calc formulas cummulative

tanx=

sinx/cosx

cotx=

cosx/sinx

secx=

1/cosx

cscx=

1/sinx

sin²x + cos²x =

1

sec²x - tan²x =

1

sin2x =

2sinxcosx

cos2x =

cos²x - sin²x

cos²x =

(1/2)(1 + cos2x)

sin²x =

(1/2)(1 - cos2x)

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))

cosAsinB

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

sinAsinB =

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

cosAcosB =

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

ln(ab) =

lna + lnb

ln(a/b)

lna - lnb

ln(aⁿ⁾

nlna

ln(1/a) =

-lna

limit

lim_x→af(x) = L

Definition of Continuity: function is continuous at x=a iff

1. f(a) exists

2. lim_x→af(x) exists

3. lim_x→af(x) = f(a)

Intermediate Value Theorem (IVT): If

1. f is continuous on the closed interval [a,b]

2. f(a) ≠ f(b)

3. k is between f(a) and f(b)

Then there exists a number c between a and b for which f(c) = k

Squeeze Theorem

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

Law of Cosines:

c² = a² + b² - 2abcosc

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C)

e^x graph

ln graph

definition of derivative formula

f’(x) = lim_h→0 [f(x+h) - f(x)] / h

Definition of derivative alternate form

f’(x) = lim_x→b[f(x) - f(b)]/x - b

Avg rate of change

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

Reasons f won’t be differentiable at point x=a:

1. f not continuous at x=a

2. the graph has a corner or cusp at x=a

3. the graph has a vertical tangent at x=a

chain rule

if h(x) = f(g(x)), then h’(x) = f’(g(x)) * g’(x)

product rule

d/dx (f * g) = f ‘g + fg’

quotient rule

d/dx (f/g) = [f’g - fg’]/g²

Linear Approximation

y = f(a) + f’(a)(x - a)

Inverse Functions

d/dx[f^-1(x)] = 1/f’(f^-1(x))

d/dx (xⁿ) =

n * x^n-1

d/dx sinx =

cosx

d/dx cosx =

-sinx

d/dx tanx =

sec²x

d/dx cscx =

-cscx * cotx

d/dx secx =

secx * tanx

d/dx cotx =

-csc²x

d/dx arcsinx =

1/√1-x²

d/dx arctanx =

1/ (1+x²)

d/dx arcsec x

1/ [ |x|√x²-1

d/dx e^x =

e^x

d/dx lnx =

1/x

d/dx log_ax

1/xlna

d/dx a^x

a^xlna

Extreme Value Theorum (EVT):

1. If f is continuous on a closed interval [a,b]

2. Then f has an absolute max and an absolute min on the interval [a,b]

critical points

f’(c) = 0 (stationary point)

f(c) DNE (singular point)

Mean Value Theorum

1. If f is continuous on [a,b]

2. differentiable on (a,b)

Then there exists a number c between a and b such that f’(c) = AVG rate change

Rolle’s Theorum

1. f is continuous on [a,b]

2. Differentiable on (a,b)

3. f(a) = f(b)

Then there is at least one number c on (a,b) such that f’(c) = 0

∫xⁿ du =

[xⁿ⁺¹] / [n+1] +C, n ≠ -1

∫1/u du =

ln|u| + C

∫e^u du =

e^u + C

∫a^udu =

a^u/lna + C

∫du/√a²-u² =

arcsin u/a + C

∫du/a²+u² =

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

∫du/u√u²-a² =

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

Total Distance Travelled

a to b ∫|v(t)|dt

Total displacement

a to b ∫v(t)dt