if f(x) is continuous on [a,b], then f(x) has an absolute max or min on the interval

Back

d/dx(cosx)

Front

-sinx

Back

a(t)<0

Front

v(t) decreasing

Back

∫sec²x dx

Front

tanx+C

Back

Mean Value Theorem

Front

if f(x) is continuous on [a,b] and differentiable on (a,b), there is at least one point (x=c) where f'(c)= F(b)-F(a)/b-a

Back

∫sinx dx

Front

-cosx+C

Back

Limit Definition of Derivative

Front

limit (as h approaches 0)= F(x+h)-F(x)/h

Back

v(t)=0

Front

p(t) is at rest or changing direction

Back

d/dx(secx)

Front

secxtanx

Back

If f''(x)<0

Front

f(x) is concave down & f'(x) is decreasing

Back

If f'(x)=0

Front

there is a max or min on f(x) [number line test]

Back

d/dx(lnu)

Front

u'/u

Back

d/dx(tanx)

Front

sec²x

Back

v(t)<0

Front

p(t) is moving left

Back

∫k dx [k IS A CONSTANT]

Front

kx+C

Back

∫csc²x dx

Front

-cotx+C

Back

Rolle's Theorem

Front

if f(x) is continuous on [a,b] and differentiable on (a,b), and if f(a)=f(b), then there is at least one point (x=c) on (a,b) [DON'T INCLUDE END POINTS] where f'(c)=0

Back

If f''(x)>0

Front

f(x) is concave up & f'(x) is increasing

Back

∫cosx dx

Front

sinx+C

Back

v(t) and a(t) has different signs

Front

speed of particle decreasing

Back

p(t), x(t), s(t)

Front

means position function

Back

Product rule of f(x)g(x)

Front

f'(x)g(x)+g'(x)f(x)

Back

d/dx(cscx)

Front

-cscxcotx

Back

∫(x^n)dx

Front

x^(n+1)∕(n+1) +C

Back

p''(t) or v'(t)

Front

a(t)= acceleration

Back

∫cscxcotx

Front

-cscx+C

Back

a(t)=0

Front

v(t) not changing

Back

d/dx(cotx)

Front

-csc²x

Back

If f'(x)>0

Front

f(x) is increasing

Back

If f''(x)=0

Front

f(x) has a point of inflection & f'(x) has a max or min

Back

d/dx(a^u)

Front

a^u(lna)(u')

Back

∫secxtanx dx

Front

secx+C

Back

If f'(x)<0

Front

f(x) is decreasing

Back

a(t)>0

Front

v(t) increasing

Back

Basic Derivative

Front

f(x^n)= nX^(n-1)

Back

∫(1/x)dx

Front

ln|x|+C

Back

Intermediate Value Theorem

Front

if f(x) is continuous on [a,b], then there will be a point x=c that lies in between [a,b]

Back

Alternate Definition of Derivative

Front

limit (as x approaches a number c)=
f(x)-f(c)/x-c x≠c

Back

1st fundamental theorem of calculus

Front

(bounded by a to b) ∫f(x)dx= F(b)-F(a)

Back

v(t)>0

Front

p(t) is moving right

Back

p'(t)

Front

v(t)= velocity

Back

d/dx(e^u)

Front

e^u(u')

Back

Continuity Rule

Front

If the limit exists (aka left limit and right limit are equal), and the limit equals the function at that point.

Back

Section 2

(23 cards)

d/dx(cot⁻¹u)

Front

-u'/(1+u²)

Back

d/dx(csc⁻¹u)

Front

u'/|u|√(u²-1)

Back

∫f(x)dx [BOUNDS ARE SAME]

Front

0

Back

2nd fundamental theorem

Front

(bounded by 1 to x)
d/dx[∫f(t)dt]= f(x)(x')

Back

average value

Front

(1/(b-a))[∫f(x)dx] [BOUNDED BY A TO B]

Back

total distance of particle

Front

∫|v(t)|dt

Back

Cross section for volume:
isosceles triangle [A=1/2s²]

Front

v= 1/2∫[f(x)-g(x)]²dx

Back

Volume (DISK)

Front

V=π∫f(x)²dx

Back

Cross section for volume:
equilateral triangle [A=√3/4s²]

Front

v= √3/4∫[f(x)-g(x)]²dx

Back

∫du/√(a²-u²)

Front

(sin⁻¹u/a)+C

Back

Displacement of particle

Front

∫v(t)dt

Back

d/dx(tan⁻¹u)

Front

u'/(1+u²)

Back

∫du/|u|√(u²-a²)

Front

(1/a)(sec⁻¹u/a)+C

Back

Area between curves

Front

A=∫f(x)-g(x) dx

Back

position of particle at specific point

Front

p(x)= initial condition + ∫v(t)dt (bounds are initial condition and p(x))

Back

Cross section for volume: square [A=s²]

Front

v=∫[f(x)-g(x)]²dx

Back

d/dx(sin⁻¹u)

Front

u'/√(1-u²)

Back

∫du/(a²+u²)

Front

(1/a)(tan⁻¹u/a)+C

Back

d/dx(cos⁻¹u)

Front

-u'/√(1-u²)

Back

Volume (WASHER)

Front

V=π∫f(x)²-g(x)²dx

Back

d/dx(sec⁻¹u)

Front

u'/|u|√(u²-1)

Back

derivative of exponential growth equation:
P(t)=Pe^kt