114)
Use your graphing calculator to graph the equation y=x5+ 3x4+x3+x2+x– 1. Zoom in
on the graph to approximate the roots of the equation. Use Newton’s method to verify your
results.
114)
115)
If y=ln x
ln x2, then find y’.
115)
116)
Suppose the number of fleas N (in thousands) in a given area is related to the number of
pounds of insecticide x sprayed on the nesting areas according to Nx – 10x+N= 300. Find
dN
dx , the rate of change of the number of fleas with respect to the number of pounds of
insecticide.
116)
117)
Use logarithmic differentiation to find dy
dx from y=
3x2+ 2x– 7
5x2– 1 73x– 5
.
117)
118)
Suppose that P, the proportion of people affected by a certain disease, is described by ln
P
1 –P= 0.9t, where t is the time in months. Find dP
dt , at the rate at which P grows with
respect to time.
118)
119)
Suppose that c= (0.2q–20)3+ 20,000 is a total cost function, where c is the total cost (in
dollars) of producing q units of a product. How fast is marginal cost changing when q=
150?
119)
120)
Find y’ if y= log 5x+ 3.
120)
20
121)
If the cost to produce x units of a product is C(x) = 0.5x2+ 9x+ 440 and the marginal cost
function is C’(x), then find C”(x), the rate of change of the marginal function with respect
to x.
121)
122)
If y=ex×ex2, then find y‘.
122)
123)
Find dy
dx if xex+ (ln x)y+y2= 3
123)
124)
The total sales x ( in hundred thousands) for a video cassette t months after the video is
released can be approximated by x=t20t
t2+ 10. Use logarithmic differentiation to find dx
dt .
124)
125)
Suppose a population is growing according to the equation P=200et. Find the rate of
growth of the population, dP
dt .
125)
126)
If f(x) =2
x2, find f”(x).
126)
127)
Differentiate: f(x) =e6x3– 5x2 + 3x– 4
127)
21
128)
If y=x2+ 3
x+ 4
3
, use logarithmic differentiation to find y’.
128)
129)
Find dy
dx if 5x + 4
3y– 4 = 6x
129)
130)
If y= (7x+5)11, then find y”’.
130)
131)
Find y’ if y= (x+1)5x.
131)
132)
Differentiate: h(x) =e8x2
4x– 5
132)
133)
Suppose you deposit $3000 in a savings account with 4.8% interest compounded monthly.
Then A, the amount of money in the account after t years, is given by A= 3000(1.004)12t.
Use logarithmic differentiation to find dA
dt .
133)
134)
The total cost for a product is given by C(x) = 212 + 3 ln(2x+ 1). Find the rate of change of
C”(x).
134)
22
135)
Find the point elasticity of the demand equation p= 15 – 0.3q for the value q= 300, and
determine whether demand is elastic, inelastic or has unit elasticity.
135)
136)
Use implicit differentiation to find y” from 3x2–xy + 5y2= 0.
136)
137)
Find dy
dx where y= ln s2– 2s+ 1
(s3+ 3s+2)5/2 .
137)
138)
If y=(ex2+ 7x– 8)4, then find y’.
138)
139)
If the total profit (in dollars) from the sale of x radios is P(x) = 50x– 0.05x2– 700 + 5 ln(x),
use Newton’s method to approximate the break–even points for profit. (Note: There are 2
break–even points; one is between 10 and 50, and the other is between 900 and 1000.)
139)
140)
Use logarithmic differentiation to find dy
dx from y= (x2+x+3)x2– 1.
140)
141)
The total cost for a product is given by C(x) = 900 + 8 ln(3x+ 7). Find the rate of change of
C”(x).
141)
142)
If the position of a particle moving along the x–axis at a given time is given by x(t) =t3,
find all the higher–order derivatives of this function.
142)
23
143)
If y=x
2x– 1 , find d2y
dx2.
143)
144)
The total revenue from the sales of a certain product are given by R(x) =400x
ln(2x+ 7) . Find
the marginal revenue.
144)
145)
If y=e2 – 4x, find d3y
dx3.
145)
146)
An experiment was set up to find a relationship between weight and systolic blood
pressure in children. It was found that p(x), the systolic blood pressure, was approximately
p(x) = 17.5(1 + ln x), where x is the child’s weight in pounds. Find p’(x).
146)
147)
Find y’ if y= ln (x2+5)5(3 –4x)4.
147)
148)
If the position of a particle moving along the x–axis at a given time is given by x(t) = 5t+
11, find all the higher–order derivatives of this function.
148)
149)
The percentage p of a certain radioactive element present after t years is given by p=
e–0.000028t. Find dp
dt .
149)
24
150)
Find dy
dx if xy2=ex+y
150)
151)
If y=e2, then find y’.
151)
152)
If f(x) =7
x, find the rate of change of f’.
152)
153)
Use your graphing calculator to graph the equation y=x3–ex+x. Zoom in on the graph
to approximate the roots of the equation. Use Newton’s method to verify your results.
153)
154)
If the point elasticity of demand for a product is –2.3 and the price of the product increases
3%, what is the approximate percentage decrease in demand?
