L'Hôpital's Rule
Evaluate:
lim[x→0] (sin(x) / x)
1
Find all c in (1,3) that satisfy the MVT in f(x)=x2 on [1,3]
c =2
Find the critical points and classify them using the first derivative test for f(x)=x3-3x2+4
- At x=0, f(x) changes from increasing to decreasing: Local Maximum
- At x=2, f(x) changes from decreasing to increasing: Local Minimum
Find two numbers whose sum is 10 and whose product is as large as possible
x=5, y=5
What is the integral of f(x) evaluated from 1 to 4, when
f(x) = 3x2 - 2
57
Evaluate:
lim[x→∞] (x/ex)
0
Find all c in (-1,3) that satisfy the MVT in f(x)=x2-2x-3 on [-1,3]
c = 1
Find the critical points and classify them using the second derivative test for f(x)=x4-4x3+6x2
Since f′′(0) > 0, x=0 is a local minimum
A farmer uses 1600ft of fencing to enclose a rectangular area which will be divided into three pens of equal size. What is the maximum total area of the three pens that he can enclose with the limited amount of fencing that the farmer has available?
80,000 ft2
What is the integral of f(x) evaluated from 0 to 1, when
f(x) = x(ex^2)
(e-1)/2
Evaluate:
lim[x→π] ((x-π)/cos(x))
0
Find all c in (1,e) that satisfy the MVT in f(x)=ln(x) on [1,e]
c = e-1
Find the extrema for f(x)=x3-6x2+9x+5 using the first derivative test
- At x=1, f(x) changes from increasing to decreasing: Local Maximum
- At x=3, f(x) changes from decreasing to increasing: Local Minimum
A box with a square base and an open top must have a volume of 32 cubic units. What dimensions minimize the surface area?
x=4, h=2
Thus, 4x4x2
What is the integral of f(x) evaluated from 0 to 1/2, when
f(x) = 1/√(1-x2)
π/6
Evaluate:
lim[x→1] ((ln(x))/(x-1))
1
Find all c in (0,2π) that satisfy the MVT in f(x)=cos(x) on [0,2π]
c = π
Use the second derivative test to classify extrema for f(x)=2x3-9x2+12x+1
- At x=1, local maximum
- At x=2, local minimum
Find the point P on the line y=4-x that is closest to the point (7,6)
P = (5/2, 3/2)
What is the integral of f(x) evaluated from 2 to 6, when
f(x) = 1/(x+1)ln(x+1)
ln(ln(7)) - ln(ln(3))
Evaluate:
lim[x→0+] (ln(x)/x)
-∞
Find all c in (1,4) that satisfy the MVT in f(x)=x3-3x2+5 on [1,4]
c = 1+√3
Identify absolute extrema for f(x)=-x3+6x2-9x+5 on [0,4]
- Absolute maximum: f(0)=f(3)=5
- Absolute minimum: f(1)=f(4)=1
You currently sell a product for $20 and sell 100 units. For every $1 increase, you lose 4 customers. What would be the MAXIMUM revenue possible?
MAX Revenue = $2,025
- Price: $22.50
- Quantity (Q*) = 90 units
What is the integral of f(x) evaluated from 1 to 2, when
f(x) = 4x3/(1+x4)
ln(17/2)