes from zero to negative (or vice
versa) than the function is decreasing (draw a
line under chart).
● Write where the function is increasing and decreasing
(should be easy to see especially with the drawn
lines).
● You will often be asked to find extrema which can be
easily found using your sign chart and the drawing
you made with it.
Derivative Tests, Concavity, and Graph Interpretation:
○ Information given by the first derivative (only if on
interval):
■ Critical numbers at f’(x)=0 or undefined
■ Function is increasing if f’(x) > 0
■ Function is decreasing if f’(x) < 0
■ First Derivative Test:
● If the critical value at f’(x) = 0 has a
change from negative to positive it’s
(relative)a minimum.
● If the critical value at f’(x) = 0 has a
change from positive to negative it’s
(relative) a maximum.
■ Turning points on first derivative grade can
be points of inflection.
○ Information given by the second derivative (only if
on interval):
■ There can be an inflection when signs
change, f’’(x) = 0, or f’’ is DNE.
● Find critical values of the second
derivative and make a sign chart with
them to find where signs change and
where inflections will occur.
■ If f’’(x) > 0 on an interval then the graph is
concave upward (tangent line lies below the
graph) and there will be a min at (c,f(c)).
■ If f’’(x) < 0 on an interval then the graph is
concave downward (tangent line lies above
the graph) and there will be a max at (c,f(c)).
■ Concavity can also be found with a graph on
the first derivative as someone can find the
slope of f’:
● If the slope of f’ is positive the graph is
concave upward.
● If the slope of f’ is negative the graph is
concave downward.
■ Second Derivative Test for Local Extrema:
● Let f’(c) = 0
● If f’’(c) > 0 then f has a local max at c.
● If f’’(c) < 0 then f has a local min at c.
● If f’’(c) = 0 the test is inclusive and now
you have to do the first derivative test to
find extrema.
● JUST USE FIRST DERIVATIVE TEST
TO FIND EXTREMA (unless specifically
told to second derivative test)
● USE SECOND DERIVATIVE WHEN
ASKED FOR CONCAVITY!!!
● May have to find POI in order to begin a
problem.
■ Graphing Calculator Tips:
● Normal view window is zoom 6
● Normal view window for trig problems is
zoom 7
● If interval is given in the problem use
that as your window
● When given two functions and asked
when they have the same slope find
both derivatives and graph them both.
Then find the intersection point to arrive
at your answer.
○ Using an intersection point can also
sometimes be used when there are
trig functions in the MVT and you
must find “C” (and it’s marked as a
calculator problem).
● When it’s a calculator problem about
finding the x-value that makes the
slope/derivative equal a constant you
should:
○ Find the derivative
○ Set it equal to the constant and
subtract the constant to the other
side.
○ Now take that function and graph it
○ NOW THE ZERO is the x-value!!!
Limits At Infinity:
● When you have a limit at infinity:
a) Use BOB0 and EATSDC
b) If BOTNA comes up try and use L’Hopital’s
Rule:
i) Only use L’Hopital’s Rule if you get an
indeterminate form and there is NO
RADICAL SIGN!
Do L’Hopital’s Rule by taking the
derivative of the numerator and
denominator and seeing if you can plug
in the given value (in this case infinity).
iii) Repeat until you get an answer.
c) You can also use fuzzy math:
i) If you have a number over infinity the
answer is approximately 0.
ii) If you have a infinity over a number the
answer is approximately infinity.
d) If you come into contact with a limit question
with a radical sign (they say) to:
ii)
i)
Divide by something such as √X2 for x
in order to simplify.
Optimization Problems:
● Find the shortest distance from the given point to a
point on the graph.
○ Use the distance formula and plug in the two
known values
○ Try and knock out one variable by making
something like this:
■ y=x2
○ Simplify and raise everything to the one-half
power
○ Differentiate
○ Now set only the numerator to 0
○ Now plug in the x-value you get into the original
function to find the corresponding y-value.
○ Now you have your point and you’re done.
