Antiderivatives
find the most general antiderivative of the function:
f(x) = x(2-x)2
The sum of 2 numbers is 16. What is the smallest possible value of the sum of their squares?
128
lim as x -> 0 of (sin4x)/(tan5x)
4/5
find the most general antiderivative for the function:
f(x) = (2+x2)/(1+x2)
F(x) = arctanx+x+C
Find the dimensions of a rectangle with area 1000m2 whose perimeter is as small as possible.
root(1000) x root(1000)
the rectangle is a square
using left & right approximations, estimate the area under the graph of f(x) = squareroot(x) from x= 0 to x=4 using 4 rectangles
left approximation: ~4.1463
right approximation: ~6.1463
lim as x -> 0 of 3x5lnx
0
evaluate the integral from 1 to 5 of 4x2dx
496/3
find f.
f''(x) = 2+x3+x6
f(x) = x2+1/20x5+1/56x8+Cx+D
A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
approx 16.65 ft
lim as x -> 1 of (2-x)tan(pi*x/2)
e2/pi
g(r) = the integral from 0 to r of squareroot(x2+4)dx
g'(r) = f(r) = squareroot(r2+4)
find the function from the given information:
f''(x) = 18x-4
f(-1) = 3, f(1) = 13
f(x) = 3x3-2x2+2x+10
A rectangular corral is being built adjacent to a river to pen in extremely fluffy and small puppies. The corral must be 16,200 square feet. What dimensions would minimize the length of the fencing?
90 ft x 180 ft
3840.875
lim as x -> infinity of (xe1/x-x)
1
evaluate the integral of (8x9+14x6-3x3+11)/x4 dx
4x3/3 + 14x3/3 - 3ln|x| - 11x-3/3 +C
find the most general antiderivative:
f(x) = 2*squareroot(x) + 6cosx
F(x) = 4/3x3/2+6sinx+C
A chocolate factory is located 1 km north of the bank of a straight river that is 2km wide. A pipeline is to be constructed from the chocolate factory to a chocolatier's shop on the south bank of the river, 6km east of the factory. The cost of laying pipe is $400,000/km over land to a point P on the north bank and $800,000/km under the river to the chocolatier's chocolate underwater storage tanks. To minimize the cost of the pipeline, where should P be located?
4.88 km from the point on the bank 1 km south of the chocolate factory
use the following table to approximate the area under the curve between 0 and 14. Use a trapezoid approximation.
381.5
lim as x -> infinity of x(ln2)/(1+lnx)
2
the integral of (9x3)/(8-x4)6 dx
9/(20(8-x4)5)+C