3.7 "Optimization"
The product is 192 and the sum is a minimum.
sqrt192 sf" and sqrt192
a. Find the differential of
y=x^6
b. use that number to approximate
(1.01)^6
(a = 1 and dx =0.01)
a.
dy=6x^5dx
b.
~~1.06
Find the increasing and decreasing intervals. Find relative extrema.
f(x)=x^2-4x+3
f(x) is decreasing on
(-oo,2)
b/c f'(x)<0
f(x) in increasing on
(2,oo)
b/c f'(x)>0
f(x) has rel min @ x=2 b/c f'(x) sign changes from - to +
Find all relative extrema.
f(x)=5+3x^2-x^3
rel min (0,5) b/c f'(x) sign changes from - to +
rel max (2,9) b/c f'(x) sign changes from + to -
Determine whether Rolle’s Theorem can be applied to f on the indicated interval. If Rolle’s Theorem can be applied, find all values of c in the interval such that f’(c)=0.
f(x) = x^(2/3) - 1
[-8,8]
Rolle's Theorem cannot be applied.
Find the length and width of a rectangle that has the given perimeter and a maximum area.
Perimeter: P units
P/4 sf" and " P/4
Use linear approximation AND differentials to estimate (2.001)5
~~32.08
Find the increasing and decreasing intervals. Find relative extrema.
f(x)=x^3-3x^2+1
f(x) is decreasing on
(0,2)
b/c f'(x)<0
f(x) is increasing on
(-oo,0)uu(2,oo)
b/c f'(x)>0
f(x) has rel min @x=2 b/c f'(x) sign changes from - to +
f(x) has rel max @x=0 b/c f'(x) sign changes from + to -
Determine POI.
f(x) = x^5
POI @ x=0 b/c f(x) exists in the original function & f''(x) sign changes.
Determine whether Rolle’s Theorem can be applied to f on the indicated interval. If Rolle’s Theorem can be applied, find all values of c in the interval such that f’(c)=0.
f(x) = x^2 - 2
[0,2]
c=1
A rancher has 200 feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions should be used so that the enclosed area will be a maximum?
25 ft & 100/3 ft
use the information to evaluate and compare 𝜟y and dy:
y=x^4+1 x=-1 𝜟x=dx=0.01
𝜟y=-0.039
dy=-0.040
compare: 𝜟y is greater than dy by 0.01.
Find the increasing and decreasing intervals. Find relative extrema.
f(x)=(x+3)/x^2
f(x) is decreasing on
(-oo,6)uu(0,oo)
b/c f'(x)<0
f(x) is increasing on
(-6,0)
b/c f'(x)>0
f(x) has rel min @ x=-6 b/c f'(x) sign changes from - to +
f(x) has no rel max b/c @ x=0, there is a vertical asymptote
Determine POI.
f(x) = x^(1/3)
POI @ x=0 b/c f(x) exists in the original function & f''(x) sign changes.
Determine whether Rolle’s Theorem can be applied to f on the indicated interval. If Rolle’s Theorem can be applied, find all values of c in the interval such that f’(c)=0.
f(x)= (x-1)(x-2)(x-3)
[1,3]
c=(6+-sqrt12)/12
A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum value?
3 in x 6 in x 6 in
Use linear approximation to approximate the value of
sqrt 1.1
sqrt1.1~~ 1.05
Find the increasing and decreasing intervals. Find relative extrema.
f(x)=|x^2-x|
f(x) in increasing on
(0, 1/2) uu (1,oo)
b/c f'(x)>0
f(x) is decreasing on
(-oo,0) uu(1/2,1)
b/c f'(x)<0
f(x) has rel min @ x=0, 1 b/c f'(x) sign changes from - to +
f(x) has rel max @x=1/2
b/c f'(x) sign changes from + to -
Determine POI.
f(x) = 1/x
f(x) has no POI b/c x=0 does not exist in the original function.
Verify that the hypothesis of the MVT is satisfied on the given interval and find all values of C that satisfy the conclusion of the theorem.
f(x) = x^3+x-4
[-1,2]
c=+-1
A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet.
32/(4+pi) sf " X " 16/(4+pi) sf" feet "
use linear approximation to find the value of
sqrt99.4
sqrt99.4~~ 9.952
Find the increasing and decreasing intervals. Find relative extrema.
f(x) = |x+1|+ |x-2|
f(x) is decreasing on
(-oo,-1)
b/c f'(x) < 0
f(x) is increasing on
(2,oo)
b/c f'(x) > 0
f(x) has no rel min or max b/c f'(x) has no f'(x)=0 and f'(x) sign doesn't change - to + or + to - at a single x value
Determine POI and find the concavities.
f(x)=x^4-4x^3
POI @ x=0,2 b/c x exists in the original function and f''(x) sign changes.
f(x) concaves down on
(0,2)
b/c f''(x)<0
f(x) concaves up on
(-oo,0)uu(2,oo)
b/c f''(x)>0
Verify that Rolle’s Theorem is satisfied on the given interval and find all the values of c that satisfy the conclusion of the theorem.
f(x) = x^3 - 3x^2 + 2x
[0,2]
c=(3+-sqrt3)/3
Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the circle to enclose the max total area?
r=2/pi, x=0
The radius of a sphere is measured to be 60in with a possible measure error of
+-0.04
Estimate the possible error in the computed volume of the sphere.
% error for radius
~~+-0.0667%
% error for volume
~~+-0.2%
Find the increasing and decreasing intervals. Find relative extrema.
f(x) = |-(x-1)(x^2-5x+6)|
f(x) is decreasing on
(-oo,-1)uu(2-(sqrt3)/3,2)uu(2+(sqrt3)/3,3)
b/c f'(x)<0
f(x) is increasing on
(-1,2-(sqrt3)/3)uu(2,2+(sqrt3)/3)uu(3,oo)
b/c f'(x)>0
f(x) has rel min @ x=-1,2,3 b/c f'(x) sign changes from - to +
f(x) has rel max @
x=2-(sqrt3)/3,2+(sqrt3)/3
b/c f'(x) sign changes from + to -
Determine POI & find the concavities.
f(x) = (x^2+1)/(x^2-4)
f(x) has no POI b/c
x=+-2
are vertical asymptotes.
f(x) is concaving down on
(-2,2)
b/c f''(x)<0
f(x) is concaving up on
(-oo,-2)uu(2,oo)
b/c f''(x)>0
Determine whether Rolle’s Theorem can be applied to f on the indicated interval. If Rolle’s Theorem can be applied, find all values of c in the interval such that f’(c)=0.
f(x) = (x^2-2x-3)/(x+2)
[-1,3]
c=-2+sqrt5