110)
The path of a projectile is given by y=12x+7x3+ 1
5x. Find an equation of the tangent to this
curve at x= 1.
110)
111)
Find dz
ds where z=
5u7+u3+ 1; u=7s2– 8s.
111)
112)
Differentiate: f(x) =(x + 1)(5 –x3)
9 –x2
112)
113)
The sales s of a product vary with the number of weeks of an advertising campaign
according to s= 3 +8
t– 4 . Use the power rule to find ds
dt .
113)
114)
A cardboard container in the shape of a cube is being filled with coffee beans. Find the rate
of change of the volume with respect to the length of a side of the cube. Evaluate this rate
of change when a side is 5 inches long.
114)
115)
Suppose that the total cost function for the production of x units of a product is given by
C(x) = 4000 + 55x+0.1x2. Find the derivative of this function.
115)
116)
Suppose the position function of an object moving along a number line is given by s=f(t)=
2t3+ 5t, where t is in seconds and s is in meters.
(a) Find the average velocity over the interval 7.5, 7.6 .
(b) Find the velocity when t= 7.5.
116)
117)
Find y if y=x4+ 4
x2.
117)
118)
Find the derivative of y= (3x5+
7x3+ 1)(5 x9+ 7 3x2+ 1).
118)
119)
Let p= 10,000 – 12 q be the demand function for a manufacturer’s product. Find the rate
of change of price p per unit with respect to quantity q. How fast is the price changing with
respect to q when q= 9? Assume that p is in dollars.
119)
120)
If the profit for a stereo company (in thousands of dollars) is given by the equation
P(x) = 25 + 10x–3x2, find the slope of the tangent to this curve at (2, 33).
120)
121)
If the profit for a toy company (in millions of dollars) is given by the equation P(x) = 4x–
x2, find the slope of the tangent to this curve at (1, 3).
121)
122)
Determine the relative and percentage rates of change of y=f(x) =3x2+ 5x+ 2 when x= 1.
122)
123)
The path of a projectile is given by y=x+1
4x2. Find an equation of the tangent to this
curve at x= 2.
123)
22
124)
The revenue R from the sale of x units of a product is R=3
x+ 20x. The number of units
sold after t weeks of advertising is x= 9 +122
t. Find dR
dt when t= 2.
124)
125)
A cylindrical tank has a radius of 3 feet. It is being filled with water. Find the rate of change
of its volume with respect to the height of the cylinder. Evaluate this rate of change when
the height is 7 ft.
125)
126)
Find y’ if y= 5(2x2– 3x+4)8.
126)
127)
Find y’ if y=x+ 1.
127)
128)
A certain factory emits sulfur dioxide into the atmosphere. The concentration C(x), in parts
per million, is given by C(x) =0.3
x where x is the distance from the plant in miles. What is
C(x)?
128)
129)
If the cost function for a manufacturer’s product is given by C=7q2
q2+ 1 + 100
, find the
marginal cost function.
129)
130)
If the profit for a telephone company (in millions of dollars) is given by the equation
P(x) = – 11.3x2+ 22x+ 1, use the derivative function on your graphing calculator to find the
slope of the tangent to this curve at x= 3.69.
130)
23
131)
If the velocity function is given by V(x) = 12x–4x2, find an equation of the tangent line to
the graph of V(x) at (1, 8).
131)
132)
If y=
3x2+ 3x+ 7, then find the rate of change of y with respect to x.
132)
133)
The concentration of salt in a solution is given by C(t) =t1/3. Find dC
dt when t= 64.
133)
134)
The demand for a product is given by p=200,000
q– 1 . Does the derivative of this function
exist for all values of q?
134)
135)
Find the slope of the curve y= 4x2– 2x + 1 at the point (3, 31).
135)
136)
Suppose that the profit p made by selling a certain product at a price x is given by p=f(x)
and the rate of change of that profit with respect to change in price is dp
dx = 10 at x= 7.
Estimate the change in the profit p if the price changes from 7.5 to 8.5.
136)
137)
Find y if y=7
3(9x+ 3).
137)
138)
Differentiate: f(x) = ( x + 5x)(2 x–3x)
138)
24
139)
Suppose the position function of an object moving along a line is given by s=f(t) =5t2– 3t
+ 7 where t is in seconds and s is in feet.
(a) Find the average velocity over the interval 5, 5.1 .
(b) Find the velocity when t= 5.
139)
140)
When a ball is dropped from 256 feet above the ground, its height H in feet above the
ground after t seconds is given by H= 256 – 16t2. Find dH
dt .