● Optimization Real World Problems (shapes):
○ Use the formula for the particular shape or you
may have to make up a formula based on the
problem
○ Knock out a variable (probably will have to use
another formula)
○ Differentiate
○ Set that to 0 and solve for one variable
○ Use that to find the other variables
Linearization/Tangent Line Approximation:
● Use the point-slope formula
● You should be given a point and a slope or be able to
find it
● Often you will have to know that a point on the inverse
of a function is opposite on the original function
○ Ex: (1,2) ---> (2,1) on inverse
● Make the equation
● Plug in the value you are approximating in for X
○ y-1=4(X-3)
● Solve for Y
● You will often be asked if your answer is an
overestimate or an underestimate.
● You can figure this out by finding the second
derivative and seeing if:
○ F’’ < 0 (overestimate)
○ F’’ > 0 (underestimate)
Inverse Functions
● Make sure the function is one-to-one
● You will typically either be given a function and an xvalue (where you’re expected to find the y-value) or a
point.
● Often you will have to know that a point on the inverse
of a function is opposite on the original function
○ Ex: (1,2) ---> (2,1) on inverse
● Use the theorem:
○ (F-1)’ (a) = 1/f’(g(x))
Integration:
● Integration=area under the curve
● Definite integrals are defined from a to b (integrals
with numbers on top and bottom).
● Indefinite integrals are not defined on an interval.
● Riemann Sums are approximations of definite
integrals:
○ When doing Riemann Sums:
i. Construct rectangles using the number of
intervals given, intervals may not be equal.
ii.
iii.
iv.
v.
vi.
vii.
viii.
ix.
x.
On the bottom, put x-values and if needed
the interval (changes) in circles.
On the top of each rectangle put f(x) or the
corresponding y-value.
From there you can use any riemann sum
method to solve: left hand rule (exclude last
term), right hand rule (exclude first term),
midpoint formula (construct midpoints to
use), or the trapezoidal rule (double middle
terms or take separate trapezoids).
You may have to separate figures into
separate terms to multiply them by a
different base then add them together.
Use corresponding formula (b*h,0.5h(b1+b2),
etc) to form your riemann sum and solve.
Only use the numbers you need (sometimes
tables are given with extra numbers but don’t
be fooled).
Make sure you only use numbers at the
ends/beginnings/middles of each rectangle.
When given a riemann sum that can be done
with a calculator DON’T stray away from the
riemann sum method.
Some riemann problems (especially word
problems) will require an extra step (such as
subtracting your answer from the original
number given).
xi.
○ You may have to determine if what you found is
an overestimate or underestimate in which case
you:
i. See if the rectangles are inscribed
(underestimate) or not (overestimate).
ii. Justify by saying what riemann sum you
used and whether the graph is concave up
or down, increasing or decreasing .
● Displacement (area under curve):
i. Just the integral of change in y.
ii. When you have a definite integral either look
at the function and figure out what the graph
would look like. Then use geometric
formulas to solve for the area.
iii. OR plug in the defined values into the
function and subtract one from the other
● s(b)-s(a)
● Know all of these theorems for integrals:
○ And These:
○ What to do with powers:
■ The integral of number*ex is that number*ex
● Integral of 5ex = 5ex
■ Powers that are multiplied (as long as the
base is the same) can be added.
■ Powers that are divided (as long as the base
is the same) can be subtracted.
■ If you see a function in parenthesis to a
power then distribute that power (as long as
it’s easy otherwise use u-substitution).
● Solving Differential Equations:
○ Given dy/dx = f(x)
○ Cross multiply
○ Take the integral of both sides to get rid of d in dy
○ You should have a general solution and DON’T
FORGET TO ADD “C” TO THE END as this is a
general solution
○ ANYTIME YOU HAVE A GENERAL SOLUTION
OF AN INDEFINITE INTEGRAL ALWAYS ADD
“C”.
○ If you were given a point at the beginning of the
problem use the y-value as y and the x-value as
x in the function to solve for “C”.
