Derivatives
derivative of 2x3
6x2
integral of x2
1/3x3+C
limx->2x^2+7
11
Selena has 10 apples that she picked off of a tree. Selena eats one apple and her friend takes 2 to go. How many apples does Selena have left?
7 apples
What is the quadratic formula?
(-b+-sqr(b2-4ac))/2a
derivative of sin(x)-cos(x)
cos(x)+sin(x)
integral of 1/x
ln|x|+C
limx->infinity 1/x + 5
5
The width of a rectangle is increasing at a rate of 2 cm/sec and its length is increasing at a rate of 3 cm/sec. At what rate is the area of the rectangle increasing when its width is 4 cm and its length is 5 cm?
22 cm2/sec
what is the area under the curve of the line y = 4/x on the interval (5, 9)? (round to three decimal places)
2.351
derivative of sin-1(x) - ln(x)
1/(sqrt(1-x2)) - 1/x
integral of 5x2-8x+5
5x3/3-4x2+5x+C
limx->-5(x^2 - 25)/(x^2 + 2x - 15)
5/4
Water flows 8ft3/min into a cylinder with radius 4 feet. How fast is the water level rising?
1/2pi ft/min
What is the average value equation?
fave=1/(b-a) intab f(x)dx
(-csc(x)cot(x)tan(x) - csc(x)sec2(x))/tan(x)
integral of 7sinx
-7cosx+C
limx->4(x3 - 64)/(16(x2 - x - 12)
3/7
An ice sculpture in the form of a sphere melts in such a way that it maintains its spherical shape. The volume of the sphere is decreasing at a constant rate of 2pi cubic meters per hour. At what rate, in square meters per hour, is the surface area of the sphere decreasing at the moment when the radius is 5 meters? (Note: For a sphere of radius r, the surface area is 4pir2 and the volume is (4/3)pir3)
(4pi)/5
What theorem is this:
f(a)<K<f(b)
a<c<b
f(c)=k
Intermediate Value Theorem
Formula for the derivative of a function
f'(a) = limh->0(f(a+h) - f(a))/h
integral of sec8/9xcosec10/9x
-9(cotx)1/9+C
limx->4(2x3 - 128)/(sqrt(x) - 2)
384
does the series 1/(n4-3), given the first term of the sequence is n = 1 and infinite terms, converge or diverge?
Converge
If f(x) is continuous on [a,b] and differentiable on (a,b) then there is such c that:
fl(c)=(f(b)-f(a))/(b-a)
Mean Value Theorem