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)