Applications of Derivatives
A guy has 2400 ft of fencing and wants to fence a rectangular field against a river, no fence is needed on the river side. What are the dimensions of the fence if area is being MAXIMIZED?
600 ft and 1200 ft
Find the area of the largest rectangle that can be inscribed in a semi circle with radius R
A=r^2
A spherical blow pop you are enjoying is losing volume at 800 mm^3/s. How fast will the blow pop lose radius when the blow pop is 20 mm across?
-.637 mm/min or -2/pi
Find the absolute max of 1-(x-1)^2/3 on [-1,2]
Compute the linearization
f(x)= sqrtx e^x-1 x=1
y=3/2 (x-1) +1
Find two numbers whose sum is 80 when product is a maximum
40 and 40
Guy has a rectangular garden with an area of 1000 m^2. On one side each meter of fencing is $90 and the other three fencing is $30. Find dimensions of a fence that minimize cost.
20sqrt5 and 50sqrt5
water is draining out of a conical tank where volume V, radius r, and height h at a constant rate over time t. Determine the rate of change of the radius with respect to t if r=2 and the height is twice the radius. Volume is changing at a rate of 2 m^3/s.
1/4pi m/s
Find the absolute min of x^2 -8ln(x) on [1,4]
(4,4.910)
Approximate using linearization
sqrt17
4 1/8
Find length and width of a rectangle that has the given perimeter and max area. Perimeter=600 yds
150 and 150
suppose x and y are differentiable on t and related to the function y=x^2+3. Find dy/dt when x=1 given dx/dt=2 when x=1.
dy/dt=4
A pebble in a pond causes concentric circles to ripple out. The radius r of the outer ripple is increasing at a constant rate of 1 ft/sec. When radius is 4 ft what is the total area of the disturbed water changing?
8pi ft^2/sec
Find the absolute min of sin(x)cos(x) on [0, pi]
(3pi/4,-1/2)
Approximate using linearization
ln(1.07)
ln(1.07) approximates to .07
Find the length and width of a rectangle that has the given area and minimum perimeter. area=100 ft^2
10 and 10
Air is being pumped into a spherical balloon; its volume increases at a rate of 100 cm^3/s. How fast is radius increasing when the diameter is 50 cm.
1/25pi cm/s
A 20 ft ladder is leaning against a house. The foot of the ladder begins to slip away from the house at a rate of 2 ft/sec. How fast is the top of the ladder sliding down the wall when the foot of the ladder is 12 ft from the house?
-1.5 ft/sec
Find the linearization L(x) of the function at a.
f(x)= x-2x^2 a=5
y=-19(x-5)-45
Approximate using linearization
(-1.999)^3
y=-7.988
Find the point on the parabola y^2=2x that is closest to (1,4)
(2,2)
A 10 ft ladder is leaning against a building where the base of the ladder is 6 ft from the wall. The rate at which the ladder is falling down the wall is 1 ft/s. At what rate is the ladder moving away from the wall?
4/3 ft/s
Find the absolute max of 2x^3 -15x^2 +24x +7 from [0,6]
(6,43)
Find the linearization L(x) of the function at a.
f(x)= tan(x) a= pi/4
y=2(x-pi/4)+1
Approximate using linearization
sin(89degrees)
y=1