112)
Find y’ if y=42x + 1.
112)
113)
If y=ln x
ln x2, then find y’.
113)
114)
Find dy
dx where y=log2(x2+ 3x+ 1).
114)
115)
Find y’ if y=e2x+1.
115)
116)
Use logarithmic differentiation to find dy
dx from y= (x2+x+3)x2– 1.
116)
117)
If x2+y2= 4, use implicit differentiation to find d2y
dx2 and simplify your answer. Note: dy
dx
should not appear in your final answer.
117)
118)
The percentage p of a certain radioactive element present after t years is given by p=
e–0.000028t. Find dp
dt .
118)
119)
If f(x) = (2x–5)4, find f”’(x).
119)
120)
Suppose that a company can produce 15,000 units when the number of hours of skilled
labor y and unskilled labor x satisfy 500 = (x+1)1/4(y+9)1/5. Find dy
dx , the rate of change of
skilled labor hours with respect to unskilled labor hours.
120)
121)
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.
121)
122)
Find y’ if y=(3x– 7)(9x– 4)(2x +7)
(3x + 4)(5x + 4)(2x + 5)
122)
123)
Find dy
dx where y= ln s2– 2s+ 1
(s3+ 3s+2)5/2 .
123)
124)
If y=x ln x, then find y(4).
124)
125)
If f(x) = 4x3+ 7x2– 5x– 3, find f”’(x).
125)
126)
In psychology, the Weber–Fechner law for stimulus response is R=k ln S
S0 where R is the
response, S is the stimulus, and S0 is the lowest level of stimulus that can be detected. Find
dR
ds .
126)
21
127)
The location of an Earth–orbiting satellite in a grid centered at the center of the Earth is
given by 5x2+ 7y2= 3000. Find d2y
dx2.
127)
128)
Differentiate: g(x) = 4xe5x– 7
128)
129)
The total revenue from the sales of a certain product are given by R(x) =2000x
ln(3x+ 10) . Find
the marginal revenue.
129)
130)
If the position of a particle moving along the x–axis at a given time is given by x(t) =t4+ 5
t3– 7t+ 19, find all the higher–order derivatives of this function.
130)
131)
Find y’ if y=x2 ln(2x– 3)
2x– 3 .
131)
132)
Find y’ if y= – 3e4x2– 5x + 3.
132)
133)
If the total profit (in dollars) from the sale of x lawn mowers is P(x) = 30x– 0.03x2– 750 +
ln(x), use Newton’s method to approximate the break–even points for profit. (Note: There
are 2 break–even points; one is between 0 and 40, and the other is between 900 and 1000.)
133)
22
134)
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 .
134)
135)
Determine the point elasticity of the demand equation p= – q+ 30, where p> 0 and q>
0.
135)
136)
Find the second derivative y” from y= (11 – 3x– 5x3)7/5 .
136)
137)
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 .
137)
138)
Find y’ if y=xex
x+ 1 .
138)
139)
Find dy
dx if xex+ (ln x)y+y2= 3
139)
23
140)
The intensity of an earthquake is measured on the Richter scale. The reading R is given by
R= log I
I0, where I is the intensity and I0 is a standard minimum intensity. If I0= 5, find
dR
dl , the rate of change of the Richter scale reading with respect to the intensity.
140)
141)
The altitude (in feet) of a rocket t seconds into flight is given by h= – t3+ 20t2+ 900t+ 10, t
0. Use Newton’s method to approximate when the rocket will hit the ground.
141)
142)
Find dy
dx if 5x + 4
3y– 4 = 6x
142)
143)
Use Newton’s method to approximate the root of x3+ 5x– 1 = 0 that lies between 0 and 1.
Continue the approximation procedure until the difference of two successive
approximations is less than 0.0001.
143)
144)
Find y’ if y= ln(x2) +ln3x.
144)
145)
Find y’ if y=e ln x2.
145)
146)
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.
146)
24
147)
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
dx .
147)
148)
Find y’ if y=3(4x– 5)7(9 –x)6
(8x– 3)5x6e–2x
148)
149)
Determine the point elasticity of the demand equation p=100
q+ 2 , where p> 0 and q> 0.
