Limits and Continuity
lim_(x->a)f(x)
exists if
lim_(x->a^-)f(x)=lim_(x->a^+)f(x)
d/dx(c)
d/dx(x^n)
d/dx(c*f(x)
0
n*x^(n-1)
c*f'(x)
Average Rate of change of f on [a,b] .
(f(b)-f(a))/(b-a)
int(a)dx
int(ax^ndx)
ax+c
a*(x^(n+1))/(n+1)+c
What is Mean Value Theorem?
If f(x) is continuous on [a,b] and differentiable on (a,b) then
(f(b)-f(a))/(b-a)=f'(c)
for some value c in (a,b)
AKA: slope of secant line = slope of tangent line
Average Rate = Instant Rate
f is concave up when ...
f''(x)>0
Calculate the area between two curves.
int_a^b(hi-lo)dx=int_a^b[f(x)-g(x)]dx
where a, b start and finish boundaries, or intersections of a bounded region.
f(x) is continuous if
lim_(x->a^-)f(x)=lim_(x->a^+)f(x)
and
lim_(x->a)f(x)=f(a)
d/dx sin(x)
d/dx cos(x)
d/dx tan(x)
cos(x)
-sin(x)
sec^2(x)
Where are critical points located for a function f ?
f'(x)=0
f'(x) DNE
int(e^udu)
int (du)/(u)
e^u+c
ln|u|+c
What is Intermediate Value Theorem?
If f(x) is continous on [a,b] then f(x) takes on every value between f(a) and f(b).
If g(x)=int_0^xf(t)dt , what conditions for f give a relative max for y=g(x) .
f changes from positive to negative.
Washer Method: Show calculation and describe each part.
\pi int_a^b(R(x)^2-r(x)^2)dx
R(x) is large radius: Difference between axis of rotation and farther equation.
r(x) is small radius: Difference between axis of rotation and nearer equation.
a,b are boundaries for calculation.
f(x) is differentiable at x=a if
f(x)
is continuous and
lim_(x->a^-)f'(x)=lim_(x->a^+)f'(x)
d/dx(f(x)*g(x))
d/dx(f(x)/(g(x)))
f'(x)g(x)+g'(x)f(x)
(f'(x)g(x)-g'(x)f(x))/(g(x))^2
Inflection points occur when ...
f''(x)
changes sign.
int sin(x)dx
-cos(x)+C
A particle is slowing down if
velocity and acceleration have opposite signs.
What conditions make a left sum an underestimate for int_a^bf(x)dx ?
If f is increasing on [a,b]
If f(x)=x^2 and g(x)=x , what is the volume created by cross sections formed from rectangles with bases perpendicular to x-axis and height 3 times the measure of each base?
3int_0^1(g(x)-f(x))^2dx
Limit Definition for f'(x) (Equation of the derivative)
lim_(h->0)(f(x+h)-f(x))/(h)
d/dx(f(g(x))
f'(g(x))*g'(x)
Vertical Asymptotes occur when:
Horizontal Asymptotes occur when:
VA:
dy/dx=#/0
HA:
lim_(x->\infty)=#
Rewrite
int 1/sqrt(2x-1)dx
with u substitution and find antiderivative.
int1/2(1/sqrt(u))du=1/2intu^(-1/2)du
1/2u^(1/2)/(1/2)+C
sqrt(2x+1)+c
f is increasing when ...
f'(x)>0
What conditions make a trapezoidal sum an over estimate for int_a^bf(x)dx ?
If f is concave up on [a,b] (f''(x)>0 for a<x<b)
Total Distance traveled for a particle traveling in a straight line.
int_(t_1)^(t_2)|v(t)|dt
Limit Definition of f'(a) (Derivative at the point x=a )
lim_(x->a)(f(x)-f(a))/(x-a)
or
lim_(h->0)(f(a+h)-f(a))/(h)
d/dx(e^u)
d/dx(ln(u))
e^u*u'
1/u*u'
Steps for solving related rates problems
Write an equation that models the situation.
Implicitly differentiate all variables that are changing with respect to time.
Plug in known values
Solve
d/dx(int_0^(x^2)sin(t)dt)
FTC part 2
sin(x^2)*2x
f has a relative min when ...
f'(x) changes from negative to positive
OR
f'(x)=0 and f''(x)>0
A particle is moving toward the origin when...
Position and Velocity are opposite signs.
Steps to solving the differential equation:
dy/dx=(x+3)/(2y)
and g(0)=2
Separate and Integrate
int 2ydy=int (x+3)dx
y^2=x^2+3x+C
Solve for C ,Solve for "y"
2^2=2^2+3(2)+C,C=6
y^2=x^2+3x-6,y=sqrt(x^2+3x-6)