154)
155)
An approximation of one root of x4
2– 2x3+ 5 is x1= 1. Apply Newton’s method twice to
find an approximation to the root of the polynomial.
155)
156)
The volume of a company’s sales y (in thousands of dollars) is related to its advertising
expenditures x (in thousands of dollars) by the equation xy – 11x+ 25y= 0. Find dy
dx .
156)
157)
The total revenue from the sales of a certain product are given by R(x) =2255x
ln(7x+ 50) . Find
the marginal revenue.
157)
25
158)
The position of an object moving on the x–axis is given by x(t) = 12t3+ 32t+3, where t is the
time in seconds. Find x”(t), the acceleration of the rock, when t= 6.
158)
159)
The demand equation for a product is p= 400 – 0.2q2. Find the rate of change of quantity
with respect to price.
159)
160)
Use your graphing calculator to graph the equation y=ex
x2+ 1
–x7– 36x2– 29. Zoom in on
the graph to approximate the roots of the equation. Use Newton’s method to verify your
results.
160)
161)
Medical research has found that between heartbeats the pressure in the aorta of a normal
adult is a function of time and can be modeled by the equation P= 95e–0.491t. Find dP
dt , the
rate at which the pressure changes with respect to time.
161)
162)
Find y’ if y=e ln x2.
162)
163)
Find y’ if y=42x + 1.
163)
164)
The Reynolds number is a quantity that relates to the flow of a liquid. If it is given by the
equation R(x) =A ln x–Bx, where x is the radius of the tube the liquid flows through and
A and B are positive constants, find dR
164)
165)
If C(x), the cost to produce x units of a product is C(x) = 6x3+ 11x+ 124 and the marginal
cost function is C’(x), then find the rate of change of the marginal function with respect to
x.
165)
26
166)
Find y’ if y=3x(4x– 6)7
(5x– 9)2(7x– 5)9
3
166)
167)
Find y’ if y= ln x– 1
x+ 1 .
167)
168)
Determine the point elasticity of the demand equation pq = 81, where p> 0 and q> 0.
168)
169)
Differentiate: f(x) =49 –x2
(x– 7)2
169)
170)
Determine the point elasticity of the demand equation (p+ 1) q+ 3 = 1000, when p= 24.
170)
171)
The loudness of sound (L, measured in decibels) perceived by the human ear depends
upon intensity levels (I) according to L= 10 log I
I0, where I0 is the standard threshold of
audibility. If I0= 1, find dL
dl , the rate of change of the loudness with respect to the standard
threshold of audibility.
171)
172)
The altitude (in feet) of a rocket t seconds into flight is given by h= – 2t3+ 114t2+ 480t+ 1, t
0. Use Newton’s method to approximate when the rocket will hit the ground.
172)
27
173)
Determine the point elasticity of the demand equation p= – 3q+ 120, where p> 0 and q>
0.
173)
174)
If the position of a particle moving along the x–axis at a given time is given by x(t) =t3+ 3
t2– 5t+ 12, find all the higher–order derivatives of this function.
174)
175)
Suppose p= 500 – 0.01q– 0.02 q is the demand equation for a product. Find the rate of
change of quantity with respect to price.
175)
176)
Differentiate: f(x) = (10x– 4) ·59x– 4
176)
177)
Find y’ if y=x5x– 1
177)
178)
Differentiate: g(x) =e2x– 5
178)
179)
The percentage p of a certain radioactive element present after t years is given by p=
e–0.00003466t. Find dp
dt .
179)
180)
An object moving along the circle x2+y2= 36. What is dy
dx , the rate of change of the
y–coordinate of the particle with respect to the x–coordinate of the particle?
180)
28
181)
A shoe company is planning a new brand of walking shoe for the market. Research shows
that x, the number of shoes sold (in thousands) at a price of $p, can be approximated by x=
–2ln p– 30
60 where
31 p. Find dx
dp .
181)
182)
Use Newton’s method to approximate the root of x3– 6x+ 1 = 0 that lies between 0 and 1.
Continue the approximation procedure until the difference of two successive
approximations is less than 0.0001.
182)
183)
Differentiate: f(x) =e
–2
x2
183)
–2
184)
Find an equation of the tangent line to the curve x2+y+y2= 13 at the point (–1, 3).
184)
185)
If f(x) =x(x2+ 9x+ 3), find the rate of change of f’.
185)
186)
The price $p and demand x for a product are related by 3x2+ 2xp + 25p3= 300,000. Find
dp
dx , the rate of change of price with respect to demand.
186)
29
187)
Find y’ if y=e1–x
x2– 1 .
187)
188)
If the total profit (in dollars) from the sale of x dishwashers is P(x) = 40x– 0.02x2– 1000 + 2
ln(x), use Newton’s method to approximate the break–even points for profit. (Note: There
are 2 break–even points; one is between 10 and 50, and the other is between 1900 and
2000.)
188)
189)
Suppose a population is growing according to the equation P=99et. Find the rate of
growth of the population, dP
dt .
189)
190)
Find dy
dx where y=log2(x2+ 3x+ 1).
190)
191)
Find y’ if y=log2(4x+ 5).
191)
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