110)
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).
110)
111)
If f(x) is defined by f(x) =x2,if x< 1
x, if 1 x, then use your graphing calculator to graph the
functions g(x) =x2 and h(x) =x to determine:
(a) if f(x) is continuous for all values of x.
(b) if f(x) is differentiable for all values of x.
111)
112)
Find the derivative of x=f(x) = (1.3t7+9.1t2– 18t+11)(2.1t8+1.8t5– 17t+ 5).
112)
113)
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)?
113)
114)
The weekly output of a certain product is Q(x) = 200x+6x2 where x is the number of
workers. Find the equation of the tangent line to this equation when there are 60 workers.
114)
115)
By direct use of the definition of derivative, find d
dx (f(x)) if f(x) =3x + 5.
115)
116)
Money is invested in an annuity that earns r percent per year, compounded monthly. The
future value S of $2000 invested in such an annuity for 5 years is S= 2,000 1 +0.01r
12
60
.
Find S’.
116)
21
117)
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 .
117)
118)
A cylindrical tank has a height of 7 inches. It is being filled with syrup. Find the rate of
change of its volume with respect to the radius of the cylinder. Evaluate this rate of change
when the radius is 2 inches.
118)
119)
Differentiate: g(x) = (5x2– 7x + 3)(7x2– 4x + 1)
119)
120)
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.
120)
121)
Suppose the number of calories of heat required to raise 1 gram of water (or ice) from
–40°C to x°C is given by f(x) =
1
2x+ 20, if –40 x< 0
x+ 100, if 0 x
. Is this function differentiable for
all values of x?
121)
122)
The average cost c of producing q units of a product is given by
c= 0.001q2– 0.2q+ 11 +15,000
q. Find the marginal cost function.
122)
123)
If a manufacturer‘s average cost equation is c=350
q– 422q+17q2, find the total cost
function c, and the marginal cost function dc
dq . What is the marginal cost when 20 units are
produced?
123)
124)
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.
124)
125)
If the revenue function is given by R(x) = 5x – 7, find the derivative of the revenue function
(this is called the marginal revenue).
125)
126)
Find the slope of the curve y=4
2 –x at the point (3, –4).
126)
127)
Find y’ if y=
4(2x+5)3.
127)
128)
Find the derivative of y= (3x5+
7x3+ 1)(5 x9+ 7 3x2+ 1).
128)
129)
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’.
129)
23
130)
The sales s of a product vary with the number of weeks of an advertising campaign
according to s= 1 +5
t+ 2 . Use the power rule to find ds
dt .
130)
131)
Find the slope of the tangent line to the curve y=x+ 2 at the point (0, 2).
131)
132)
Find y’ if y=x2+ 1.
2x3– 1 .
132)
133)
The reaction R to the injection of a drug is a function of the dosage x according to the
equation R(x) =x2450 –x
4. Find the value of the derivative of this function when x= 10.
133)
134)
Find y’ if y=1
34x2– 3
.
134)
135)
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.
135)
136)
Find y if y=3
4x4/3 –1
5x1/4 +x–5/6.
136)
24
137)
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 = 8 at x= 3.
Estimate the change in the profit p if the price changes from 3.5 to 4.5.
137)
138)
Find all values of x for which the curve y=x3+x2 has slope 1.
138)
139)
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.
139)
140)
Find y’ if y=2
3 – 4x.
140)
141)
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.
141)
142)
By direct use of the definition of a derivative, find d
dx f(x) if f(x) =1
x+ 6 .
142)
143)
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.
143)
144)
The concentration of salt in a solution is given by C(t) =t1/3. Find dC
dt when t= 64.
144)
25
145)
Find all values of x for which the curve y= 6x2+ 4x– 5 has slope 2.
145)
146)
Find the slope of the curve y= 2 x at the point (4, 4).
146)
147)
Find y if y=x2+ 2x + 3
x.
147)
148)
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
8.
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 3
milligrams.
148)
149)
If u= (w+ 1) w+ 1, find the rate of change of u with respect to w when w= 3.
149)
150)
Differentiate: f(x) = ( x + 5x)(2 x–3x)
150)
151)
Find y if y=1
x3.
151)
152)
If y=f(x) =x2– 2x+ 2, find the (a) relative and (b) percentage rate of change of y when x=
4.
