Limits
lim_(x to 4)(sqrt(x)-2)/(4-x)
-1/4
f(x)=x^2e^x
f'(x)=2xe^x+x^2e^x
The dimensions of the largest garden enclosed by at most 24 ft of fencing.
What is 6ftx6ft?
There is a hole in the bottom of a cylindrical can of radius 2mm. Water is leaking out of the can at a rate of 8mm3/s. The rate of change of the height of the water in the can is ____mm/s.
What is
-2/pi (mm)/s
The domain of
f(x)=(x+3)/((x+3)sqrt(1-x)
What is
(-infty, -3) \cup (-3, 1)
lim_(x to 0)(sin(2x))/x
2
f(x)=e^(x^2+x)/e^(x^3-x^2+x+1
e^(-x^3+2x^2+1)(-3x^2+4x)
or
(e^(x^3-x^2+x+1)e^(x^2+x)(2x+1)-e^(x^2+x)e^(x^3-x^2+x+1)(3x^2-2x+1))/(e^(x^3-x^2+x+1))^2
The maximum volume of an open box with a square bottom that has a surface area of at most 108 in2
What is V=54in3?
A sphere of ice is melting equally across the surface so that the radius of the sphere is decreasing at a rate of 1/20 m/s. When the radius of the sphere is 5 m, the volume of the sphere is changing at ____.
Volume of Sphere:
V=4/3pir^3
What is -5pi m3/s?
The value of c for which the following function is continuous
f(x)={(x^2+cx+2, x leq 2),((c-x)^2, x>2):}
What is c=1?
lim_(x to infty)(sqrt(16x^3+9x^8+15))/(4x^4+3x^3
3/4
g(x)=sqrt(tan(sin(cos(2x^2+e^(3x))))
g'(x)=1/2(tan(sin(cos(2x^2+e^(3x)))))^(-1/2)(sec^2(sin(cos(2x^2+e^(3x)))))
(cos(cos(2x^2+e^(3x))))(-sin(2x^2+e^(3x)))(4x+3e^(3x))
The dimensions of identical squares cut from the corners a piece of 7x15 in cardboard so when the flaps are folded up it forms an open box of maximum volume.
x=1.5in
A box with a square base of area 125 in2 is being filled with rice. The height of the rice in the box is increasing at 1/15 in/s. The rate at which the volume of the rice in the box is changing is ____.
What is 25 in3/s?
The value of c for which the following function is continuous
f(x)={(c^2+x^2, x leq 2),(x(3/2-c), x>2):}
What is c=-1?
lim_(x to 0)tan(x)csc(x)
1
y=x^(ln(x^2+x)
dy/dx=y[ln(x^2+x)/x+ln(x)(2x+1)/(x^2+x)]
=x^(ln(x^2+x)[ln(x^2+x)/x+ln(x)(2x+1)/(x^2+x)]
The minimum cost required to make a closed box whose length is 3 times its width, volume is 36 ft3 and the cost of cardboard is $0.50 per square foot.
What is $36?
A book of height 5 in is leaned against a wall with the bottom originally 1 in away from the bottom of the wall. As the book slides down the wall, the bottom of the book moves away from the wall at a rate of 0.5 in/s. After 4 seconds the rate the top of the book is sliding down the wall is ____.
What is -3/8 in/s?
The values of a and b for which the following function is continuous
f(x)={(ax+6, x<-4),((x+2)^2, -4 \leq x \leq 0), ((b+x)^2-bx, x>0):}
What is a=1/2 and b=-2?
lim_(x to -infty)(3x^7-15x^3+23)/(4x^(7/2)-2x^3+1)^2
No Solution
y=arctan(x^2+3x+ln(x/e^x))
dy/dx=(2x+2+1/x)/sec^2(y)
The dimensions (radius and height) of a closed can of volume 54pi mm3 made of the least amount of material (ie trying to minimize surface area of can).
Note:
Volume of a cylinder:
V=pir^2h
Surface area of an open cylinder:
A=2pirh
Area of a circle:
pir^2
What is r=3mm, h=6mm
A rocket is being launched straight up and an engineer watches the launch from 25 miles away. If the angle of elevation from her position is changing at a rate of 3 degrees per second the rate at which the elevation of the rocket is changing when the angle of elevation is 30 degrees is ______.
What is 100 miles/second?
The domain of
g(x)=tan(x)
What is for
\cup_(n in mathbb{Z})(pi/2+n, pi/2+(n+1))
or
{x in mathbb{R}: x ne pi/2+n, forall n in mathbb{Z}}
or
{x in mathbb{R}: cos(x) ne 0}