Calculus formulas

never get a 60 on a formula quiz again

arc length

arc length

when y = f(x) --> L = ∫ sqrt[1 + (f'(x))^2] dx from A to B

when x = g(y) --> L = ∫ sqrt[1 + (g'y)^2] dy from A to B

integration by parts

∫ u dv = uv - ∫ v du

sin 2x

2sinxcosx

cos 2x

cos^2 x - sin^2 x

cos^2x (Power to Double Angle)

1/2(1+cos2x)

sin^2x (Power to Double Angle)

1/2(1-cos2x)

sin(-x)

-sinx

cos(-x)

cosx

tan(-x)

-tanx

|x|

x if x>=0 -x if x<0

sin(A+B)

sinAcosB+cosAsinB

sin(A-B)

sinAcosB-cosAsinB

cos(A+B)

cosAcosB-sinAsinB

cos(A-B)

cosAcosB+sinAsinB

sinAcosB

1/2[sin(A-B)+sin(A+B)]

sinAsinB

1/2[cos(A-B)-cos(A+B)]

cosAsinB

1/2[sin(A+B)-sin(A-B)]

cosAcosB

1/2[cos(A-B)+cos(A+B)]

Distance Formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

Midpoint Formula

(x₁+x₂)/2, (y₁+y₂)/2

ln (ab)

ln a + ln b

ln (a/b)

ln a - ln b

ln (a^n)

n ln a

ln (1/a)

-ln a

One-sided limits

lim x--> 2^- (approaches from the left) lim x--> 2^+ (approaches from the right)

Definition of a Limit

the limit of f(x), as x approaches a, equals L iff lim x--> a^- (f(x))= lim x--> a^+ (f(x))

Limit Laws

lim (f±g) = lim f ± lim g lim (c ⋅ f) = c ⋅ lim f lim (fg) = lim f ⋅ lim g lim (f/g) = lim f / lim g for lim g ≠ 0 lim √f(x) = √lim f(x) lim c = c lim f(g(x)) = f lim g(x)

Definition of a Vertical Asymptote

lim x-->a+ = +-∞ or lim x-->a- = +-∞

Definition of Horizontal Asymptote

lim x-->+∞ = a or lim x-->-∞ = a

Limits of the Ratios of Two Functions

if f grows faster than g then limx-->∞ g(x)/f(x) = 0 and limx-->∞ f(x)/g(x) = ∞

Definition of Continuity

1. f(a) exists.

2. lim x→a f(x) exists.

3. lim x→a f(x) = f(a)

Intermediate Value Theorem

if:

1. f is continuous on [a,b]

2. f(a)≠f(b)

3. k is a number between f(a) and f(b) then : there exists a number c on [a,b] such that f(c)=k

Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) and limx→a f(x) = limx→a h(x) = L, then limx→a g(x) = L

lim x->∞ e^x

∞

lim x->-∞ e^x

0

Lim x-> 0+ lnx

-∞

lim x->∞ lnx

∞

Lim x->0 (e^(x)-1) / x

=1

lim of sinx/x as x approaches 0

1

Lim of (1-cosx)/x as x approaches 0

0

lim as x approaches +-infinity (1+c/x)^x

e^c

lim as x approaches 0+ (1+cx)^1/x

e^c

difference quotient

f'(x)=lim h->0 (f(x+h)-f( x))/h

alternate form

f'(c) = lim x->b (f(x)-f(b))/(x-b)

Normal Line

a line perpendicular to a tangent line at the point of tangency

average rate of change

Slope of secant line between two points, use to estimate instantanous rate of change at a point

f(b)-f(a)/b-a

Reasons a function would not be differentiable at a point

1. f not continuous at x=a

2. The graph of f has a "corner" or "cusp" at x=a

3. The graph of f has a vertical tangent at x=a

pos, velocity, speed, acceleration

positions: s(t) v(t)=velocity=s'(t) |v(t)|= speed a(t)=v'(t)=s''(t)=acceleration

at rest, speeding up, slowing down

1. patricle at rest when v(t)=0

2. speed is increasing if: v and a are same sign

3. speed is decreasing if: v and a are different signs

average velocity

same as average rate of change of the position

Product Rule

f'g+fg'

Quotient Rule d/dx(f/g)

(f'g-fg')/g^2

Chain Rule

d/dx f(g(x)) = f'(g(x)) g'(x)

Power Rule

d/dx x^n = nx^n-1

y' a^x

a^x(ln a)

y'(sinx)

cosx

y'(cosx)

-sinx

y'(tanx)

sec^2(x)

y'(cscx)

-cscxcotx

y'(secx)

secxtanx

y'(cotx)

-csc^2(x)

Linear Approximation

y = f(a) + f'(a)(x-a)

y'(arcsinx)

1/sqrt(1-x^2)

y'(arctanx)

1/(1+x^2)

y'(arcsecx)

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

y'(lnx)

1/x

y'(logax)

1/xlna

lim x-> -infinity arctanx

-pi/2

lim x-> infinity arctanx

pi/2

inverse function derivative

1/(f'(f-1(x))

derivative of cos-1

-1/sqr (1-x^2)

derivative of csc-1

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

derivative of cot-1

-1/(1+x^2)

L'Hopital Rule

this rule can only be used directly when you get the indeterminate forms; take derivatives of top and bottom separately

Extreme Value Theorem

1. If f is continuous on [a,b] 2.then f has an absolute maximum and an absolute minimum on [a,b].

Definition of a Critical Point

A critical point of f is a number c such that either

1. f'(c) = 0 (stationary point)

2. f'(c) does not exist (singular points)

Candidates test for extreme values on a closed interval

1. find the y values of critical points in the interval [a,b]

2. find y values of endpoints, a and b

3. the largest y-value is the maximum and the smallest yvalue is the minimum

Mean Value Theorem

if f(x) is continuous and differentiable then slope of tangent line equals slope of secant line at least once in the interval (a, b)

f '(c) = [f(b) - f(a)]/(b - a)

Rolle's Theorem

Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0

1st Derivative Test

f(c) is classified as:

1. If f'(x) changes from - to + at c, then f has rel. min at

2. If f'(x) changes from + to - at c, then f has rel. max at

3. If f'(x) is + on both sides of c or - on both sides of c⟶f(c) is neither max nor min

Test for Concavity

1. If f''(x) > 0 f is concave up

2. If f''(x) < 0 concave downward on I.

inflection point

concavity changes; is probable if second derivative is zero

Test for Inflection Points

f" changes signs f" = 0 or dne

2nd derivative test

If f'(c) \= 0 and f"(c)

∫ x^n du

x^(n+1)/n+1 + C n does not equal -1

∫ 1/u du

ln |u| + C

∫e^u du

e^u + C

∫sin u du =

-cos u + C

∫cos u du =

sin u + C

∫sec^2udu

tanu+c

∫cscucotu du =

-cscu + C

∫secutanu du

sec u + C

∫csc^2udu

-cotu+c

△x (rieman sum)

△x = b - a / n (for intervals)

LRAM

left rectangular approximation method

RRAM

right rectangular approximation method

MRAM

midpoint rectangular approximation method

Lower vs Upper Sum

use he smallest y - value for the heights

use the largest y -value for the heights

TAM

trapezoid rule: bases are y - values, height is △x

∫du/√(a² + u²)

sin^-1(u/a)+C