152)
26
153)
The typing speed of a certain computer student varies with the number of hours of practice
x according to S= 12 0.6x+ 5. Find S’.
153)
154)
If the revenue function is given by R(x) = 11x+ 9, find the derivative of the revenue
function (this is called the marginal revenue).
154)
155)
If the output function is given by Q(x) = 150 +3x2, find the slope of the curve at the point
where x= 25.
155)
156)
By direct use of the definition of derivative, find d
dx (f(x)) if f(x) = 3x2 + 5x– 4.
156)
157)
Find y’ if y=x+ 2
x– 3
4
.
157)
158)
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.
158)
159)
If an object moves horizontally according to x = 5t– 2 and vertically according to y= 2x2+
x, find its vertical speed, dy
dt .
159)
27
160)
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.
160)
161)
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.
161)
162)
Find the slope of the curve y= 3x2–x at the point (2, 10).
162)
163)
Find the rate of change of y=x(x2+ 9x+ 3) with respect to x.
163)
164)
Find y’ if y=x+ 1.
164)
165)
Find y if y=x–2–x+x–4/7.
165)
166)
The demand D for a product at price x is given by D=35
x– 1 +40. Find d
dx
35
x– 1 + 40 and
discuss what happens at x= 1.
166)
167)
A stereo company sells 300 car stereos per month at a price of $280 per car stereo. Market
research indicates that they can sell one additional stereo for each $1 they reduce the price.
In this case the total revenue is given by: R(x) = (300 +x)(280 –x) where x is the number of
additional stereos above 300. Find the marginal revenue function using the product rule.
167)
168)
Find y if y=5
44x
.
168)
169)
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).
169)
170)
If a manufacturer‘s average cost equation is c=38
q+ 400 – 25q+11q2, find the total cost
function c, and the marginal cost function dc
dq . What is the marginal cost when 8 units are
produced?
170)
171)
A manufacturer will sell N(x) cars after spending $x thousand on advertising, where N(x) =
1000 –3780
x. Find the equation of the tangent line to this curve when x= 20.
171)
172)
If the cost C of removing p percent of the particulate pollution from the exhaust gases at an
industrial site is given by C(p) =6200p
100 –p, find C’(p).
172)
173)
Determine the relative and percentage rates of change of y=f(x) =3x2+ 5x+ 2 when x= 1.
173)
29
174)
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)?
174)
175)
Find the slope of the curve y=4x
x+ 2 at the point where x= 3.
175)
176)
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.
176)
177)
If the profit for a car company (in thousands of dollars) is given by the equation 400 –6x2,
find the slope of the tangent to this curve at (1, 394).
177)
178)
Find dy
dx where y=
79x3– 8x+ 5.
178)
179)
The typing speed of a certain computer student varies with the number of hours of practice
x according to S= 10 0.4x+ 1. Find S’.
179)
180)
Differentiate: f(x) =(x + 1)(5 –x3)
9 –x2
180)
30
181)
Let p= 405 –2q3 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= 4? Assume that p is in dollars.
181)
182)
If y=(6u2–7)3 and u=(9 – 2x)5, then by direct use of the chain rule find dy
dx and evaluate
when x= 5.
182)
183)
Find y if y=1
2–1
2x.
183)
184)
If the concentration of salt in a particular saline solution is given by the function S(t) =
1.565, find dS
dt .
184)
185)
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.
185)
186)
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.
186)
187)
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.
187)
188)
Find v if v=t5/7(t2/7+ 3t).
188)
189)
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).
189)
190)
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’.
190)
191)
Find the slope of the curve y=2x+ 5
x– 3 at the point (4, 13).
191)
192)
Differentiate: f(x) = 4x5x– 7
192)
193)
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 .
193)
194)
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.
194)
32
195)
If the profit for a cat food company (in hundred of dollars) is given by the equation P(x) =
178x–3x2, use the derivative function on your graphing calculator to find the slope of the
tangent to this curve at x= 243.
195)
196)
If the cost function for a manufacturer’s product is given by C=7q2
q2+ 1 + 100
, find the
marginal cost function.
196)
197)
If the demand equation is D(x) = 2 +4
x, then find an equation of the tangent to this curve
when x= 3. Use your graphing calculator to verify that this line is tangent to D(x) when x=
3.