140)
141)
Suppose that the demand for x number of hats (in thousands) is determined by the price p
of the hats as follows: x= 10 +200
2p. Find dx
dp without using the quotient rule.
141)
142)
If the distance in feet of a ball thrown from a bridge is given by D(t) = 30t+16t2 where t is
the number of seconds after the ball is thrown, then the velocity of the ball is found by D(t).
142)
143)
Find an equation of the tangent line to the curve y= (x–2)2 when x= 0.
143)
144)
Two children are selling cups of lemonade for $0.25 each. If they sell q cups of lemonade,
the total revenue r is given by r= 0.25q. Find the marginal revenue.
144)
145)
Find y’ if y=2x+ 3
7 – 5x.
145)
146)
A particle travels along a straight line path according to the equation of motion s=t2+ 9
where t is in seconds and s is in meters. Find the velocity of the particle at t= 4.
146)
25
147)
The demand D for a product at price x is given by D=1000
x– 1. Find d
dx
1000
x– 1 and
discuss what happens at x= 0.
147)
148)
Differentiate: f(x) =3x2 + 6
5x2– 2x
148)
149)
If the formula for the wind–chill, W, at 15° is given by W= 55.174 – 2.15s– 23.18 s, find
the derivative of this formula.
149)
150)
Differentiate: g(x) = (x– 8)(2x– 7)(x + 5)
150)
151)
Use the Chain Rule to find dy
dx where y=7u11 – 8u+ 5; u=9x2– 11x+ 5.
151)
152)
Find y if y=
5x3x2
152)
153)
If the cost C of removing p percent of the impurities from the waste water in a
manufacturing process is given by C(p) =7700p
100 – 2p, find C’(p).
153)
26
154)
The position of an object dropped from a height of 400 feet is given by: y(t) = 400 –16t2.
Find the rate of change of y with respect to t and evaluate it when t= 1 and t= 3. Interpret
your results.
154)
155)
Find y if y=5
44x
.
155)
156)
Find y if y=3x
5+4
3.
156)
157)
Find the rate of change of the area of a square with respect to its one side. Evaluate it when
side = 4 units.
157)
158)
Let p= 20 +1
q be the demand function for a manufacturer’s product. Find the rate of
change of price p per unit with respect to quantity q. How fast is the price changing with
respect to q when q= 2? Assume that p is in dollars.
158)
159)
Let f(x) =4x2– 3x+ 1.
(a) Find f(x).
(b) Evaluate f (1).
(c) Find an equation of the tangent line to the graph of y=f(x) at the point (1, 2).
159)
27
160)
Find y if y=2
3x
.
160)
161)
If the velocity function is given by V(x) = 7x2+ 5x+ 1, find an equation of the tangent line
161)
162)
Find the slope of the curve y= (x2+ 2x–2)4 at the point (0,16).
162)
163)
The demand D for a product at price x is given by D= – 5.2 x+ 22. Find d
dx –5.2 x+ 22
and graph the result on your graphing calculator. What is the behavior of the graph when
x= 0?
163)
164)
A pizza maker sells large pizzas for $13 each. If he sells q large pizzas, the total revenue r is
given by r= 13q. Find the marginal revenue.
164)
165)
Suppose the position function of an object moving along a number line is given by s=f(t)=
t3+ 4t, where t is in seconds and s is in meters.
(a) Find the average velocity over the interval 5, 5.1 .
(b) Find the velocity when t= 5.
165)
166)
Suppose the cost of a product can be approximated by c=13
x– 2x+ 22. Find the derivative
of this function to determine whether or not this function is continuous for all values of x.
166)
167)
Find y’ if y=x2
x3+ 4 .
167)
168)
A radio station charges its advertisers $75 per commercial. Each advertiser must purchase
lat least 100 commercials. The station offers to discount the per–commercial charge by
$0.30 for each additional commercial purchased above 100. The revenue for the radio
station is given by: R(x) = (100 +x)(75 – 0.3x) where x is the number of additional
commercials above 100. Find the marginal revenue function using the product rule.
168)
169)
The concentration of acid in a buffer solution is given by A= 0.8. Graph this function on
your graphing calculator and describe its slope.
169)
170)
The demand equation for a manufacturer’s product is p= 400 –q2, where p is the price per
unit for q units. Find the marginal revenue function.
170)
171)
Find y’ if y=1
34x2– 3
.
171)
172)
Find y’ if y=5(4x–1)2
2.
172)
173)
The proportion P of voters who recognize the name of a certain candidate after t months of
advertising is given by P(t) =18t
t2+ 80. Find the equation of the tangent line to this curve
when x= 1. Use your graphing calculator to verify that this line is tangent to P(t) when x=
1.