149)
150)
Find y’ if 3x2– 7y2= 8.
150)
151)
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.
151)
152)
Suppose a population is growing according to the equation P=200et. Find the rate of
growth of the population, dP
dt .
152)
153)
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.
153)
25
154)
Use logarithmic differentiation to find dy
dx from y=xx3.
154)
155)
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.
155)
156)
If y=x2(x+1)4
x2+ 4 , use logarithmic differentiation to find y’.
156)
157)
Find dy
dx if 4x2y– xey=x–y + 2
157)
158)
If y= (4x–3)432x+ 1, use logarithmic differentiation to find y’.
158)
159)
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).
159)
160)
The percentage p of a certain radioactive element present after t years is given by p=
e–0.00003466t. Find dp
dt .
160)
161)
The location of an Earth–orbiting satellite in a grid centered at the center of the Earth is
given by 8x2+ 11y2= 4995. Find d2y
dx2.
161)
162)
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.)
162)
163)
Suppose the demand equation for the manufacturer’s product is p= 100e–0.04q, where p is
the price per unit for q units. Find the marginal revenue function.
163)
164)
The total cost for a product is given by C(x) = 900 + 8 ln(3x+ 7). Find the rate of change of
C”(x).
164)
165)
Find y’ if y=ln x
x.
165)
166)
Find y’ if y=(2x+ 3)3x– 4
166)
167)
If y=(ex2+ 7x– 8)4, then find y’.
167)
168)
Find y’ if y=x5x– 1
168)
27
169)
Find an equation of the tangent line to the curve x2+y2+xy = 16 at the point (0, 2).
169)
170)
Find dy
dx if y2x + 4 + ln(x + 2) =xy3
170)
171)
An approximation of one root of x4– 8x+ 4 = 0 is x1= 1. Apply Newton’s method twice to
find the approximation x3.
171)
172)
The altitude (in feet) of a rocket t seconds into flight is given by h= – t3+ 80t2+ 750t+ ln t+
200, t
0. Use Newton’s method to approximate when the rocket will hit the ground.
172)
173)
For an isosceles right triangle, the length of the hypotenuse h is related to the length of a
side x by
2x2=h2. Find dh
dx , the rate of change of the hypotenuse with respect to to a side.
173)
174)
Find y’ if y= log 5x+ 3.
174)
175)
If y=e2x2–3, then find y” at x= 0.
175)
176)
If f(x) =ex2+1, find f”(x).
176)
28
177)
Find dy
dx if xy2=ex+y
177)
178)
If p=qe2q – 4, find the rate of change of p with respect to q when q= 2.
178)
179)
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 – 30x+ 14y= 0. Find dy
dx .
179)
180)
If y=e2 – 4x, find d3y
dx3.
180)
181)
Use implicit differentiation to find y” from 3x2–xy + 5y2= 0.
181)
182)
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 .
182)
183)
At a volleyball game concession stand, the profit P from selling x number of T–shirts (in
hundreds) is given by P= 8x– 4x ln x. Find dP
dx . Use your graphing calculator to graph
both P and dP
dx on the same set of axes. Verify that when the graph of P reaches its peak, the
graph of dP
dx crosses the x–axis.
183)
29
184)
If C(x), the cost to produce x units of a product, is C(x) = 4x3+ 13x+ 8 and the marginal
cost function is C’(x), then find the rate of change of the marginal function with respect to x
when x= 9.
184)
185)
If y=f(x) =ex2+ 3, find the (a) relative and (b) percentage rate of change of y when x= 0.5.
185)
186)
If f(x) =x(x2+ 9x+ 3), find the rate of change of f’.
186)
187)
Differentiate: h(x) =e8x2
4x– 5
187)
188)
Determine the point elasticity of the demand equation p= – 3q+ 120, where p> 0 and q>
0.
188)
189)
Find y’ if y= ln ln(2x+ 3) .
189)
190)
Find y’ if y=x3 ln(4x+ 5).
190)
191)
The total cost for a product is given by C(x) = 1750 + 300 ln(x+ 2). Find the rate of change
of C”(x).
191)
30
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