MA141 FINAL WE LOCKED IN

Epsilon-Delta definition of a limit

the limit as x approaches a of f(x) = L for every epsilon (change in x) > 0 there is a delta (change in y) > 0 such that 0 < | x - a | < delta then | f(x) - L | < epsilon

L’Hopital Rule

when 0 infinity → limit as x goes to a is g(x)/(1/(fx))
when 1^infinity, infinity^0, 0^0, → e^ (limit as x goes to a of g(x) * ln(f(x)))

Continuity Definition

if f(n) exists, limit as x goes to n of f(x) exists, that limit = f(n), then the function is continuous

Squeeze Theorem

f(x) <= g(x) <= h(x), if the limit as x goes to c for f(x) and h(x) both equal L then limit as x goes to c for g(x) is also L

Intermediate Value Theorem

f is continuous on [a,b] and L is between f(a) & f(b)
then there is a n where a < n < b so that f(n) = L

Limit of the tangent line

m = limit as x goes to n ( f(x) - f(n) ) / ( x - n )

Types of Discontinuity

removable discontinuity - limits on both side exist and are equal but the function’s value at that point is different (a hole on a line)

jump discontinuity - left and right limits exist but are different

Horizontal Asymptote

limit as x goes to + or - infinity of f(x) = L then y=L is a horizontal asymptote

Vertical Asymptote

limit as x goes to a of f(x) = + or - infinity then x=a is a vertical asymptote

Sin x derivative

Cos x

Cos x derivative

-Sin x

Tan x derivative

Sec²x

Sec x derivative

(Sec x)(Tan x)

Cot x derivative

-Csc² x

Csc x derivative

(-Csc x)(Cot x)

Arcsin x derivative

1 / sqrt(1-x²)

Arccos x derivative

-1 / sqrt(1-x²)

Arctan x derivative

1 / (1 + x²)

Arccot x derivative

-1 / (1 + x²)

Arcsec x derivative

1 / ( |x| sqrt(x²-1) )

Arccsc x derivative

-1 / ( |x| sqrt(x²-1) )

e^x derivative

e^x

ln(x) derivative

1/x

a^x derivative

(a^x)(lna)

log_ax derivative

1 / ( x * lna)

Implicit Differentiation

when equation is in terms of x and y:

take derivative of both sides
solve for dy/dx
!!y is a function of x!!

Logarithmic Differentiation

when f has x in base and exponent:

take natural log of both sides and use rules to simplify
use implicit differentiation
solve for dy/dx
!!y is a function of x!!

limit as theta goes to 0 for (sin theta) / theta

= 1

limit as theta goes to 0 for (cos theta - 1) / theta

= 0

Linearization

L_C(x) = f(c) + f’(c)( x - c )
f(x) is about L_C(x)

Differentials

dy = f‘(c)Δx
f(x + Δx) ≈ f(x) + f’(x)Δx
Δy ≈ dy

Extreme Value Theorem

if a function is continuous on [a,b] then it has an absolute max and min

Fermat’s Theorem (critical points)

if f’(c) = 0 or DNE a function has local min or max @ x = c

Mean Value Theorem

f is continuous on [a,b] and differentiable on (a,b)
then there is a c in (a,b) such that f’(c) = f(b) - f(a) / b - a

Rolle’s Theorem

f is continuous on [a,b] and differentiable on (a,b)
then there is a c in (a,b) such that f’(c) = 0 (horizontal tangent line)

Finding global max and min on [a,b]

1) find f’(c) = 0 or DNE
2) plug critical points, a, b into f(x)
3) compare values

First Derivative Test

c is a critical point
f’ goes from + to -, local max @ x = c
f’ goes from - to +, local min @ x = c

Points of inflection

tells when the graph is concave up or down
f’’(c) = 0 or DNE

Second Derivative Test

c is a crit point, f’(c) = 0 for all of these
f’’(c) > 0, local min @ x = c (concave up)
f’’(c) < 0, local max @ x = c (concave down)
f’’(c) = 0, SDT fails, use FDT

Area and Riemann Sum

left and right: xi = a + iΔx
middle: xi = a + (i-1/2)Δx
Δx = b - a / n

Limit Definition of Definite Integral

right points: ∫_a^b f(x)dx = limit as n goes to infinity of n Σ i = 1 (f(x_i)Δx)
left points: ∫_a^b f(x)dx = limit as n goes to infinity of n-1 Σ i = 0 (f(x_i)Δx)
mid points: ∫_a^b f(x)dx = limit as n goes to infinity of n Σ i = 1 (f(m_i)Δx)

Upper and Lower Bounds for Estimating Definite Integrals

m <= f(x) <= M
then
m(b-a) <= ∫_a^b f(x) dx <= M(b-a)

Definite Integral Properties

∫_a^b c dx = c(b-a)
∫_a^a f(x) dx = 0
∫_a^c f(x) dx = ∫_a^b f(x) dx + ∫_b^c f(x) dx

FTC-1

f is continuous on [a,b]
G(x) = ∫_a^x f(t) dt for all x in [a,b]
G(x) is an antiderivative of f(t)
(basically just plug in the function of x into f(t) to get the antiderivative)

FTC-II

F is an antiderivative on [a,b]
∫_a^b f(x) dx = F(b) - F(a)

Integration by Parts

LIATE (log, inverse trig, algebraic, trig, exponential) ← for u values
uv - ∫vdu

Type-I Region

y = f(x) and y = g(x)
2 vertical lines x=a and x=b
f(x) >= g(x) then
∫_a^b f(x) - g(x) dx = Area

Type-II Region

x = f(y) and x = g(y)
2 horizontal lines y=a and y=b
f(y) >= g(y) then
∫_c^d f(y) - g(y) dy = Area

Disk Formula

rotation around horizontal line: ∫_a^b [f(x)]² dx = Area
rotation around vertical line: ∫_c^d [f(y)]² dy = Area

Washer Method

f is outer function and g is inner function
rotation around horizontal line: ∫_a^b [f(x)]² - [g(x)]² dx = Area
rotation around vertical line: ∫_a^b [f(y)]² - [g(y)]² dy = Area

Shell Method

rotation around vertical line: ∫_a^b 2pi r(x)h(x) dx = Area (r(x) = x if around x=0)
rotation around horizontal line: ∫_c^d 2pi r(y)h(y) dy = Area (r(y) = y if around y=0)

cos(ax) antiderivative

1/a * sin(ax) + c

sin(ax) antiderivative

-1/a * cos(ax)

sec²(ax) antiderivative

1/a * tan(ax)

sec(ax)tan(ax) antiderivative

1/a * sec(ax)

csc²(ax) antiderivative

-1/a cot(ax)

csc(ax)cot(ax) antiderivative

-1/a * csc(ax)

What inverse trig antiderivatives ARE known?

arcsin, arctan, arccos, arcsec

e^x antiderivative

e^x

1/x antiderivative

ln|x|

a^x antiderivative

a^x/lnx