Summary All Units

Question 1

Definition of a limit?

Answer 1

The value a function approaches as its input approaches a specific value, but never actually reaches it

Question 2

Direct Substitution

Answer 2

Plug in the value x is approaching.
real # = answer
#/0 = DNE
0/0 or inf./inf. = keep going

Question 3

Algebra Skills

Answer 3

Factor out
rationalize
common denominator

Question 4

FEPL

Answer 4

Use for determining whether numerator or denominator inc/dec faster.Use when 0/0 or inf./inf
Factorial
Exponential
Polynomial
Logarithmic

Question 5

W.M.M. (What Matters Most)

Answer 5

Limit technique where you simplify to most ‘important’ terms on top and bottom. Use when 0/0 or inf./inf

Question 6

Hole - Removable discontinuity

Answer 6

lim (x->c) F(x) must exsist
F(c) can be defined or und.
the limit cannot equal F(c)

Question 7

Non-removable discontinuity

Answer 7

There is a vertical asymptote at x=c
lim(x->c) F(c) DNE
F(c) can be defined or und.
Can also be a jump.

Question 8

Continuity

Answer 8

At x=c f(c) exsists
lim (x->c) f(x) exsists
f(c) = the limit

Question 9

Derivative

Answer 9

slope of a tan line @ a point, nX^(n-1) <– basically
alt. def = lim (x->a) (f(x)-f(a))/ (x - a)
trad. def = F(a)= lim(h->0) (f(a+h) - f(a))/ h

Question 10

Relations Between f(x) f’(x) and f’‘(x)

Answer 10

f(x) | f’(x) | f’‘(x)
cc up | inc | pos
cc down | dec | neg
(POI) HTL | 0
Inc | pos
dec | neg
HTL | 0

Question 11

F’(c) is undef if…

Answer 11

-discontinuity
- different “left” and “right” tan lines (sharp turn, cusp)
-vertical tan line

Question 12

Notation For
First Derivative?
Second?

Answer 12

f’(x) = dy/dx

f’‘(x) d^2y/dx^2

Question 13

d/dx [sinx] = ?

Answer 13

cosx

Question 14

d/dx [cosx] = ?

Answer 14

-sinx

Question 15

product rule
d/dx [fg] = ?

Answer 15

= f’(g) + (f)g’

Question 16

quotient rule
d/dx [ f/g] = ?

Answer 16

= (f’(g) - (f)g’)/g^2

Question 17

d/dx [secx] = ?

Answer 17

= secxtanx

Question 18

d/dx [cscx]= ?

Answer 18

= -cscxcotx

Question 19

d/dx [ tanx] = ?

Answer 19

sec^2x

Question 20

d/dx [cotx] = ?

Answer 20

-csc^2x

Question 21

Equation of the line tangent to the graph of f

Answer 21

y - f(a) = f’(a) (x - a)
or
y - y1 = m (x-x1)

Question 22

Normal Line

Answer 22

A line that is perpendicular to a tan line at a given point

Question 23

Chain Rule
d/dx [f(g(x))] = ?

Answer 23

f’(g(x))(g’(x)0

Question 24

d/dx [lnx] = ?

Answer 24

(x’)1/x

Question 25

d/dx [e^x] = ?

Answer 25

(x')e^x

Question 26

d/dx [logaX] =

Answer 26

(X')1/xlna

Question 27

d/dx [ a^x] = ?

Answer 27

(a^x)(lna )(x')

Question 28

Critical #'s on graph of f(x)

Answer 28

when f'(x) = 0 or when f'(x) undef

Question 29

SIGN CHART

Answer 29

chart used to determine points at which y changes from inc to dec or y' pos to neg ( and vise versa, good for dtermining relatives max)

Question 30

POI

Answer 30

Where f(x) changes curvature f''(x) = 0 f''(x) undef

Question 31

First Derivative Test

Answer 31

when f'(c) changes signs and critical value (sign chart) f has a rel. max. @ x=a if it changes from + to - f has a rel. min. @ x=a if it changes from - to + (always works)

Question 32

Second Derivative test

Answer 32

Criticals values, find concavity f has a rel. max. @x=a if f'(0) = 0 AND f''(x) < 0 f has a rel. min. @x=a if f'(0) = 0 AND f''(x) > 0 (doesn't always work)

Question 33

Average Velocity from t=p to t=q

Answer 33

v(avg) = dy/dx y(p) - y(q)/p-q slope of a sec line on position graph

Question 34

instantaneous velocity at time t

Answer 34

v(t) = y' slope of a tan line on position graph

Question 35

Avg. acceleration from t= p to t= q

Answer 35

a(avg) = v(p) -v(q) / p-q slope of a sec line on velocity graph

Question 36

instantaneous acceleration at time t

Answer 36

a(t) = v'(t) slope of tan line on vel. graph

Question 37

Speed

Answer 37

Magnitude of velocity |velocity| .

Question 38

Speed is inc when... Speed is dec when...

