Optimization
A box (with a lid) must have the minimum surface area possible while possessing a volume of 100cm3, What are the dimensions of the box?
A 15m ladder is resting against the wall. The bottom is initially 10m away from the wall and is being pushed towards the wall at a rate of 1/4 m/s. How fast is the top of the ladder moving up the wall 12 seconds after we start pushing?
0.1319 m/s
What is the absolute max value of...
1/5x3 + 1/2x2 + 3 when -4>x>0?
(-1.667, 3.463)
What is the Derivative of...
2x2 + 7x + 15
4x + 7
Lim x -> 2 (12x2 - 3x + 8)
Lim x -> 2 = 50
Let x and y be positive whole numbers, 3x + 4y = 80
(6x + 7)(4y - 3) = g
What values of x and y give the maximum values of g?
Two people on bikes are separated by 350 meters. Person A starts riding north at a rate of 5 m/s and 7 minutes later Person B starts riding south at 3 m/s. At what rate is the distance separating the two people changing 25 minutes after Person A starts riding?
d′ =7.9958m/s
What is the absolute min value of...
6x2 + 1/2x + 3?
(-0.0417, 2.9896)
What is the Derivative of...
45x4 + 16x3 + 75x2 + 92x
180x3 + 48x2 + 150x + 92
Lim f(x), x -> -3
f(x) = (4x + 6)/(t2 + 1)
Lim f(x), x -> -3 = -3/5
We want to construct a cylindrical can with a bottom but no top that will have a volume of 30 cm3. What are the dimensions of the can that will minimize the surface area of the can?
r=2.1216cm, h=2.1215cm
Air is being pumped into a round balloon at a rate of 5 cm3/min. What is the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm?
r′=1/80π cm/min
(The π is a pi symbol)
What are the Critical Points of...
3x3 + 5x2 + 3
Are they Min's or Max's, are they Local or Absolute?
(-1.111, 5.058) - Local Max, and (0, 3) - Local Min
What is the Derivative of...
(9x2 + 12x)(32x3 + 9x)
1440x4+1536x3+243x2+216x
Lim f(x), x -> -6
f(x) = 7 - 4x (x < 1), x2 + 2 (x ≥ 1)
Lim f(x), x -> -6 = 31
We want to build a box whose length is 6 times the width and the box will enclose 20cm3. The cost of the material of the sides is $3/cm2 and the cost of the top and bottom is $15/cm2. What are the dimensions of the box that will minimize the cost?
W=0.7299cm, L=4.3794cm, H=6.2568
A tank of water in the shape of a cone is leaking water at a constant rate of 2m3/hour. The base radius of the tank is 5m and the height of the tank is 14m.
At what rate is the depth of the water in the tank changing when the depth of the water is 6m?
h′= −98/225π =−0.1386
(The π is the pi symbol)
What are the critical points of...
1/2x5 + x4 + 1/2x3 + 2x2 + 2
Are they min's or max's?
(-1.771, 6.622), Local Max
(0, 2) Local Min
What is the Derivative of...
(6x2 + 15)/(3x + 12)
Lim f(x), x -> -5
f(x) = (x2 - 25)/(x2 + 2x - 15)
Lim f(x), x -> -5 = 5/4
We have 45m2 of material to build a box with a square base and no top. What are the dimensions of the box that will maximize the enclosed volume?
W/L = 3.8730m, H = 1.9365m
Suppose that we have two resistors connected in parallel with resistances R1 and R2 measured in ohms (Ω). The total resistance, R, is equal to...
1/R = 1/R1 + 1/R2
Suppose that R1 is increasing at a rate of 0.4 Ω/min and R2 is decreasing at a rate of 0.7Ω/min. At what rate is R changing when R1=80Ω and R2=105Ω?
R′ = −0.002045Ω/min
What sections of...
1/2x5 + x4 + 1/2x3 + 2x2 + 2 (-2 > x > 0)
Are concave up, concave down?
Concave up (-0.8855 > x > 0)
Concave down (-2 > 0 > -0.8855)
What is the Derivative of...
((12x2)(14x3))/(32x)
21x3
Lim f(x), x -> ∞
f(x) = 4x7 - 18x3 + 9
Lim f(x), x -> ∞ = ∞