173)
174)
If y=f(x) =x2– 2x+ 2, find the (a) relative and (b) percentage rate of change of y when x=
4.
174)
175)
The supply function for a computer company is given by f(x) = 3 x+ 20. Rewrite this
function so that all terms are in the form xn.
175)
176)
If y=t– 2
t– 7
5
, find dy
dt .
176)
177)
Differentiate: g(x) =
57x2 + 6x– 4
177)
178)
If the height of an eaglet sitting in a nest is given by y= 324, find dy
dt .
178)
179)
Find all points x for which the curve y= 2x2+ 3x+ 5 has a tangent line that is
perpendicular to the line y=x + 15.
179)
180)
The demand D for a product at price x is given by D= – 3x+ 100. Find d
dx –3x+ 100
and discuss what happens at x= 0.
180)
181)
Find the slope of the curve y=4x
x+ 2 at the point where x= 3.
181)
30
182)
Find all values of x for which the curve y=x3+x2 has slope 1.
182)
183)
One hour after x milligrams of a particular drug are given to a person, the change in body
temperature T(x), in the degrees Fahrenheit, is given approximately by T(x) =x21 –x
11 .
The rate at which T changes with respect to size of dosage x, T’(x), is called the sensitivity of
the body to the dosage. Use the product rule to find the slope of T(x) when the dosage is 7
milligrams.
183)
184)
When a ball is dropped from the top of a building, its speed after t seconds is given by 32t.
What is d
dt (32t)?
184)
185)
If y=4u2– 13u+ 3 and u=7x3+5x2+ 4x–14, then by direct use of the chain rule find dy
dx
and evaluate when x= 1.
185)
186)
If the concentration of salt in a particular saline solution is given by the function S(t) =
1.565, find dS
dt .
186)
187)
By direct use of the definition of derivative, find d
dx (f(x)) if f(x) = 3x2 + 5x– 4.
187)
188)
A hot dog vendor sells chili dogs for $2.75 each. If she sells q chili dogs, the total revenue r
is given by r= 2.75q. Find the marginal revenue.
188)
189)
Suppose the position function of an object moving along a number line is given by s=f(t)=
t3+ 2t2+ 5, where t is in seconds and s is in meters.
(a) Find the average velocity over the interval 2, 2.01 .
(b) Find the velocity when t= 2.
189)
190)
By direct use of the definition of derivative, find d
dx (f(x)) if f(x) =3x + 5.
190)
191)
The typing speed of a certain computer student varies with the number of hours of practice
x according to S= 21 0.3x. Find S’.
191)
192)
Find the equation of the tangent line to the graph of the curve y= 6x5/3 + 3x1/3 + 1 at the
point (1, 10).
192)
193)
If y=3u3–2u2– 5u– 6 and u=4x2– 2x–13, then by direct use of the chain rule find dy
dx
and evaluate when x= 2.
193)
194)
By using the definition of a derivative, find f(x) =1
2x+ 1 .
194)
195)
Find all values of x for which the curve y= 6x2+ 4x– 5 has slope 2.
195)
32
196)
The derivative of the revenue function (the marginal revenue function) for a product is
given by 4
x2+ 1 . Use this information to determine if the revenue equation is continuous
for all values of x.
196)
197)
If a manufacturer’s average cost is given by c= 0.001q2– 0.12q+ 7 +7000
q find
(a) the marginal cost function;
(b) evaluate it when 50 units are produced.
197)
198)
If an object moves horizontally according to x = 3t– 5 and vertically according to y= 3x2,
find its vertical speed, dy
dt .
198)
199)
Find y if y=4x3–6x2+ 7x– 8.
199)
200)
If the amount of money hidden in your great–aunt’s mattress is given by the equation A(t)
= $643, find dA
dt .
200)
201)
An object moves along the ground according to the equation f(x) =x23. Find f(x).
201)
202)
Suppose that c= 0.008q3–2.4q2+ 80q+ 20, 400 is a cost function, where c is the total cost in
dollars of producing q units of a product. Find the marginal cost when q= 10.
202)
203)
Find y if y= 0.39x–2–0.17x–1+ 3.927 + 1.9x.
203)
33
204)
The sales s of a product vary with the number of weeks of an advertising campaign
according to s= 9 +11
2t+ 3 . Use the power rule to find ds
dt .
204)
205)
The height of a boulder just sitting on a 30–foot cliff is given by y= 32. Graph this function
on your graphing calculator and describe its slope.
205)
206)
A cone has a height of 10 inches. It is being filled with ice cream. Find the rate of change of
its volume with respect to the radius of the base of the cone. Evaluate this rate of change
when the radius is 12 inches.