Answer 38

v(t) and a(t) have same signs v(t) and a(t) have different signs

Question 39

Implicit differentiation

Answer 39

deriving with respect to variable and proerpyl denoting it eg derivative of 2y = x^2 wrt x 2dy/dx = 2x

Question 40

d/dx [ arcsinu] =

Answer 40

(u') x 1/√1-x^2 bound from (-pi/2, pi/2)

Question 41

d/dx [arccosu] =

Answer 41

(u')x -1/√1-x^2 bound from (0, pi)

Question 42

d/dx [arctanu] =

Answer 42

(u')x 1/x^2 +1 boundn from (-pi/2 , pi/2)

Question 43

d/dx [ arccotu] =

Answer 43

(u') x -1/x^2 +1

Question 44

d/dx [arcsecu] =

Answer 44

(u'1) x 1/|x|√x^2-1

Question 45

d/dx [arccsc] =

Answer 45

(u')x -1/|x|√x^2-1

Question 46

pythagorean theorem

Answer 46

a^2 + b^2 = c^2

Question 47

perimeter / circumference

Answer 47

c= 2rpi

Question 48

trigonemtic ratios sin(x) cos(x) tan(x)

Answer 48

s = o/h c = a/h t = o/a

Question 49

Area of... Parellelogram triangle circle

Answer 49

A=bh A=(1/2)bh A= pi(r^2)

Question 50

Volume of... prism cylinder

Answer 50

V=bh v=pi(r^2)(h)

Question 51

Steps for related rates

Answer 51

1. Identify rates, relationship, and moment 2. Implicityly differentiate relationship with respect to time, t 2. plug in knows and solve for unknown quanity

Question 52

Rates? Moment? Relationship?

Answer 52

d/dt the time that is given t(initial) = 0, find some at a specific time, t the equation you use, use this to differentiate

Question 53

Absolute Extrema

Answer 53

Highest/ lowest point on graph 1) find f' 2) Critical values 3) create a candidates test

Question 54

Is VS AT

Answer 54

the abs. min/max IS c, IT occurs at x= a

Question 55

Optimization Guidelines

Answer 55

1. Use the secondary equation to write the primary equation 2. Differentiate and find critical points 3. (If more than one critical point, do a canidates test) 4. Be sure to answer question asked

Question 56

Primary Equation (in regard to optimization)

Answer 56

What you want to minimize or maximize

Question 57

Secondary Equation (In regard to optimiaztion)

Answer 57

what you know about the variables' relationship

Question 58

L'hopitals Rule lim (x->c) f(x)/g(x)

Answer 58

Use if 0/0 or inf./inf. occurs when finding limit take the derivative of numerator and a denominator seperately: lim(x->c) f'(x)/g'(x)

Question 59

LRAM

Answer 59

overestimates when f is dec underest. when f is inc

Question 60

RRAM

Answer 60

overest when f is inc underest when f is dec

Question 61

MRAM

Answer 61

overest when f is cc down underest when f is cc up

Question 62

TSAM

Answer 62

overest when f is cc up underest when f is cc down

Question 63

Indefinite integral

Answer 63

∫x+2dx = notation (1/n+1)x^n+1 +C ∫x+2dx = 1/2x^2 +2x + C MUST ADD PLUS C WITH INDEFINITE INTEGRALS

Question 64

∫cosu du =

Answer 64

du= sinu + C

Question 65

∫-sinu du =

Answer 65

du= cosu + C

Question 66

∫sec^2u du

Answer 66

tanu +C

Question 67

∫-csc^2u du

Answer 67

cotu + C

Question 68

∫secutanu du

Answer 68

secu + C

Question 69

∫-cscucotu du

Answer 69

cscu + C

Question 70

BUT ∫sinu du =

Answer 70

-cosu + C (antiderivative)

Question 71

Find the particular soln.

Answer 71

intergrate and then Plug into given points to find C

Question 72

SOLVE the differential equation

Answer 72

intergrate, solve for c, and then solve for final answer

Question 73

Definite Integrals

Answer 73

Find area under curve bounded ∫ with bounds at bottom and top (from a to b) f(b)-f(a) integrate, then plug in bounds (top to bottom) and subtract the two

Question 74

First fundamental theorem of calculus

Answer 74

bound from a to b ∫f(x)dx = F(b) -F(a), as long as f(x) is continuous equivalently bound from a to b ∫g'(x) dx = g(b)-g(a)

Question 75

U-Sub

Answer 75

Needed when you have a chain rule in an integral, you want to substitue the inner most prod with u, take its derivative, and divide the new du by it. It should ideally cancel out other variables to make it easily integrateable

Question 76

Integrating Quaotients: 1.Is the denominator a monomial? 2. Are the numerator and denominator both true polynomials? - is the degree of the denom. > then the numer.

