Apps. of Derivatives
Identify the inflection point(s) of f(x)= x3-3x2
answer: (1,-2)
Solve.
(83x-5)+5=9
x= 17/9
ex+xex-2xex^2
How long will it take $12,000 to grow to $15,000 if the investment earns interest at the rate of 5%/year compounded monthly?
t= ln(5/4)/12ln(12.05/12) years
What are the critical numbers of the function. f(x)= 4x/1+x2
critical numbers= x=-1,1
Solve.
2(e3x-5)-7=9
ln(8)+5/3
What is the derivative of f(x)=(e-2x+3)2
-4(e-2x+3)(e-2x)
A culture with 8600 bacteria has grown to 10,000 bacteria after 1 hour.
a. After how many hours will it take the bacteria to double?
rate= ln(50/43)
answer: t=ln(2)/ln(50/43)
When is the function concave downward?
f(x)= 1/x4
In the long run, y=e1-3x+4 will reach?
a. y=4
b. y= infinity
c. y= -infinity
d. y= -4 a
y= 4
What is the derivative of f(x)= 5/3e-7x+1
105e-7x(1/(3e-7x+1)2)
A farmer has 4000yds of fencing and wants to construct a fence for his backyard. His yard has three sides and one side is covered by a river. What are the max dimensions of the fence he should build?
Max dimensions:
L= 2000
W=1000
For what values of x is the function increasing. f(x)= 1-x2/x
Expand and simplify: ln(x1/2/x2(cuberoot(3x+7))
1/2ln(x)-2ln(x)-1/3ln(3x+7)
What is the equation of the tangent line for the function. f(x)= xe-2x at the inflection point?
y=(-1/e2)x+(2/e2)
How long will it take an investment of $6000 to grow to $7000 if the investment earns interest at the rate of 7.5% compounded continuously?
t= ln(7/6)/0.075 years
What are the relative extrema of the function. (use second derivative test)
f(x)= 4x3-3x2
relative max= (0,0)
relative min= (1/2,-1/4)
(-infinity,3/2)
Where is the function concave up?
f(x)= xe-x
increasing: (2, infinity)
You want to construct a rectangular box with a square base. The box has a total volume of 30ft2. The material for the base costs 0.40 cents, side costs 0.10 cents, top costs 0.20 cents. What are the dimensions of the box that can be constructed at the min. cost.
Dimensions: 201/4x201/4x201/2