the differential $d x$ is an independent variable that can be assigned any nonzero real number; the differential $d y$ is defined to be $d y = f ' (x) d x$

given a differentiable function $y = f ' (x),$ the equation $d y = f ' (x) d x$ is the differential form of the derivative of $y$ with respect to $x$

the linear function $L (x) = f (a) + f ' (a) (x - a)$ is the linear approximation of $f$ at $x = a$

the relative error expressed as a percentage

the error that results in a calculated quantity $f (x)$ resulting from a measurement error dx

given an absolute error $\text{Δ} q$ for a particular quantity, $\frac{\text{Δ} q}{q}$ is the relative error.

tangent line approximation (linearization)

since the linear approximation of $f$ at $x = a$ is defined using the equation of the tangent line, the linear approximation of $f$ at $x = a$ is also known as the tangent line approximation to $f$ at $x = a$

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Calculus Volume 1 · 4.2 Linear Approximations and Differentials

Linear Approximations and Differentials: practice problems

21 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 47
Determine the necessary conditions such that the linear approximation function is constant. Use a graph to prove your result.
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Exercises 49
When is the linear approximation exact?
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Exercises 51
For the following exercises, find the linear approximation $L (x)$ to $y = f (x)$ near $x = a$ for the function.
$f (x) = \frac{1}{x}, a = 2$
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Exercises 53
For the following exercises, find the linear approximation $L (x)$ to $y = f (x)$ near $x = a$ for the function.
$f (x) = sin x, a = \frac{π}{2}$
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Exercises 55
For the following exercises, find the linear approximation $L (x)$ to $y = f (x)$ near $x = a$ for the function.
$f (x) = sin^{2} x, a = 0$
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Exercises 57
For the following exercises, compute the values given within 0.01 by deciding on the appropriate $f (x)$ and $a,$ and evaluating $L (x) = f (a) + f^{'} (a) (x - a) .$ Check your answer using a calculator.
[T] $sin (0.02)$
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Exercises 59
For the following exercises, compute the values given within 0.01 by deciding on the appropriate $f (x)$ and $a,$ and evaluating $L (x) = f (a) + f^{'} (a) (x - a) .$ Check your answer using a calculator.
[T] $(15.99)^{1 / 4}$
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Exercises 61
For the following exercises, compute the values given within 0.01 by deciding on the appropriate $f (x)$ and $a,$ and evaluating $L (x) = f (a) + f^{'} (a) (x - a) .$ Check your answer using a calculator.
[T] $sin (3.14)$
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Exercises 63
For the following exercises, determine the appropriate $f (x)$ and $a,$ and evaluate $L (x) = f (a) + f^{'} (a) (x - a) .$ Calculate the numerical error in the linear approximations that follow.
[T] $cos (0.01)$
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Exercises 65
For the following exercises, determine the appropriate $f (x)$ and $a,$ and evaluate $L (x) = f (a) + f^{'} (a) (x - a) .$ Calculate the numerical error in the linear approximations that follow.
[T] $(1.01)^{- 3}$
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Exercises 67
For the following exercises, determine the appropriate $f (x)$ and $a,$ and evaluate $L (x) = f (a) + f^{'} (a) (x - a) .$ Calculate the numerical error in the linear approximations that follow.
[T] $8.99$
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Exercises 69
For the following exercises, find the differential of the function.
$y = x cos x$
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Exercises 71
For the following exercises, find the differential of the function.
$y = \frac{x ^{2} + 2}{x - 1}$
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Exercises 73
For the following exercises, find the differential and evaluate for the given $x$ and $d x .$
$y = \frac{1}{x + 1},$ $x = 1,$ $d x = 0.25$
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Exercises 75
For the following exercises, find the differential and evaluate for the given $x$ and $d x .$
$y = \frac{3 x ^{2} + 2}{x + 1},$
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Exercises 77
For the following exercises, find the differential and evaluate for the given $x$ and $d x .$
$y = x^{3} + 2 x + \frac{1}{x},$
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Exercises 79
For the following exercises, find the change in volume $d V$ or in surface area $d A .$
$d A$ if the sides of a cube change from $x$ to $x + d x .$
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Exercises 81
For the following exercises, find the change in volume $d V$ or in surface area $d A .$
$d V$ if the radius of a sphere changes from $r$ by $d r .$
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Exercises 83
For the following exercises, find the change in volume $d V$ or in surface area $d A .$
$d V$ if a circular cylinder of height 3 changes from $r = 2$ to $r = 1.9 cm .$
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Exercises 85
For the following exercises, find the change in volume $d V$ or in surface area $d A .$
A pool has a rectangular base of 10 ft by 20 ft and a depth of 6 ft. What is the change in volume if you only fill it up to 5.5 ft?
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Review Exercises 531
True or False? Justify your answer with a proof or a counterexample. Assume that $f (x)$ is continuous and differentiable unless stated otherwise.
Find the linear approximation $L (x)$ to $y = x^{2} + tan (π x)$ near $x = \frac{1}{4} .$
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Key terms · 7

