Limits
Lim x approaches 5 ((x+10)/(x^2-100))
-1/5
y=7/2x^3
-21/2x^4
d/dx(c)=0
A constant
Water leaks into a circular pool. The radius of the pool is increasing at a rate of 4 cm/min. How fast is the area of the pool increasing when the radius is 5 cm?
40pi cm^2/min
Int (5x+ 7)dx
(5x^2)/2+7x+C
Lim x approaches 2 ((2x-2)/(x-2))
DNE
y=5/8x^-3
15/8x^4
d/dx(x^n)=nx^(n-1)
Power Rule
Find the rate of change of the area A, of a circle with respect to its circumference C.
C/2pi
Int 0 to 1 (5x)dx
5/2
Lim x approaches inf ((sin(2x))/x^2)
0
y=cbrt(x)+3
1/(3cbrt(x^2))
d/dx[f(x)g(x)]=f(x)g’(x)+g(x)f’(x)
Product Rule
A conical cup is 4cm across and 6 cm deep. Water leaks out of the bottom at the rate of 2cm^3/sec. How fast is the water level dropping when the height of the water is 3 cm.
-2/pi cm/s
Int -3 to 3 (2v^2/3)dv
12
Lim x approaches 0 ((sin^2(4x))/(x^2cos(2x)))
16
y=4/x^2+1/x^3
(-8x+3)/x^4
d/dx[f(x)/g(x)]=(g(x)f’(x)-f(x)g’(x))/g(x)]^2
Quotient Rule
The radius of a right circular cylinder is increasing at the rate of 4 cm/sec but its total surface area remains constant at 600cm^2. At what rate is the height changing when the radius is 10 cm.
-16cm/s
Int ((t^2+2)^2/t^4)dt
x-4/x-4/(3x^3)+C
Lim x approaches b ((4a^2-x^2)/(2a+x))
2a-b
y=6/sqrt(4x)-3/x^2
-3/(2xsqrt(x))+6/x^3
d/dx f(g(x))= f’(g(x))g’(x)
Chain Rule
In a right triangle, leg x is increasing at the rate of 2 m/s while leg y is decreasing so that the area of the triangle is always equal to 6 m^2. How fast is the hypotenuse changing when x=3 m.
-14/15 m/s
Int e to 2e (cosx - 1/x)dx
sin(2e)-ln(2)-sin(e)