Answer 76

1.Yes! Then just divide it out 2. Yes! No! Start with long division

Question 77

∫u^n du =

Answer 77

1/(n+1) x (u^n+1 ) +C

Question 78

∫a^u du=

Answer 78

(a^u)x1/lna + c

Question 79

∫1/u du =

Answer 79

ln|u| + C

Question 80

∫e^u du =

Answer 80

e^u + C

Question 81

∫tanu du =

Answer 81

-ln|cosx|+ C

Question 82

∫secu du =

Answer 82

ln|secx + tanx| +C

Question 83

∫cotu du =

Answer 83

ln|sinx|+ C

Question 84

∫cscu du =

Answer 84

-ln|cscx + cotx|+ C

Question 85

∫1/1^2 + u^2 du =

Answer 85

arctanu+ C

Question 86

∫1/(1^2 - u^2)^1/2 du =

Answer 86

arcsinu + C

Question 87

∫1/(u)(u^2 - 1^2)^1/2 du=

Answer 87

arcsec|u|+ C

Question 88

∫1/a^2 + u^2 du =

Answer 88

(1/a)arctan(U/a) + C

Question 89

∫1/(a^2 - u^2)^1/2 du =

Answer 89

arcsin(u/a) + C

Question 90

∫1/(u)(u^2 - a^2)^1/2 du =

Answer 90

(1/a)arcsec|u/a| + C

Question 91

degree of denominator > degree of numer. BY MORE THAN ONE

Answer 91

likely inverse trig function

Question 92

Heaveside method (2x^2 - 13x + 3)/(x+1)(x-1)(x-2)

Answer 92

A/(x+1) + B/(x-1) + C/(x-2) 2. make into common denominator and simplify 3. plug in the value for each letter (a= -1 b=1 c=2) and solve for each respectively 4. Plug back into the letters in first step

Question 93

Partial Fractions

Answer 93

Needed when the denom. is factorable

Question 94

Integration By Parts

Answer 94

Given: ∫f(x)g(x) dx =? Choose variables u and v such that: u= f(x) dv=g(x)dx Originall changes to -> ∫udv =? ∫u dv = uv - ∫v du

Question 95

How to choose U for IBP

Answer 95

L)og I)nverse trig A)ny alg (poly, power) T)rig E)xponential

Question 96

Table method for IBP

Answer 96

Cross arrow down, start with + and alt.

Question 97

Improper Integrals

Answer 97

1. upper or lower bound is inf./-inf. respectively 2. any bound that makes the function discontinuous 3. any values w/in the bounds that make the function discontinuous

Question 98

Converges when..

Answer 98

the function approaches a value as x apparoaches inf. or -inf. lim (x-> inf.) = #

Question 99

diverges when

Answer 99

lim f (x->inf.) DNE

Question 100

2nd Fundamental theorem of calculus: Common Case

Answer 100

bound from a to x ∫f(t) dt = f(x)

Question 101

2nd FTC Gneral case

Answer 101

bound from u to v ∫f(t)dt = v'f(v) - u'f(u)

Question 102

how to find missing point given another

Answer 102

f(b) = f(a) + bound a to b ∫f"(t) dt

Question 103

from a to b ∫rate dt =

Answer 103

value - value

Question 104

net distance from a to b|

Answer 104

p(b) -p(a) = bound a to b ∫V(t)dt

Question 105

total distance

Answer 105

from a to b ∫|v(t)|dt

Question 106

avg. vel from a to b

Answer 106

[v(b) - v(a)] / b - a

Question 107

avg speed

Answer 107

|velcotiy|

Question 108

avg rate of chance from a to b

Answer 108

v(b) - v(a) / b - a

Question 109

avg value from a t o b

Answer 109

bound a to b ∫F(x)dx / b-a

Question 110

IVT

Answer 110

If f is continuous, then for any value k between f(a) and f(b) there is some c where f(c) =k

Question 111

EVT

Answer 111

If f is continuous then f(x) has an abs min/max value on [a,b]

Question 112

MVT

Answer 112

If f is continiuous and differential , then there is some c on a < c < b such that f'(c) = f(b) - f(a) / b - a

Question 113

Rolles Theorem

Answer 113

if f is continuous and differential, then there is some c such that f'(c) = 0

Question 114

MVTI

Answer 114

If f is continuous, then there is some c, a

Question 115

Area Between Curves

Answer 115

from a to b ∫(top-bottom) dx or from a to b ∫(right - left) dy

Question 116

volume b/ cross-section

Answer 116

find area in terms of b, then plug (top-bottom) for b, then integral then boom done

Question 117

Disc method

Answer 117

bound a to b pi∫ r^2 dx or dy

Question 118

washeer method

Answer 118

A = piR^2 - pir^2 bound a to b v=∫A dy or dy

Question 119

Arc length formulas

Answer 119

l = bound a to b ∫[I^2 + (dy/dx)^2]^1/2 dx I= (dx/dx) l = bound a to b ∫[I^2 + (dx/dy)^2]^1/2 dy I=dy/dx

Question 120

eulers method

Answer 120

intial condition: (xi, yi) given point step size: h (change of x, the wifth of each x) number of steps: n how many itteriation we use x -> x + vx | y-> y +vy | y'= dy/dx =/ vY/vX vY=/ y'(vX)

Question 121

proprotionality y=kx -> y= k/x z = k(xy)

Answer 121

direct inverse jointly

Question 122

shortcut dy/dx = ky

Answer 122

y=De^kx

Question 123

logisitcs eq

Answer 123

dy/dx = k(y) [1-y/L] y= L/[1 + be^-kx] k= constant of proportionality L = carry capacity b= consant