○ Now substitute that value in for “C” in the
equation and you have found THE particular
solution.
● Slope Fields:
○ Graphs of slopes
○ What do I do?
● You can integrate with a visual graph:
○ Use a geometrical formula to find the area of the
shape formed from the definite integral.
○ Don’t forget that area above the x-axis is positive
and area below the x-axis is negative. THIS
REMAINS TRUE WHEN ADDING THE AREAS
TOGETHER.
● Adding/Subtracting/Working with multiple integrals:
○ Integrals will be given
○ Be careful here
○ Draw a number line
○ Check to see if the given integrals have their
numbers in the right spots (a,b).
○ If not switch them and multiply by a negative
○ THEN make a problem (addition or subtraction
usually) that you can solve
○ Alternatively you can add the two smaller bounds
to equal the largest bound and solve from there.
○ Watch out since that answer may be
multiplied/divided/modified in some way.
● Using the Calculator to find integrals:
○ Graph:
■ Enter the function in the “y=” section (where
you enter functions to be graphed).
■ Go to the “graph” tab
■ Hit “2nd” and “trace” followed by “7”
■ Type in the applicable x-values followed by
hitting enter (a,b).
■ Done.
○ Regular:
■ Hit “math”, “9”
■ Specify “a” and “b”
■ Enter the function
■ Enter “X” after d
■ Hit enter
■ The Calculator with solve for a number.
○ AND USE THE CALC AS MUCH AS YOU CAN
■ To get absolute value signs - “math” “>”
“abs.”
● Integrating rational functions:
○ Separate the numerator into pieces that can be
placed over the denominator (only works if the
denominator is one term).
○ Many times you can bring the denominator to the
top and change the power to a negative, then
solve normally.
● Read directions carefully!
● Does your answer supply the problem with what it
asked for?
● Look out for unexpected details and extra steps
(especially in word problems).
● Don’t forget about adding “C” on indefinite integrals.
● The Fundamental Theorem of Calculus:
● Part 1:
○ Gives exact area under curve using calculus
○
○ Integrate the function and then subtract the
function evaluated at b and then a
○ DON’T use the FTOC on absolute value
functions.
■ Instead use geometry to solve absolute
value functions (probably should have
addressed this earlier)
■ Draw the graph
■ Break up the graph on the interval into
geometric pieces who’s area can be found
(usually triangles)
■ Add up the areas.
● Part 2:
○ Displacement = add positive areas and account
for negative areas
■ Or take integral as normal
○ Distance = all areas are made positive
■ Take integral with absolute value signs (can
be done on calculator)
MVT for Integrals:
● You will find a c-value
○
○ Integrate on the defined interval to get a value
○ f(c) = that value
○ You can then rewrite f(c) as the function but
replacing the variable (like x) with c.
○ Solve for C
Average value of a function:
○
○ Just like taking a normal defined integral just with
the extra beginning part.
○ Do this when you see “average value” in the
problem (not average rate because that is just
change in y over change in x).
○ WORK DOWN so you don’t forget the fraction.
U-Substitution:
● Use U-substitution when you can’t do anything else to
solve the integral (simplify, power rule, distribution,
etc.)
● Usually two functions are multiplied or divided
● Use U-substitution especially when big exponents are
involved.
○ Start by finding which function you’ll go after to
make your “u”
■ EX: u=(x+2)
○ Usually the inner function or a power that’s a
function is a good place to look.
○ Differentiate the u equation
○ Try and make “du” look like the rest of the
function you’re integrating typically by multiplying
the other side by the inverse of the coefficient
getting in your way.
■ You may have to divide by a number
■ Carry that final fraction over to the outside of
the integral.
■ DO NOT FORGET THIS! CHECK FOR IT!
○ Integrate (everything that you had originally) with
“u” replacing the function you assigned to “u”
○ Plug that function back in for “u” after you
integrate.
○ Don’t forget to add “C”
● The special case of u-substitution:
● When you have an extra variable and can’t get du to
match the rest.
