MAT 191 / Calculus I Key Information

Limit

f(x) is defined on some open interval that contains c, except possibly at c itself.

Lim x->c f(x) = L

"the limit of f(x) equals L, as x approaches c"

This means we can make values of f(x) arbitrarily close to L by restricting x to be close to c on either side of c but not equal to c.

One-Sided Limits

Lim x->c- f(x) = L

"the limit of f(x) equals L, as x approches c from the left"

meaning if we can make values of f(x) close to L by taking x to be sufficiently close to c with x less than c.

Lim x->c+ f(x) = L

"the limit of f(x) equals L, as x approaches c from the right"

meaning if we can make values of f(x) close to L by taking x to be sufficiently close to c with x more than c.

Identifying if a limit exists:

1. if the limit as x approaches c from the right = the limit as x approaches c from the left.

Definition of an Infinite Limit

Suppose f(x) is defined on some open interval that contains c, except possibly at c itself. We then write:

lim x-> c f(x) = infinity

if the values of f(x) can be made arbitrarily large by taking x to be sufficiently close to c, but not equal to c, likewise we write:

lim x-> f(x) = -infinity

if the values of f(x) can be made arbitrarily large negative by taking x to be sufficiently close to c, but not equal to c.

Definition of Vertical Asymptote

The vertical line x = c is called a vertical asymptote of the curve y = f(x) if at least one of the following statements is true:

lim x-> c f(x) = infinity

lim x-> c- f(x) = infinity

lim x-> c+ f(x) = infinity

lim x-> c f(x) = -infinity

lim x-> c- f(x) = -infinity

lim x-> c+ f(x) = -infinity

Limit Law 1

lim x -> c [f(x) + g(x)] = lim x-> c f(x) + lim x-> c g(x) = L + M

let lim x-> c f(x) = L

& lim x-> c g(x) = M

Limit Law 2

lim x -> c [f(x) - g(x)] = lim x-> c f(x) - lim x-> c g(x) = L - M

let lim x-> c f(x) = L

& lim x-> c g(x) = M

Limit Law 3

lim x->c k f(x) = k lim x->c f(x) = k*L

let k be a real number

let lim x-> c f(x) = L

Limit Law 4

lim x -> c [f(x) g(x)] = (lim x-> c f(x)) (lim x-> c g(x)) = L * M

let lim x-> c f(x) = L

& lim x-> c g(x) = M

Limit Law 5

lim x -> c f(x) / g(x) = lim x-> c f(x) / lim x-> c g(x) = L / M , provided M does not = 0

let lim x-> c f(x) = L

& lim x-> c g(x) = M

Limit Law 7

lim x->c [f(x)]^n = [lim x->c f(x) ] ^n

Limit Law 8

lim x->c k = k

let k be a real number

Limit Law 9

lim x->c x = c

Limit Law 10

lim x->c ^n sqrt f(x) = ^n sqrt lim x->c f(x) = ^n sqrt L

Direct Substitution Property

If f(x) is a polynomial, a rational function, or any of the six trigonometric functions and c is in the domain of f, then lim x->c f(x) = f(c)

Squeeze/Sandwich Theorem

suppose we have 3 functions f,g,h defined on an open interval containing x=c, except possibly at c itself. If f(x) < or = to g(x) < or = to h(x) for all x in the open interval and lim x->c f(x) = L = lim x->c h(x), then lim x->c g(x) =L

Definition of Continuity

A function f is continuous at x = c if lim x->c f(x) = f(c)

Continuity Check List

1. f(c) MUST be defined (closed circle at x=c)

2. a limit MUST exist ( lim x->c- f(x) = lim x->c+ f(x))

3. f(c) MUST EQUAL lim x->c f(x)

Important Theorem (forgot name :( )

if functions f,g are continuous at x=c , then ea. of following are also continuous at x=c:

1. f +/- g

2. f*g

3. k*f(x) {k is a constant}

4. f/g {provided g(c) does not = 0}

Another theorem

suppose f is continuous at x=b. if lim x-> g(x) = b then lim x->c f(g(x)) = f (lim x->c g(x))

Intermediate value theorem

suppose f is continuous on [a,b] Let M be any value b/w f(a) and f(b), then there exists an x-value c such that a

Correlary to IVT

Suppose f is continuous on [a,b] If f(a) and f(b) have opposite signs, then there exists an x-value c such that a

Removable Discontinuity

lim x->c f(x) exists but either f(c) is undefined or f(x) exists but isn't = limit

jump discontinuity

lim x->c+ f(x) = L (exists) & lim x->c- f(x) = M (exists) BUT L not = to M

limit from right does not equal limit from left

infinite discontinuity

vertical asymptote

3 ways a limit can fail to exist

1. jump

2. asymptote

3. oscillation

e^x

3 ways a function can fail to be differentiable

1. vertical tangent line

2. sharp corner / cusp

3. discontinuity (hole, jump, asymptote)

3 ways a function can fail to be continuous

1. asymptote

2. hole

3. jump

Surface Area of a Cone

πr(r + sqrt(h^2 + r^2))

