Worksheet 6 Answers (AC 1.3, 1.5.1)
pdf
keyboard_arrow_up
School
Cornell University *
*We aren’t endorsed by this school
Course
1110
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
4
Uploaded by CorporalSardine2209
The Derivative at a Point (Active Calculus
1.3, 1.5.1) Solutions
Learning Objectives:
You should be able to...
Calculate the derivative of
f
at
x
=
a
.
Interpret the derivative of
f
at
x
=
a
graphically and in specific contexts.
Today we are focusing on the following definition:
Derivative at a point.
The derivative of
f
at
x
=
a
, denoted
f
′
(
a
), is the instantaneous rate of change
of
f
at
x
=
a
. That is,
f
′
(
a
) =
IV
x
=
a
= lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
,
provided this limit exists.
1. Let
f
(
x
) =
1
x
.
(a) Sketch the graph of
f
(
x
). Sketch a line on your graph whose slope is
f
′
(2). Based on this, is
f
′
(2)
positive or negative?
Answer
.
The graph of
f
(
x
) should be very familiar to you; here it is (in black), along with the
line tangent to the graph at
x
= 2 (in blue), which has slope
f
′
(2):
−
2
2
−
2
2
This line has negative slope, so
f
′
(2) is negative
.
(b) Calculate
f
′
(2) exactly using the definition of the derivative. Is this consistent with part
(a)
?
Answer
.
−
1
4
, which is negative as predicted.
2. Consider the function
f
(
x
) graphed below.
Adapted from materials created by the Cornell Active Learning in Math team and Harvard University’s Math 1a.
1/
4
Worksheet 6:
The Derivative at a Point (Active Calculus
1.3, 1.5.1) Solutions
MATH 1110
−
3
−
2
−
1
1
2
3
−
1
1
2
3
4
5
(a) By drawing tangent lines, estimate the values of
f
′
(
−
1),
f
′
(0),
f
′
(1), and
f
′
(2).
Answer
.
We can estimate the slope of the tangent lines by using (rough) points that lie on the
tangent line:
f
′
(
−
1)
≈
2
−
0
−
1
.
5
−
(
−
0
.
5)
=
2
−
1
=
−
2
f
′
(0) = 0
f
′
(1)
≈
2
−
0
1
.
5
−
0
.
5
= 2
f
′
(2)
≈
4
−
0
2
−
1
= 4
(b) As you may have suspected, the function graphed above is
f
(
x
) =
x
2
.
Compute
f
′
(1) using the
limit definition of derivative. Is this close to your estimate from part
(a)
?
Answer
.
f
′
(1) = 2
3. A company manufactures rope, and the total cost of producing
r
feet of rope is
c
(
r
) dollars.
(a) Interpret the statement “
c
(2000) = 800” in a sentence.
Answer
.
Producing 2,000 feet of rope costs
800.
(b) What are the units of
c
′
(
r
)?
Interpret the statement “
c
′
(2000) = 0
.
35” in a sentence.
(This
2/
4
Worksheet 6:
The Derivative at a Point (Active Calculus
1.3, 1.5.1) Solutions
MATH 1110
statement can also be written as
dc
dr
r
=2000
= 0
.
35.)
Answer
.
$
feet
. After producing 2,000 feet of rope, the next foot of rope will cost approximately
0.35.
(c) Suppose that
c
(2000) = 800 and
dc
dr
r
=2000
= 0
.
35. Use these values to estimate
c
(2100). Explain
your thinking.
Answer
.
c
(2100)
≈
$835
.
(d) Do you expect
c
′
(2000) to be less than, equal to, or greater than
c
′
(3000)? Why?
Answer
.
We can make an argument in all cases. For instance, we might expect
c
′
(2000)
> c
′
(3000)
because production may be more efficient on longer lengths of rope; or it may be the opposite and
the extra length is more difficult to manage (perhaps because of the increased weight) and so
c
′
(2000)
< c
′
(3000).
(e) You hear someone claim that
dc
dr
r
=5000
=
−
0
.
1. Is this possible? Why or why not?
Answer
.
That does not seem possible; this would mean that producing rope longer than 5,000
feet would cost less than producing a rope of shorter length.
Because it involves effort (whether
time or resources) to produce rope,
dc
dr
should be positive.
4. Gromit has been growing a giant squash for Tottington Hall’s annual Giant Vegetable Competition. He
has carefully tracked his squash’s length and weight.
Let
w
(
ℓ
) be the squash’s weight in kg when its
length is
ℓ
cm.
(a) What are the units of
w
′
(
ℓ
)?
Answer
.
Positive since, as the length of the squash increases, we would expect it to be heavier
as well.
(b) Suppose
w
′
(70) = 8. Which one of the following is the most reasonable conclusion?
A. It takes 70 days for the squash to grow to be 8 kg.
B. When the squash is 70 cm long, it weighs 8 kg.
3/
4
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
Worksheet 6:
The Derivative at a Point (Active Calculus
1.3, 1.5.1) Solutions
MATH 1110
C. When the squash is 70 cm longer than its current length, it will weigh 8 kg more than it
currently does.
D. When the squash grows from 70 cm to 71 cm, it will gain about 8 kg in weight.
Answer
.
D
is correct.
(c) Interpret the statement
w
′
(105) = 10 in words.
Answer
.
We can say something like, “When the squash is 105 cm long, the instantaneous rate
of change of the squash’s weight is 10 kg/cm.” We can also translate this into a more “everyday”
statement like, “If the squash grew from 105 cm to 106 cm, we’d expect it to gain about 10 kg.”
In economics:
the cost of production
c
(
x
) is a function of
x
, the number of units produced, and
the marginal cost of production
,
c
′
(
x
), is the rate of change of cost with respect to
x
.
5. Oftentimes, economists are interested in the additional cost
; i.e., the cost of producing exactly one
additional unit. Suppose that it costs
c
(
x
) = 25 + 16
x
−
0
.
04
x
2
dollars for a restaurant to produce
x
burgers.
(a) Compute the actual (exact) cost of producing the 101
st
burger.
Answer
.
The cost of producing the 101
st
burger is
c
(101)
−
c
(100) =
7.96
.
(b) Use a derivative to approximate the marginal cost of making the 101
st
burger, having made 100
burgers. How does this compare to the exact cost?
Answer
.
The cost of the 101
st
is approximately
c
′
(100) = $8.
4/
4