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Find the linear approximation of the following function, and determine if it is…
Related Topics
Wize University Calculus 1 Textbook > Applications of Differentiation
Linearization (Linear Approximation)
4 Activities
Find the linear approximation of the following function, and determine if it is an over/under approximation.
f
(
x
)
=
x
3
4
+
x
3
f(x)=\frac{x^3}{4+x^3}
f
(
x
)
=
4
+
x
3
x
3
at
x
=
2.1
x=2.1
x
=
2.1
3.1
2
\frac{3.1}{2}
2
3.1
, under approximation
2.1
2
\frac{2.1}{2}
2
2.1
, over approximation
3.1
3
\frac{3.1}{3}
3
3.1
, under approximation
2.1
3
\frac{2.1}{3}
3
2.1
, over approximation
I don't know
Check Submission
More Linearization (Linear Approximation) Questions:
Use linearization to construct a linear approximation to
x
4
\sqrt[4]x
4
x
for
x
x
x
values near
1
1
1
, given that
f
′
(
x
)
=
1
4
x
−
3
4
f'(x)=\frac14x^{-\frac34}
f
′
(
x
)
=
4
1
x
−
4
3
.
Linear Approximation
Using a linear approximation, estimate
f
(
4.1
)
f(4.1)
f
(
4.1
)
, given that
f
(
4
)
=
2
f(4)=2
f
(
4
)
=
2
and
f
′
(
x
)
=
x
2
+
20
f^{\prime}(x)=\sqrt{x^2+20}
f
′
(
x
)
=
x
2
+
20
. If
f
(
x
)
f(x)
f
(
x
)
is concave up near this point, what does that tell us about our estimate?
Linear Approximation
Use a suitable linear approximation to estimate
3.99
\sqrt{3.99}
3.99
. Is your estimate likely to be an over- or under-estimate? HINT: Use
f
(
x
)
=
x
f(x)=\sqrt{x}
f
(
x
)
=
x
, which gives us
f
′
(
x
)
=
1
2
x
f'(x)=\frac1{2\sqrt{x}}
f
′
(
x
)
=
2
x
1
and
f
′
′
(
x
)
=
−
1
4
x
2
3
f''(x)=-\dfrac{1}{4\sqrt[3]{x^2}}
f
′′
(
x
)
=
−
4
3
x
2
1
.
Linear Approximation of Given Function
Use a linear approximation to estimate
f
(
4.1
)
f(4.1)
f
(
4.1
)
, given that
f
(
4
)
=
2
f(4)=2
f
(
4
)
=
2
and
f
′
(
x
)
=
x
2
+
20
f^{\prime}(x)=\sqrt{x^2+20}
f
′
(
x
)
=
x
2
+
20
.
Linear Approximation
Using a linear approximation, estimate
f
(
4.1
)
f(4.1)
f
(
4.1
)
, given that
f
(
4
)
=
2
f(4)=2
f
(
4
)
=
2
,
f
′
(
x
)
=
x
2
+
20
f^{\prime}(x)=\sqrt{x^2+20}
f
′
(
x
)
=
x
2
+
20
Linear Approximation
Use a suitable linear approximation to estimate
3.99
\sqrt{3.99}
3.99
.
Practice: Linear Approximation
Consider the function
f
(
x
)
=
sin
(
x
)
f(x) = \sin(x)
f
(
x
)
=
sin
(
x
)
a) Find the linear approximation of the function
f
(
x
)
f(x)
f
(
x
)
near
x
=
0
x = 0
x
=
0
.
Written Answer 5
Use linear approximation at a suitable close value to estimate
arctan
(
0.9
)
\arctan(0.9)
arctan
(
0.9
)
. Your answer may be left in terms of fractions.
Linear Approximation
Approximate
(
28
)
1
/
3
(28)^{1/3}
(
28
)
1/3
using a linear approximation of the function
h
(
x
)
=
x
1
/
3
h(x) = x^{1/3}
h
(
x
)
=
x
1/3
Let
f
(
x
)
=
cos
(
3
x
)
+
x
−
1
f(x)=\cos{(3x)}+x-1
f
(
x
)
=
cos
(
3
x
)
+
x
−
1
. Approximate
f
(
π
3
+
1
5
)
f(\frac{\pi}{3}+\frac{1}{5})
f
(
3
π
+
5
1
)
using linear approximation. Is this overestimation or underestimation of the actual value? Estimate the max error bound for this approximation.
