Limits
limx->0 1/x
DNE
d/dx (x(x-7)+6)
2x-7
∫4x dx
2x^2 +C
What is the condition to use IVT?
Continuity
Find the area between y=-4x^2+6 and the x and y axis.
4.667
limx->3- (x^2-2)/(x-3)
-infinity
d/dx (tan(x)+sec(x))^2
2(tan(x)+sec(x))*(sec^2(x)+tan(x)sec(x))
∫(5x^2+6x-7)dx |0, 4
126 2/3
What are the conditions to use MVT?
Continuity and Differentiability
Find the area between y=-4/(x^2-7) and y=2x in the first quadrant.
3.640
The three limit rules for continuity.
1. The limit must exist
2. f(a) must exist
3. f(a) must equal the limit
d/dx (4arcsin(3x))
12/((3x)^2+1)
∫4e^(4x)
e^(4x)+C
Can you use IVT on the function f(x)=4x^2+6x^3+2x-7?
Find the volume of y=-x^2+6x+2 in the first quadrant using semi-circle cross-sections.
167.931
The limit definition of a derivative.
limh->0 (f(x+h)-f(x))/h
or limx->a (f(x)-f(a))/(x-a)
Find dy/dx of 3xy+4y^2=2x
dy/dx=(2-3y)/(3x+8y)
∫((cos(ln(x)))/x) dx
sin(ln(x))+C
On f(x)=4x^2-7 [-1,6], does k=4?
Yes, -3<4<137
Find the volume of the area between x=4 and y=3x^2 and the x-axis rotated around the x-axis
1843.2 pi
limx->4 (x^2-3x-4)/(x-4)
5
A circle's radius is increasing at a rate of 4 m/s (A=pir^2). At what rate is the area increasing when it has a radius of 6m.
dA/dt=48pi
∫(4x(ln(2x))) dx
2x^2ln(2x)-x^2+C
When does f(x)=2x^3-4x+8 satisfy MVT on [-2, 2].
x= ±√(4/3)
Find the area between y=5x^2+6x+2, y=34, and the y-axis in the first quadrant rotated around the x-axis.
922.667 pi