Volume of a Cone

1/3πr^2h

Area of Circle

πr^2

Volume of a Cube

x^3

Quadratic Formula

(-b ± √(b² - 4ac))/2a

Circumference of a circle

2πr or 2d

volume of a cylinder

πr^2h

surface area of a cylinder

2πr^2 + 2πrh

d/dx cscx

-cscxcotx

d/dx cotx

-csc^2x

d/dx sinx

cosx

d/dx arcsinx

1/(sqrt(1-x^2))

d/dx secx

secxtanx

d/dx cosx

-sinx

d/dx lnx

1/x

d/dx arccscx

-1/(|x| sqrt(x^2 - 1))

d/dx logbx or b^y = x

1/ (x ln b)

d/dx tanx

sec^2x

d/dx arcsecx

1/(|x| sqrt(x^2-1))

d/dx arccosx

-1/(sqrt(1-x^2))

d/dx arccotx

-1/ (1 + x^2)

d/dx arctanx

1/(1+x^2)

surface area of sphere

4πr^2

volume of sphere

4/3 πr^3

Pythagorean Theorem

a²+b²=c²

Logarithm power rule

log b (x^y) = y * log b (x)

special trig functions

lim x->0 sinx/x = 1 or lim x->0 x/sinx = 1

lim x->0 1-cosx/x = 0 or lim x->0 cosx - 1 / x = 0

ln(e)

1

ln(1)

0

ln(0)

DNE

e^1

e

e^0

1

when should we use logarithmic differentiation ?

when taking a derivative of a variable raised to a variable

When should we replace y's when differentiating ?

during logarithmic differentiation and sometimes when taking 2nd implicit differentiation

logarithm product rule

log b (x * y) = log b (x) + log b (y)

logarithm quotient rule

log b (x/y) = log b (x) - log b (y)

when is a particle speeding up?

when the signs of the velocity and acceleration are the same

when is a particle at rest ?

when velocity is equal to zero

In related rates: what is a derivative

a derivative is a rate of change in x with respect to y so look for words like, "increasing", "decreasing", "rate", or "changing"

ch 4 . Local(relative) Max Definition

given that a function f is defined on an open interval then,

f has a local max at f(c) if f(c) >/= f(x) for all x-values on interval

there should NOT be more than one local max on an interval b/c y-value should only be one value

(highest value on an interval)

local(relative) min definition

given that a function f is defined on an open interval then,

f has a local min at f(c) if f(c)

Global(absolute) Max Definition

given a function f is defined on its DOMAIN

f has a global max at f(c) if f(c) >/= f(x) for all x in the domain of f

global(absolute) min

given a function f is defined on its DOMAIN

f has a global min at f(c) if f(c)

T/F every local max is a global max

F

T/F every global max is a local max

T

T/F given a function on closed interval , f must contain a global max

F, the function must be CONTINUOUS

extreme value theorem

every CONTINUOUS function on a CLOSED interval has a max and min value on that interval

A function has a critical point at x=c if ...

1. c is in the domain of the function

2. f'(c) = 0 or f'(c) = DNE

Random Theorem

if f has a local max or min when x=c , then c is a critical point

T/F a critical value means that there is a local max or min

F ; just b/c there is a critical value does NOT mean that it is a local max/min

Rolle's Theorem

Suppose f is continuous on [a,b] and differentiable on (a,b) then if f(a) = f(b) then there exists an x-value c where a

Mean Value Theorem

Suppose f is a function if

1. f is cont. on [a,b]

2. f is differentiable on (a,b)

then there exists an x-value c where a

(a,b)

open interval (endpoints dont matter)

[a,b]

closed interval (endpoints matter)

Definition of increasing

f is increasing on an interval, I , if f(x1) < f(x2) for all x 1, x2 in I where x1

Definition of decreasing

f is decreasing on an interval, I , if f(x1) > f(x2) for all x 1, x2 in I where x1

Theorem for inc/dec

suppose f is defined on an interval , I ,

1. if f(x) > 0 (positive) for all x in I, then f is increasing on I

2. if f(x) < 0 (negative) for all x in I, then f is decreasing on I

3. if f(x) = 0 for all x in I, then f is constant on I

Horizontal Asymptotes/ Litmits at infinty : BOBO BOTN

Bigger on bottom = 0

Bigger on top = none

Indeterminate Forms where L'Hospital's rule applies

L'Hospital's Ready: after direct substitution yields ---> 0/0 or infinity/infinity

Must Rewrite the x-variable to create a fraction: after direct substitution yields infinity - infinity or 0 infinity ... rewrite such as: x ln(x) ---> ln(x)/(1/x), then try direct substitution which should yield 0/0 or infinity/infinity

must use log properties to work towards 0/0 or infinity/infinity : after direct substitution yields 1^infinity or 0^0 or infinity^0

delta X

(b-a)/n

Rsubi

a + deltaX*i

Antiderivative

a function F is an antiderivative of f on an interval I if F'(x) = f(x) for all x in I.

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