○ Define “u” (find the part that you want to be u)
○ Differentiate u to get du and try and match to the
rest of the function as closely as you can.
○ If you plug in “u” and try to integrate you’ll find
that you still have “x” as well
○ To get rid of “x” you can return to your original “u”
equation and solve for “x”
○ Then plug that in for “x” and distribute
○ Integrate
○ Plug in function for “u” and don’t forget to add “c”
Integrals of logarithms and exponentials:
●
●
(u-sub was already used to get
to this point)
●
(u-sub was already used to get to
this point)
●
●
●
(u-sub was already used to get
to this point)
● ln(6)-ln(4) =ln(4/6) or ln(⅔)
● 0.5ln(x) = ln(x)0.5
● When finding an integral of a function over a function
and you can’t cancel anything out you must use long
or synthetic division to simplify the integral.
How to find position (or when given an initial value) of an
object:
● ONLY when asked to find position use:
● Add the initial value to the DEFINED integral (usually
starting at 0 since a and b won’t always be clearly
given)
The Second Fundamental Theorem of Calculus (2nd
FTOC):
●
● The a and b values here could be x2 and x3 for
example.
●
○
○
○
○
Plug in functions
Take derivative to cancel out the integral
Don’t forget to chain rule
If you ever do a problem where the lower limit is
a function and the upper limit is a constant you
must switch the a and b and make the integral
negative.
○ Don’t forget to rewrite trig functions to a power
and put () around the function and put the power
outside the ().
● The 2nd FTOC on Graphs:
○ Once you see something along the lines of
○ g’(x) = f(x)
○ The graph you are seeing is probably g’(x), the
graph of the derivative.
○ When asked to find g(a value) plug that value in
for x (the upper limit) and replace t with it.
■ Make sure to show this step on tests
■
■ If asked to find g’(a value) just look at the
graph and find the corresponding y-value
■ If asked to find g’’(a value) find the slope of
the graph at that value
Particle Motion Using Integration:
● Finding when two particles have the same velocity:
○ Make sure both particle functions are in the
velocity form (if not integrate or differentiate to
get them there).
○ Then graph both on a calculator to find where
they intersect
○ Find that intersection point using “2nd” “trace” “5”
and putting the cursor over the intersection point
and hitting enter a few times.
○ That will be the time where the velocities are both
equal
● You will also commonly encounter position, average
velocity, finding extrema, finding intervals where
function is increasing or decreasing, and finding
where the speed of the particle is increasing and
decreasing
○ Review this material in Free Response Particle
Motion context
● When asked to find ABSOLUTE extrema use the
candidates test and plug in values using
■
● Relative extrema can still be found using the graph of
g’(x)
● A tangent line to F (F will be the antiderivative) will
represent the derivative of that function.
○ F = usually antiderivative
○ F = usually function
○ Solve by using:
○
○ This will give you slope when you plug in just the
b-value to the function
○ An integral form a (x-value) to a will yield a yvalue of 0.
● When asked where F is increasing or decreasing:
○ Make sure your function is velocity
○ Set it to 0 and find the critical values
○ Make a sign chart to find the intervals on which
the function is increasing or decreasing
○ Use the justification of f’ < or > 0.
Justifications for Free Response (mostly particle motion)
questions:
● See white sheets
Check For:
● FOR STUPID ERRORS SOLVE THE QUESTION
AGAIN IN YOUR HEAD!!!
● Did you read correctly and use the right given
numbers
● If you carried the fraction through when using usubstitution or average value
● Correct justification for Free Response Questions
● If you forgot to change u-1 to 1/u BEFORE integrating
as this probably meant you forgot to make it lnIxI.
● If you changed u back to x in a u-sub problem
(especially undefined)
● If you changed a and b in a DEFINED u-sub problem.
● Putting the correct numbers (specifically fractions) on
simple integral problems
● I you added “C” to any INDEFINITE integral
● If you chain ruled correctly (always think about chain
ruling)