197)
198)
Find f(u) where f(u) =u5+u11
u3
198)
199)
By direct use of the definition of a derivative, find d
dx f(x) if f(x) =2x2– 3x.
199)
200)
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.
200)
201)
Suppose the position function of an object moving along a number line is given by s=f(t) =
2t2+ 11, where t is in seconds and s is in meters.
(a) Find the average velocity over the interval 3, 3.1 .
(b) Find the velocity when t= 3.
201)
202)
Suppose that m employees of a company produce a total of q units of a product per day,
where q= 100m–m2, and that the demand equation for the product is p=3000
q+ 100 . Find the
marginal revenue product when m= 10.
202)
203)
The average cost c of producing q units of a product is given by c=4q
q+ 2 +10,000
q. Find the
marginal cost function.
203)
204)
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.
204)
205)
Find y’ if y=1
(2 –3x)4.
205)
206)
If the velocity function is given by V(x) =7x2– 22.63, find an equation of the tangent line to
the graph of V(x) at (3.51, 63.6107). Use your graphing calculator to verity that the graph of
V(x) and the tangent line are at the point (3.51, 63.6107).
206)
207)
Find dz
ds where z=
5u7+u3+ 1; u=7s2– 8s.
207)
208)
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).
208)
209)
The revenue R from the sale of x units of a product is R=2
x+ 10x. The number of units
sold after t weeks of advertising is x= 7 +12
t. Find dR
dt when t= 3.
209)
210)
If the revenue for a certain company is given by R(x) = 500, find dR
dx .
210)
211)
Suppose that the demand for candy at a price of $p per pound is given by the equation
D(p) = 40 +720
p. Find D(p) when p= 10.
211)
212)
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.
212)
213)
The position of an object thrown upward at a speed of 32 feet/sec from a height of 200 feet
is given by: y(t) = 200 + 32t–16t2. Find the rate of change of y with respect to t and
evaluate it when t= 1 and t= 2. Interpret your results.
213)
214)
Find y if y=x(x+ 2).
214)
215)
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.
215)
35
216)
Find y’ if y=5(4x–1)2
2.
216)
217)
By direct use of the definition of derivative, find d
dx (f(x)) if f(x) =1
x + 6 .
217)
218)
Money is invested in an annuity that earns r percent per year, compounded monthly. The
future value S of $10,000 invested in such an annuity for 6 years is S= 10,000 1 +0.01r
12
72
.
Find S’.
218)
219)
If a manufacturer‘s average cost equation is c= 200 +48q3, find the total cost function c, and
the marginal cost function dc
dq . What is the marginal cost when 17 units are produced?
219)
220)
Find y if y=2
x4+x4
2.
220)
221)
Find y if y=e.
221)
222)
Find y if y=
5x3x2
222)
36
223)
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).
223)
224)
Find the slope of the curve y= (x2+ 2x–2)4 at the point (0,16).
224)
225)
Find y if y=7x4–5x3+ 4x– 6.
225)
226)
If y=t– 2
t– 7
5
, find dy
dt .
226)
227)
When a ball is thrown downward at a speed of 35 feet/s from a height of 1000 feet, its
height H in feet after t seconds is given by H= 1000 – 35t –16t2. Find dH
dt .
227)
228)
The demand D for a product at price x is given by D= 8 x. Find d
dx 8x and discuss
what happens at
x= 0.
228)
229)
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.
229)
230)
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.
230)
231)
If the velocity function is given by V(x) =9x2+ 12.3, find an equation of the tangent line to
the graph of V(x) at (2.47, 67.2801). Use your graphing calculator to verity that the graph of
V(x) and the tangent line are at the point (2.47, 67.2801).
231)
232)
Differentiate: f(x) = (3x2 + x)(8x– 7)
232)
233)
The revenue R from the sale of x units of a product is R=9
x+ 18x. The number of units
sold after t weeks of advertising is x= 20 +6
t. Find dR
dt when t= 6.
233)
234)
Find an equation of the tangent line to the curve y= (x–2)2 when x= 0.
234)
235)
If the amount of money hidden in your great–aunt’s mattress is given by the equation A(t)
= $643, find dA
dt .
235)
236)
Differentiate: h(x) = (4x– 5) x
236)
237)
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.
237)
238)
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.
238)