206)
207)
Find the slope of the curve y= 3x2–x at the point (2, 10).
207)
208)
When a ball is thrown upward at a speed of 60 feet/s from a height of 500 feet, its height H
in feet after t seconds is given by H= 500 + 60t –16t2. Find dH
dt .
208)
209)
The concentration of salt in a solution is given by C(t) =t1/2. Find the rate of change of the
concentration dC
dt .
209)
210)
Suppose that the profit p made by selling a certain product at a price x is given by p=f(x)
and the rate of change of that profit with respect to change in price is dp
dx = 6 at x= 6.
Estimate the change in the profit p if the price changes from 6 to 6.5.
210)
211)
If the profit for a shoe company (in millions of dollars) is given by the equation
P(x) = 6x–2x2, find the slope of the tangent to this curve at (1, 4).
211)
212)
Find the slope of the curve y=x + 1 at the point (3, 2).
212)
213)
If the revenue function is given by R(x) = 5x – 7, find the derivative of the revenue function
(this is called the marginal revenue).
213)
214)
Find the derivative of y =7.1x2– 9.1x+ 2
2.1x2– 1.3x+ 5 .
214)
215)
The total revenue r for selling x number of bicycles is given by r= 380x. Determine the
relative and percentage rates of change of revenue with respect to x when 4 bicycles are
sold.
215)
216)
The cost C of obtaining water that contains p percent impurities is given by C(p) =120,000
P
– 1200. Rewrite this function so that all terms are in the form pn.
216)
217)
The weight W of a tree limb is given by W=3t0.422, where t is time. Find the relative rate
of change of W with respect to t.
217)
218)
Find y’ if y=x2(2x–5)5.
218)
219)
Find the equation of the tangent line to the graph of the curve y= 0.1x3+ 0.2x+ 0.0095 at
the point (0, 0.0095).
219)
35
220)
Find the slope of the tangent line to the curve y=x+ 2 at the point (0, 2).
220)
221)
Find y if y=x2+ ln 2.
221)
222)
Find y’ if y=1
(7x2– 3x+11)2
222)
223)
If f(x) =
3x, then find f’ (x) and determine whether or not f’(x) exists for all values of x.
Use your graphing calculator to graph f(x) . Use this graph to describe the behavior of the
tangent line to f(x) at
x= 0.
223)
224)
Differentiate: g(x) = (5x2– 7x + 3)(7x2– 4x + 1)
224)
225)
Find the derivative of y=
1
x–7x
x2+1
2
x–3x
x2+ 1
.
225)
36
226)
Money is invested in an annuity that earns r percent per year, compounded monthly. The
future value S of $3000 invested in such an annuity for 30 years is S= 3000 1 +0.01r
12
360
.
Find S’.
226)
227)
The position of an object thrown upward at a speed of 8 feet/sec from a height of 10 feet is
given by: y(t) = 10 + 8t–16t2. Find the rate of change of y with respect to t and evaluate it
when t= 0.25. Use your graphing calculator to graph y(t). Use the graph to interpret the
behavior of the object when t= 0.25.
227)
228)
The volume V of a spherical cell is given by V=4
3r3, where r is the radius. Find the rate of
change of volume with respect to the radius when r= 3 ×10–5 cm. Give your answer in
terms of .
228)
229)
Money is invested in an annuity that earns r percent per year, compounded monthly. The
future value S of $500 invested in such an annuity for 4 years is S= 500 1 +0.01r
12
48
. Find
S’.
229)
230)
Differentiate: f(x) = 4x5x– 7
230)
231)
If the velocity function is given by V(x) = 6x2+ 13, find an equation of the tangent line to
the graph of V(x) at (3, 67).
231)
37
232)
Differentiate: h(x) = (4x– 5) x
232)
233)
Find the slope of the curve y= 2 x at the point (4, 4).
233)
234)
If an object moves horizontally according to x = 2t+ 1 and vertically according to y= 2x3,
find its vertical speed, dy
dt .
234)
235)
Find y’ if y= (x3– 10x+ 2)(x2+ 7x+ 1).
235)
236)
The position of an object dropped from a height of 200 feet is given by: y(t) = 200 –16t2.
Find the rate of change of y with respect to t and evaluate it when t= 1 and t= 2. Interpret
your results.
236)
237)
Let f(x) =2x2– 6x+ 7.
(a) Find f ‘(x).
(b) Evaluate f ‘(2).
(c) Find an equation of the tangent line to the graph of y=f(x) at the point (2, 3).
237)
238)
Find y if y=2
x4+x4
2.
238)