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