Calculus BC Memory Sheet

sin² (x) + cos² (x)

1

cot²(x) + 1

csc²(x)

½ (1 - cos2θ)

sin² (θ)

tan² θ + 1

sec² θ

½ (1+ cos 2θ)

cos² θ

d/dx (uv)

u v’ + vu”

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

f’(g(x)) g’(x)

d/dx (ln u)

u’ / u

d/dx (sin u)

(cos u) u’

d/dx (tan u)

(sec²u) u’

d/dx (sec u)

(sec u tan u) u’

d/dx (arcsin u)

u’ / sqrt(1-u²)

d/dx (arctan u)

u’ / 1 + u²

d/dx (arcsec u)

u’ /( abs(u) sqrt(u² - 1))

d/dx a^u

a^u ln(a) u’

d/dx (uⁿ)

nu^n-1 u’

d/dx abs(u)

u/(abs(u)) u’

d/dx e^u

e^u u’

d/dx (cot u)

-(csc² u) u’

d/dx (csc u)

-(csc u cot u) u”

d/dx (arccos u)

-u’ / sqrt(1-u²)

d/dx (arccot u)

-u’ / (1+ u²)

d/dx (arccsc u)

-u’ / ( abs(u) sqrt(u² -1))

∫ k f(u) du

k ∫ f(u) du

∫ tan u du

-ln |cos u| + C = ln |sec u| + C

∫ sec u du

ln |sec u + tan u| +C

∫ sec² u du

tan u +C

∫ cot u du

ln |sin u| +C

∫ csc u du

-ln |csc u + cot u| +C

∫ csc² u du

-cot u +C

∫ sec u tan u du

sec u +C

∫ du/ sqrt(a² - u²)

arcsin u/a +C

∫ du/ usqrt(u² -a²)

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

∫ csc u cot c du

-csc u +C

∫ du/ (a² + u²)

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

∫ a^u du

a^u (1/ ln(a)) +C

∫ u dv

uv - ∫ v du

(px + q) / ((x-a) (x-b))

(A/ (x-a)) + (B/ (x-b))

(px² + qx +r) / ((x-a) (x-b) (x-c))

(A/(x-a)) + (B/(x-b)) + (C/(x-c))

Continuity

A function y=f(x) is continuous at x=a if: 1. f(a) exists, 2. lim (as x—>a) f(x) = f(a)

AROC

Δy / Δx; slope secant to the graph; multiple points

IROC

(a, b) —> f’(a); slope tangent to the graph; one point

f’(x)

lim (h—>0) f(x+h)-f(x) / h

f’(a)

lim (x—>a) f(x)-f(a) / x-a

if f’(x) > o on an interval

then f is increasing on the interval

if f’(x) < 0 on an interval

then f is decreasing on the interval

if f’’(x) > 0 on an interval

then f’ is increasing on the interval and f is concave up

if f’’(x) < 0 on an interval

then f’ is decreasing on the interval and f is concave down

lim(n—>∞) (1+ 1/n)ⁿ

e

lim(n—>0) (1+n)^1/n

e

Mean Value Theorem

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

Intermediate Value Theorem

if f is continuous on [a, b] with f(a) ≠ f(b) and k is a number between f(a) and f(b), there exists at least one number c in (a, b) for which f(c) = k

Critical Number

a number c in the interior of the domain of a function is called a critical number of f if either f’(c)=0 of f’(c) is not defined

Point of Inflection

A point of the graph of a function where there is a tangent line and the concavity changes

The 1st Derivative Test

if c is a critical number and if f’ changes sign at x=c, then:

1) f has a local minimum at x=c if f’ is negative to the left of c and positive to the right

2) f has a local maximum at x=c if f’ is positive to the left of c and negative to the right

