Question
5.The total cost to produce x boxes of cookies is C dollars, where C = 0.0001 x 3 0.03 x 2 +4 x +400. In
5.The total cost to producexboxes of cookies isCdollars, whereC= 0.0001x30.03x2+4x+400.
Intweeks,production is estimated to bex=1500+ 100tboxes.
(a)
Find the marginal costC'(x).
C'(x) =
(b)
Use Leibniz's notation for the chain rule,dC
dt
=dC
dx
dx
dt
,
to find the rate with respect totimetthat the cost is changing.
dC
dt
=
(c)
Use the results from part (b) to determine how fast costs are increasing (in dollars per week) whent=4weeks.
dollars per week
6.The formula for the volume of a sphere isS=4
3
r3,
wherer(in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.
(a)
Supposer=1
(t+ 1)2
1
11
,
wheretis time in minutes. Use the chain rule,dS
dt
=dS
dr
dr
dt
,
to find the rate at which the snowball is melting.
dS
dt
=
(b)
Use the results from part (a) to find the rate (in ft3/min) at which the volume is changing att= 1 min.
(Round your answer to four decimal places.)
ft3/min
1.Consider the functiony=f(x).
f(x) =3x1,x=4
(a)
Finddf
dx
atx=4.
df
dx
=
(b)
Findx=f1(y).
f1(y) =
(c)
Use part (b) to finddf1
dy
aty=f(4).
df1
dy
=
2.Consider the functiony=f(x).
f(x) =81x2, 0x9,x=6
(a)
Finddf
dx
atx=6.
df
dx
=
(b)
Findx=f1(y).
f1(y) =
(c)
Use part (b) to finddf1
dy
aty=f(6).
df1
dy
=
3.Find(f1)'(a).
f(x) = tan1(x) +8x2,a= 0
(f1)'(a) =
4.Finddy
dx
.
y=sin1(x4)
dy
dx
=
5.Finddy
dx
.
y= (1 + cot1(x))7
dy
dx
=
6.Finddy
dx
.
y=tan1
4x2
dy
dx
=
2.An object moves according to the position functiony=1
t+5
,t0.
Find the velocity as a function oft.
v(t) =
3.In general, the profit function is the difference between the revenue and cost functions,P(x) =R(x)C(x).
Suppose the price-demand and cost functions for the production of cordless drills is given respectively byp=1430.03x
andC(x) =74,000+ 65x,
wherexis the number of cordless drills that are sold at a price ofpdollarsper drill andC(x)
is the cost (in dollars) of producingxcordless drills.
(a)
Find the marginal cost function,MC(x).
MC(x) =
(b)
Find the revenue,R(x),
and marginal revenue,MR(x),
functions.
R(x)
=MR(x)
=
(c)
FindR'(1000)
in dollars per drill.
R'(1000) =
dollars per drill
Interpret the results.
At a production level of1000cordless drills, revenue is
---Select---
increasing
decreasing
at a rate of the absolute value ofR'(1000)
dollars per drill.
FindR'(4200)
in dollars per drill.
R'(4200) =
dollars per drill
Interpret the results.
At a production level of4200cordless drills, revenue is
---Select---
increasing
decreasing
at a rate of the absolute value ofR'(4200)
dollars per drill.
(d)
Find the profit,P(x),
and marginal profit,MP(x),
functions.
P(x)
=MP(x)
=
(e)
FindP'(1000)
in dollars per drill.
P'(1000) =
dollars per drill
Interpret the results.
At a production level of1000cordless drills, profit is
---Select---
increasing
decreasing
at a rate of the absolute value ofP'(1000)
dollars per drill.
FindP'(4200)
in dollars per drill.
P'(4200) =
dollars per drill
Interpret the results.
At a production level of4200cordless drills, profit is
---Select---
increasing
decreasing
at a rate of the absolute value ofP'(4200)
dollars per drill.
5.Findf'(x)
for the function.
f(x) =6x14x4+17
x+ 1
f'(x) =
6.Consider the function.
f(x) =x2x12+5x+2,a= 0
(a)
Evaluatef'(a).
f'(a) =
7.Find the equation of the tangent line to the graph off(x) = (4xx2)(4xx2) atx= 1.
y=
2.Consider the graph of a functiong(x).Find the pointcat which the function has a jump discontinuity but is right-continuous.
c=
What value should be assigned tog(c) to makegleft-continuous atx=c?
g(c) =
3.Find wheref(x) =
18x2x3
is continuous.
As a composition of continuous functions,f(x)
will also be a continuous function whereverf(x)
is defined.
f(x)
is defined whenh(x) =18x2x3=x2(18x)0.
x20
for allx,
so we check where(18x)0.
This is true forx.
Find the set of all points at whichf(x)
is continuous.
(, 18]
[18,)
[18,)
(,)
(,18]
5.Assume thatlim
x7
f(x) =6,lim
x7
g(x) =9,
andlim
x7
h(x) =8.
Use these three facts and the limit laws to evaluate the limit.
lim
x7
g(x)f(x)
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