Let
f
(
x
)
=
cos
(
3
x
)
+
x
−
1
f(x)=\cos{(3x)}+x-1
f
(
x
)
=
cos
(
3
x
)
+
x
−
1
. Approximate
f
(
π
3
+
1
5
)
f(\frac{\pi}{3}+\frac{1}{5})
f
(
3
π
+
5
1
)
using linear approximation. Is this overestimation or underestimation of the actual value?
The linearization of
f
(
x
)
=
t
a
n
x
f(x)=tanx
f
(
x
)
=
t
an
x
at
x
=
π
4
x=\frac{\pi}{4}
x
=
4
π
is
Linear Approximation
Approximate
9
3
.
\sqrt[3]{9}.
3
9
.
Linear Approximation
Use a suitable linear approximation to estimate
1.99
5
1.99^5
1.9
9
5
Linear Approximation
Use a suitable linear approximation to estimate
3.99
\sqrt{3.99}
3.99
.
Linear Approximation
Use a linear approximation to estimate
9.2
\sqrt{9.2}
9.2
using
f
(
x
)
=
3
+
x
.
f(x)=\sqrt{3+x}.
f
(
x
)
=
3
+
x
.
Linear Approximation
The linear approximation
L
(
x
)
L(x)
L
(
x
)
of the function
f
(
x
)
=
x
f(x)=\sqrt{x}
f
(
x
)
=
x
at
x
=
16
x=16
x
=
16
is
Practice: Linear Approximations
Practice: Linear Approximations
Approximate
ln
(
1.01
)
\ln\left(1.01\right)
ln
(
1.01
)
and
ln
(
0.98
)
\ln\left(0.98\right)
ln
(
0.98
)
.
final114
The linear approximation
L
(
x
)
L(x)
L
(
x
)
of the function
f
(
x
)
=
x
f(x)=\sqrt{x}
f
(
x
)
=
x
at
x
=
16
x=16
x
=
16
is
Approximate
x
3
2
+
x
2
3
x^{\frac{3}{2}}+x^{\frac{2}{3}}
x
2
3
+
x
3
2
at
x
=
1.1
x=1.1
x
=
1.1
using a linear approximation.
final114
The linear approximation
L
(
x
)
L(x)
L
(
x
)
of the function
f
(
x
)
=
x
f(x)=\sqrt{x}
f
(
x
)
=
x
at
x
=
16
x=16
x
=
16
is (enter L=...)
Linear Approximation: Estimate using Given Values
Use a linear approximation to estimate
f
(
2.01
)
f(2.01)
f
(
2.01
)
given that
f
(
2
)
=
1
f(2)=1
f
(
2
)
=
1
and
f
′
(
2
)
=
2
f^{\prime}(2)=2
f
′
(
2
)
=
2
.
Linear Approximation
Use a suitable linear approximation to estimate
8.01
2
3
8.01^{\frac{2}{3}}
8.0
1
3
2
.
Linear Approximation: Estimating a Cube Root
Use a linear approximation to estimate
65
3
\sqrt[3]{65}
3
65
. Then use the same linearization equation to estimate
28
3
\sqrt[3]{28}
3
28
. Is it still a good approximation? Why?
Linear Approximation of Given Function
Use a linear approximation to estimate
f
(
4.1
)
f(4.1)
f
(
4.1
)
, given that
f
(
4
)
=
2
f(4)=2
f
(
4
)
=
2
and
f
′
(
x
)
=
x
2
+
20
f^{\prime}(x)=\sqrt{x^2+20}
f
′
(
x
)
=
x
2
+
20
.
Linear Approximation: Estimating a 4th Root
Use a suitable linear approximation to estimate
68.82
1
/
4
68.82^{1/4}
68.8
2
1/4
(leave it as an exact value if you're not allowed a calculator on the exam).
Use linearization to construct a linear approximation to
x
4
\sqrt[4]x
4
x
for
x
x
x
values near
1
1
1
.
Use a linear approximation to estimate
99.9
\sqrt{99.9}
99.9
.
Concept Clarifier
Use linear approximation to estimate the value of
60
3
\sqrt[3]{60}
3
60
.
Linear Approximation
The linear approximation
L
(
x
)
L(x)
L
(
x
)
of the function
f
(
x
)
=
cos
x
f(x)=\cos{x}
f
(
x
)
=
cos
x
at
x
=
π
3
x=\frac{\pi}{3}
x
=
3
π
is
Linear Approximation
Using a linear approximation, estimate
f
(
4.1
)
f(4.1)
f
(
4.1
)
, given that
f
(
4
)
=
2
f(4)=2
f
(
4
)
=
2
and
f
′
(
x
)
=
x
2
+
20
f^{\prime}(x)=\sqrt{x^2+20}
f
′
(
x
)
=
x
2
+
20
. If
f
(
x
)
f(x)
f
(
x
)
is concave up near this point, what does that tell us about our estimate?
