Pre-calculus/FTA Review
For any rational function, where the highest exponent in the numerator < the highest exponent in the denominator then
lim_(x->+-oo) f(x)=
0
What's the product Rule?
d/dx (fg)
fg' + gf'
f'g + fg'
What's the average rate of change of f(x) in [a,b].
(f(b)-f(a))/(b-a)
The Intermediate Value Theorem States that if f is cont on [a,b] and f(a) does not equal f(b), then...
there is a c on [a,b] where f(a) <f(c) < f(b)
Find
int sinxdx =
int cosxdx =
int sec^2x dx =
int sinxdx = -cosx +c
int cosxdx = sinx + c
int sec^2x dx = tanx + c
sin^2x + cos^2x =
1
What's the quotient rule?
d/dx(f/g)
(gf' - fg')/g^2
What's the equation of the tangent line?
y - y_1 = m(x-x_1)
If f and g are inverses, then g'(x) =
1/(f'(y))
What's
int 1/x dx =
int 1/x dx = ln x + c
Find the following:
ln 1
ln e
ln 0
ln 1 = 0
ln e = 1
ln 0 = und
What's the chain rule?
d/dx(f(g(x))
f'(g(x))* g'(x)
When f'' > 0, then f is...
Concave up
What's the formula for disk method if the radius is perpendicular to the x-axis?
piint_(a)^b f(x)-g(x)dx
What's the general solution of
dy/dt = ky
y = Ce^(kt
Find the following:
sin 0
sin pi/6
sin pi/4
sin pi/3
sin pi/2
sin 0 = 0
sin pi/6 = 1/2
sin pi/4 = sqrt2/2
sin pi/3 = sqrt3/2
sin pi/2 = 1
What are the derivative rules for:
d/dx(sinx)
d/dx(cosx)
d/dx(tanx)
d/dx(sinx) = cosx
d/dx(cosx) = -sinx
d/dx(tanx) = sec^2x
Speed is equal to
int|v(t)| dt
What is the fundamental Theorem of calculus?
int_a^b f(x)-g(x) dx =
int_a^b f(x)-g(x) dx = F(b) - F(a)
Find
int (du)/(sqrt(a^2-u^2)
int(du)/(sqrt(a^2 + u^2)
int (du)/(sqrt(a^2-u^2)) = arcsin(u/a) +c
int(du)/(sqrt(a^2 + u^2)) = 1/a arctan(u/a) + c
Find the following:
cos 0
cos pi/6
cos pi/4
cos pi/3
cos pi/2
cos 0 = 1
cos pi/6 = sqrt3/2
cos pi/4 = sqrt 2/2
cos pi/3 = 1/2
cos pi/2 = 0
Find the derivatives of:
d/dx(secx)
d/dx(cscx)
d/dx(cotx)
d/dx(secx) = secxtanx
d/dx(cscx) = -cotxcscx
d/dx(cotx) = -csc^2x
changes from negative to positive
How are Rolle's Theorem and Mean Value Theorem (MVT) different?
Rolle's Theorem also states that f(a) = f(b) but MVT does not!
To find the volume of a solid using cross sections that are isosceles right triangles where the legs are the base is....
int_(a)^b b^2/4 dx
Find the following:
tan 0
tan pi/6
tan pi/4
tan pi/3
tan pi/2
tan 0 = 0
tan pi/6 = sqrt3/3
tan pi/4 = 1
tan pi/3 = sqrt 3
tan pi/2 is undefined
What's the derivative of
d/dx(sin^-1(x))
d/dx(tan^-1(x))
d/dx(sec^-1(x))
d/dx(sin^-1(x)) = 1/(sqrt(1-x^2))
d/dx(tan^-1(x)) = 1/(1+x^2)
d/dx(sec^-1(x)) = 1/(|x|sqrt(1-x^2))
The point of inflections of f are the __________ of f'
If f(x) is increasing on [a,b], then the Left Hand Riemann Sum is....
an underestimation of the actual area
Suppose I(t) is the rate which water enters a tank and O(t) is the rate at which water exits the same tank. Both are measured in gallons/hr. If there are 30 gallons of water in the tank initially, then what is the formula to find the amount of water in the tank at any given time t?
A(t) = 30 + int_0^t I(t) - O(t) dt