2nd Derivative Test

1) if f’(c)=0 and f’’(c) >0, then f has a local minimum at c

2) if f’(c)=0 and f’’(c) <0, then f has a local maximum at c

Trapezoidal Rule (if interval is consistent)

if a function f is continuous on the closed interval [a, b] where [a, b] has been partitioned into n subintervals [x₀, x₁], [x₁, x₂], …, [x_n-1, x_n] each of length (b-a)/n, then ∫[a, b] f(x) dx ≈ (b-a)/2n [f(x₀) +2f(x₁)+…+2f(x_n-1)+f(x_n)] if interval is not consistent then find area of the each trapezoid and add them up

Average Value

1/(b-a) ∫ [a, b] f(x) dx

Euler’s Method

for a function, containing the point (x₁, y₁), the point (x_2,y₂) = (x₁+∆x, y₁+∆y) which is also on the curve may be approx. by letting ∆ y = ∆ x (dy/dx (x₁, y₁))

Arc Length = distance traveled along curve/perimeter

l = ∫[a,b] sqrt(1+(f’(x))²) dx or l = ∫[a,b] sqrt(1+ (dy/dx)²) dx

d²y / dx²

(d/dt [dy/dx]) / (dx/dt)

r cos θ

x

rsin θ

y

tan θ

y/x

r²

x² + y²

Area in Polar Coordinates

A= 0.5∫[α,β] [f(θ)]² dθ = 0.5∫[α,β] r² dθ

dy/dθ

r’sinθ + rcosθ

dx/dθ

-rsinθ + r’cosθ

derivative in polar form

(dy/dθ) / (dx/dθ)

Fundamental Theorems of Calculus

1) d/dx ∫[a,u] f(t) dt = f(u)u’, where u is a function of x

2) ∫[a,b] f(x) dx = F(b) - F(a), where F’(x) = f(x)

∫[a,b] c ⋅ f(x) dx

c ⋅ ∫[a,b] f(x) dx

∫[a,a] f(x) dx

0

∫[a,b] f(x) dx

-∫[b,a] f(x) dx

∫[a,b] f(x) dx

∫[a,c] f(x) dx + ∫[c.b] f(x) dx, where f is continuous on an interval containing a, b, and c

if f(x) is an odd function

then ∫[-a,a] f(x) dx = 0

if f(x) is an even function

then ∫[-a,a] f(x) dx = 2∫[0,a] f(x) dx

if f(x) ≥ 0 on [a,b]

then ∫[a,b] f(x) dx ≥ 0

if g(x) ≥ f(x) on [a,b]

then ∫[a,b] g(x) dx ≥ ∫[a,b] f(x) dx

what series qualifies for nth-term test?

∑[n=1, ∞] a_n

What shows convergence on nth-term test?

nothing; test does not test for convergence

What shows divergence on nth-term test?

lim(n→∞) a_n ≠ 0

what series qualifies for geometric series test?

∑[n=0, ∞] arⁿ

What shows convergence on geometric series test?

|r| < 1

What shows divergence in geometric series?

|r| ≥ 1

Geometric series test comment:

sum = a/ 1-r

series for p-series?

∑[n=1, ∞] 1/n^p

when does converge for p series?

p > 1

en does diverge for p-sereis?

p ≤ 1

p-sereis comments

can be verified by integral test

series for alternating series

∑[n=1, ∞] (-1)^(n-1) a_n

when does alternating series converges?

0 < a_n+1< a_n AND lim[n→∞] a_n= 0

when does alternating series diverege

never

alternating series comments

|R_n| ≤ a_n+1

when to sue integral test

1) f must be positive, decreasing, and continuous

2)∑[n=1, ∞] a_n , a_n=f(n)

when does integral test converege

∫[1,∞] f(x)dx converges

when does integral test deivereg

∫[1,∞] f(x)dx diverges

integral test comment

R_n < ∫[n,∞] f(x)dx

sereis for ratio test

Σ[n=1, ∞] a_n

when does ratio test converege

lim[n→∞] |a_n+1 / a_n| < 1

when does ratio test diverege

lim[n→∞] |a_n+1 / a_n| > 1