Approximate
arctan
(
0.9
)
\arctan\left(0.9\right)
arctan
(
0.9
)
by linear approximation.
Use a linear approximation to approximate the value of the function by using the given information.
f
(
2
)
=
1
,
f
′
(
2
)
=
2
f(2)=1, f^{\prime}(2)=2
f
(
2
)
=
1
,
f
′
(
2
)
=
2
Approximate
f
(
2.01
)
f(2.01)
f
(
2.01
)
. Enter your answer to two decimal places.
Use a linear approximation to estimate the value of the following function at the given point.
f
(
x
)
=
2
x
+
1
x
3
+
1
\displaystyle f(x)=\frac{2x+1}{x^3+1}
f
(
x
)
=
x
3
+
1
2
x
+
1
Approximate
f
(
1.1
)
f(1.1)
f
(
1.1
)
.
Use a linear approximation to estimate
82
1
/
4
82^{1/4}
8
2
1/4
.
Approximate
x
3
2
+
x
2
3
x^{\frac{3}{2}}+x^{\frac{2}{3}}
x
2
3
+
x
3
2
at
x
=
1.1
x=1.1
x
=
1.1
using linear approximation about
x
0
=
1
x_0=1
x
0
=
1
.
Use a linear approximation to estimate the given quantity.
8.01
2
/
3
8.01^{2/3}
8.0
1
2/3
Practice: Relative Error
A farmer used 100m of fence material to create a square pen to hold in his animals. However, there's a possible error of 2m in the length of the fence he used.
a) Use a linear approximation to estimate the maximum possible absolute error in the area of the pen. Include units in the answer.
b) Use the previous linear approximation to find the maximum relative error in the area. Include units in your answer.
Practice: Linear Approximation
Consider the function
f
(
x
)
=
sin
(
x
)
f(x) = \sin(x)
f
(
x
)
=
sin
(
x
)
a) Find the linear approximation of the function
f
(
x
)
f(x)
f
(
x
)
near
x
=
0
x = 0
x
=
0
.
Linear Approximation
Consider the function
f
(
x
)
=
4
−
x
f(x) = \sqrt{ 4 - x}
f
(
x
)
=
4
−
x
Practice: Relative Error
A farmer used 100m of fence material to create a square pen to hold in his animals. However, there's a possible error of 2m in the length of the fence he used.
a) Use a linear approximation to estimate the maximum possible absolute error in the area of the pen. Include units in the answer.
b) Use the previous linear approximation to find the maximum relative error in the area. Include units in your answer.
Practice: Linear Approximation
Consider the function
f
(
x
)
=
sin
(
x
)
f(x) = \sin(x)
f
(
x
)
=
sin
(
x
)
a) Find the linear approximation of the function
f
(
x
)
f(x)
f
(
x
)
near
x
=
0
x = 0
x
=
0
.
Linear Approximation and Trigonometric Derivative
a) Show that for
f
(
x
)
=
arcsin
(
x
)
f(x) = \arcsin(x)
f
(
x
)
=
arcsin
(
x
)
, that
f
′
(
x
)
=
1
1
−
x
2
f'(x) = \frac{1}{\sqrt{1 - x^2}}
f
′
(
x
)
=
1
−
x
2
1
.
b) Use linear approximation at a suitable close value to estimate
arcsin
(
0.1
)
\arcsin(0.1)
arcsin
(
0.1
)
. Your solution may be left in terms of fractions.
Consider the function
f
(
x
)
=
4
−
x
f(x) = \sqrt{ 4 - x}
f
(
x
)
=
4
−
x
a) Find the linear approximation of the function
f
(
x
)
f(x)
f
(
x
)
near
x
=
0
x = 0
x
=
0
.
Using a linear approximation, estimate
g
(
1.1
)
g(1.1)
g
(
1.1
)
, given that
g
(
x
)
=
f
(
x
2
+
1
)
g(x)=f(x^2+1)
g
(
x
)
=
f
(
x
2
+
1
)
and
f
(
2
)
=
f
′
(
2
)
=
1
f(2)=f^{\prime}(2)=1
f
(
2
)
=
f
′
(
2
)
=
1
.
Use a linear approximation to estimate
8.01
2
/
3
8.01^{2/3}
8.0
1
2/3
For the following exercises, use a linear approximation to estimate the given quantities.
8.01
2
/
3
8.01^{2/3}
8.0
1
2/3