Calc gold sheets

∫ du

u+c

∫ uⁿ du

(uⁿ⁺¹/n+1)+c

∫ u^-1 du

ln|u| +c

∫ cos(u) du

sin(u) +c

∫ sin(u) du

-cos +c

∫ sec²(u) du

tan(u) +c

∫ csc²(u) du

-cot(u) +c

∫ sec(u)tan(u) du

sec(u) +c

∫ csc(u)cot(u) du

-csc(u) +c

∫ e^u du

e^u +c

∫ a_^u du

a^u/ln(a) +c

∫ ln(u) du

u ln(u)-u +c

∫ log_a(u) du

(u ln(u)-u)/ ln(a) +c

∫ tan(u) du

ln|sec(x)| +c or -ln|cos(x)| +c

Euler’s Method (notation)

x_new = x_old+∆x

y_new= y_old+∆y(^dy/dx|x_old,^yold)

Solving differential equations

1. seperate the variable

2. integrate both sides

3. if general solution→+c, if particular solution,→find c

4. solve for y

Simpson’s rule

S=h/3(Y₀+4Y₁+2Y₂+… 2Y_n-1+Y_n)

Mean Value Theroem

if f is continuous on the interval [a,b] and differentiable on (a,b), then there exists some value x=c on (a,b) where f’(c)=(f(b)-f(a))/(b-a)

when finding antiderivaties

don’t forget +c

Inflection points

where concavity changes; f’’(x) changes signs

solving related rates

differentiate implicitly with respect to time (d/dt)

area by Riemann sums (rectangles)

_k=1∑ⁿf(C_k)·∆X_k LRAM, RRAM,MRAM

Integral Notation

_a∫^b f(x)dx

Integrals

Area between the curve and x-axis

_a∫^b c dx

c(b-a)

_a∫^b k·f(x)dx

k·_a∫^b f(x)dx

_a∫^b (f(x)±g(x))dx

_a∫^b f(x)dx ± _a∫^b g(x)dx

Average Value of a Function

1/(b-a) · _a∫^b f(x)dx

MVT of integrals

if f is continuous on [a,b], then at some point c in [a,b] f(c)=1/(b-a) · _a∫^b f(x)dx

Fundamental theorem of Calc (Part 1)

d/dx · _a∫^x f(t)dt = f(x) if

1. derivative of an integral

2. derivative variable matches upper limit

3. lower limit is a constant

Fundamental theorem of Calc (Part 2)

_a∫^b f(x)dx= [F(x)]^b_a=F(b)-F(a)

Trapezoid Rule

T=h/2(Y₀+2Y₁+2Y₂…+2Y_n-1+Y_n)

d/dx(u·v) (product rule)

u·dv/dx + v·du/dx

d/dx(u/v) (quotient rule)

(v·du/dx - u·dv/dx)/v²

Position

Velocity

Acceleration

s(t)

v(t)=s’(t)

a(t)=v’(t)=s”(t)

Speed

|v(t)|

d/dx sin(x)

cos(x)

d/dx cos(x)

-sin(x)

d/dx tan(x)

sec_²(x)

d/dx cot(x)

-csc²(x)

d/dx sec(x)

sec(x)tan(x)

d/dx csc(x)

-csc(x)cot(x)

Derivative of aCompund Function (chain rule)

f’(g(x))·g’(x)

Slope of Parametric Equations

dy/dx = (dy/dt)/(dx/dt)

Implicit differentiation

1. dy/dx everything

2. anytime you have a “y”, tack on a dy/dx

3. put all the dy/dx on the same side and solve for dy/dx

d/dx sin^-1(u)

1/√/(1-u²)·dy/dx

d/dx cos^-1(u)

-1/√/(1-u²)·dy/dx

d/dx tan^-1(u)

1/√(1+u²) · du/dx

d/dx cot^-1(u)

-1/√(1+u²) · du/dx

d/dx sec^-1(u)

1/(|u|·√(u²-1)) · du/dx

d/dx csc^-1(u)

-1/(|u|·√(u²-1)) · du/dx

Derivatives for inverses (let f(x) and g(x) be inverses)

the slopes are reciprocals at inverse points

^df-1/dx|_x=f(a) = ^1/(df/dx)|_x=a

d/dx e^u

e^u · du/dx

d/dx a^u

a^u ·ln u · du/dx

d/dx ln u

1/u · du/dx

d/dx log_a u

1/(u·ln a) ·du/dx

Extreme Value Theorem

if a function is continuous on a closed interval [a,b] the function will have both a max and a min

critical points (if f is continuous)

any interior point where f’(x) = 0 or DNE

Max and Min can occur at

endpoints or critical points if f’(x) changes sign

lim_x→c f(x)

if f(x) is nice(no holes/asymptotes)

let x=c, substitute and simplify

if lim_x→c- f(x) = lim_x→c+ f(x)

then lim_x→c exists

Finding horizontal asymptotes

end behavior; find lim_x→∞ f(x) and lim_x→-∞ f(x)

Finding Vertical Asymptotes

Find x-values not in domain

the limit at that spot from either side = ±∞