The radius r of a circle is increasing at a rate of 3cm per minute. Find the rates of change of the area when r=6cm
36π cm^2/min
100
Find any critical numbers of the function
f(x)=(x^2)(x-3)
x=0
x=2
100
What theorem is this?
"if f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that
f'(c)=(f(b)-f(a))/(b-a)"
The Mean Value Theorem (MVT)
100
Determine the open intervals on which the graph is concave upward or downward
g(x)=(3x^2) - x^3
CU:(-infinity,-1), (1,infinity)
CD:(-1,1)
100
Find two positive numbers that satisfy the given requirements
The sum is S and the product is the maximum
S/2 and S/2
200
The radius r of a circle is increasing at a rate of 3cm per minute. Find the rates of change of the area when r=24
144π cm^2/min
200
Find any critical numbers of the function
g(t)=t(4-t)^1/2, t<3
t=8/3
200
What theorem is this?
"Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If
f(a)=f(b)
then there is at least one number c in (a,b) such that
f'(c)=0"
Rolle's Theorem
200
Find the points of inflection and discuss the concavity of the graph of the function
f(x)=(x^3)-(6x^2)+12x
POI:(2,8)
CD:(-infinity,2)
CU:(2,infinity)
200
Find two positive numbers that satisfy the given requirements
The product is 192 and the sum of the first plus 3 times the second is a minimum
24 and 8
300
A spherical balloon is inflated with gas at the rate of 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is 30cm
2/9π cm/min
300
Locate the absolute extrema (min and max) of the function on the closed interval
f(x)=2(3-x), [-1,2]
Min:(2,2)
Max:(-1,8)
300
Does Rolle's theorem apply to the following function? Explain
f(x)=l1/xl, [-1,1]
No, f is not continuous on the interval [-1,1]
300
Find the points of inflection and discuss the concavity of the graph of the function
f(x)=x/((x^2)+1)
Find two positive numbers that satisfy the given requirements
The sum of the first and twice the second is 100 and the product is maximum
50 and 25
400
A spherical balloon is inflated with gas at the rate of 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is 60cm
1/18π cm/min
400
Locate the absolute extrema (min and max) of the function on the closed interval
f(x)=cosπx [0,1/6]
Min:(1/6,(3^1/2)/2)
Max:(0,1)
400
Determine whether the mean value theorem can be applied to f on the closed interval [a,b]. If the MVT can be applied, find all the values of c in the open interval (a,b) such that f'(c)=(f(b)-f(a))/(b-a)
f(x)=x^2, [-2,1]
f'(-1/2)=-1
400
Find all relative extrema. Use the second derivative test where applicable.
f(x)=(x^4)-(4x^3)+2
Relative Minimum at (3,-25)
400
Find the length and width of a rectangle that has the give perimeter and a maximum area.
P:100meters
L=W=25meters
500
A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end. Water is being pumped into the pool at 1/4 cubic meter per minute, and there is 1 meter of water at the deep end. What percent of the pool is filled?
12.5%
500
Locate the absolute extrema (min and max) of the function on the closed interval
y=4/x+tan(πx/8), [1,2]
Min:(2,3)
Max:(1,2^1/2+3)
500
Determine whether the mean value theorem can be applied to f on the closed interval [a,b]. If the MVT can be applied, find all the values of c in the open interval (a,b) such that f'(c)=(f(b)-f(a))/(b-a)
f(x)=sinx, [0,π]
f'(π/2)=0
500
Find all relative extrema. Use the second derivative test where applicable.
f(x)=cos(x)-x [0,4π]
No relative extrema because f is not increasing
500
A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?