Limits
Derivatives
Integrals
Area and Volume
Formulas, Identities, and Derivatives
100
lim_(x->3)=-4
100
Find dy/dx:
y=3x^2+2x+10
dy/dx=6x+2
100
Evaluate.
int 1/3x^2+xdx
x^3+2x^2+c
100
What is the area of f(x)=x2 on the interval (2,5)
A=39
100
Fundamental Theorem of Calculus.
int_a^bf(x)dx=F(b)-F(a)
200
The limit as x approaches 5 does not exist because the limit from the left does not equal the limit from the right.
200
Find H'(4)
H'(4)=28
200
Evaluate the improper integral.
int_0^oo 9e^(-9x)dx
int_0^oo 9e^(-9x)dx=1
200
What is the area between f(x)=√x and g(x)=x2 on the interval (0,1)?
int_0^1 sqrt x -x^2dx=1/3
200
Average value of a function.
1/(b-a) int_a^b f(x)dx
300
Evaluate the limit:
lim_(x->-4)(7x+28)/(x^2+x-12)
lim_(x->-4)=-1
300
Find dy/dx:
y=sin^-1(3x)
dy/dx=3/sqrt(1-9x^2)
300
Evaluate the integral:
int (5+lnx)^5/xdx
1/6(5+lnx)^6+c
300
Use LRAM to estimate the area on the interval (0,π) for f(x)=sinx in exact terms.
A=π/(2sqrt2)+π/4
300
Quotient Rule.
d/dxu/v=(u'v-v'u)/v^2
400
Evaluate the limit in simplest form:
lim_(x->π/4)cos(2x)/(cos(x)-sin(x))
lim_(x->π/4)=sqrt2
400
Find dy/dx:
cos(x^2)=xe^y
dy/dx=(-2xsin(x)^2-e^y)/(xe^y)
400
Evaluate the improper integral:
int_-oo^oo (2x)/(x^2+1)^2dx
int_-oo^oo (2x)/(x^2+1)^2dx=0
400
Write the integral for finding the volume listed below:
V=πint_0^3 e^(2y)dy
400
Integration by parts.
int udv=uv-int vdu
500
lim_(t->0)(cos(5t)-1)/(e^t-t-1)
lim_(t->0)=-25
500
Find dy/dx:
y=5x^(5x)
dy/dx=25x^(5x)(lnx+1)
500
Find F(x):
int 1/(x^2+6x+8)dx
1/2ln((x+2)/(x+4))+c
500
A balloon's radius is increasing at a rate of 0.5 ft/sec. At the instant the radius is 4 ft, what is the rate of change of volume of the ballon?
dV/dt=32π ft^3/sec
500
Arc Length formula.
s=int_a^b sqrt(1+(dy/dx)^2