differential: the differential $d x$ is an independent variable that can be assigned any nonzero real number; the differential $d y$ is defined to be $d y = f ' (x) d x$
differential form: given a differentiable function $y = f ' (x),$ the equation $d y = f ' (x) d x$ is the differential form of the derivative of $y$ with respect to $x$
linear approximation: the linear function $L (x) = f (a) + f ' (a) (x - a)$ is the linear approximation of $f$ at $x = a$
percentage error: the relative error expressed as a percentage
propagated error: the error that results in a calculated quantity $f (x)$ resulting from a measurement error dx
relative error: given an absolute error $\text{Δ} q$ for a particular quantity, $\frac{\text{Δ} q}{q}$ is the relative error.
tangent line approximation (linearization): since the linear approximation of $f$ at $x = a$ is defined using the equation of the tangent line, the linear approximation of $f$ at $x = a$ is also known as the tangent line approximation to $f$ at $x = a$

Learn 4.2 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

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Linear Approximations and Differentials: practice problems

21 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 47

Determine the necessary conditions such that the linear approximation function is constant. Use a graph to prove your result.

Exercises 49

When is the linear approximation exact?

Exercises 51

For the following exercises, find the linear approximation $L (x)$ to $y = f (x)$ near $x = a$ for the function.

$f (x) = \frac{1}{x}, a = 2$

Exercises 53

For the following exercises, find the linear approximation $L (x)$ to $y = f (x)$ near $x = a$ for the function.

$f (x) = sin x, a = \frac{π}{2}$

Exercises 55

For the following exercises, find the linear approximation $L (x)$ to $y = f (x)$ near $x = a$ for the function.

$f (x) = sin^{2} x, a = 0$

Exercises 57

For the following exercises, compute the values given within 0.01 by deciding on the appropriate $f (x)$ and $a,$ and evaluating $L (x) = f (a) + f^{'} (a) (x - a) .$ Check your answer using a calculator.

[T] $sin (0.02)$

Exercises 59

[T] $(15.99)^{1 / 4}$

Exercises 61

[T] $sin (3.14)$

Exercises 63

For the following exercises, determine the appropriate $f (x)$ and $a,$ and evaluate $L (x) = f (a) + f^{'} (a) (x - a) .$ Calculate the numerical error in the linear approximations that follow.

[T] $cos (0.01)$

Exercises 65

For the following exercises, determine the appropriate $f (x)$ and $a,$ and evaluate $L (x) = f (a) + f^{'} (a) (x - a) .$ Calculate the numerical error in the linear approximations that follow.

[T] $(1.01)^{- 3}$

Exercises 67

For the following exercises, determine the appropriate $f (x)$ and $a,$ and evaluate $L (x) = f (a) + f^{'} (a) (x - a) .$ Calculate the numerical error in the linear approximations that follow.

[T] $8.99$

Exercises 69

For the following exercises, find the differential of the function.

$y = x cos x$

Exercises 71

For the following exercises, find the differential of the function.

$y = \frac{x ^{2} + 2}{x - 1}$

Exercises 73

For the following exercises, find the differential and evaluate for the given $x$ and $d x .$

$y = \frac{1}{x + 1},$ $x = 1,$ $d x = 0.25$

Exercises 75

For the following exercises, find the differential and evaluate for the given $x$ and $d x .$

$y = \frac{3 x ^{2} + 2}{x + 1},$

Exercises 77

For the following exercises, find the differential and evaluate for the given $x$ and $d x .$

$y = x^{3} + 2 x + \frac{1}{x},$

Exercises 79

For the following exercises, find the change in volume $d V$ or in surface area $d A .$

$d A$ if the sides of a cube change from $x$ to $x + d x .$

Exercises 81

For the following exercises, find the change in volume $d V$ or in surface area $d A .$

$d V$ if the radius of a sphere changes from $r$ by $d r .$

Exercises 83

For the following exercises, find the change in volume $d V$ or in surface area $d A .$

$d V$ if a circular cylinder of height 3 changes from $r = 2$ to $r = 1.9 cm .$

Exercises 85

For the following exercises, find the change in volume $d V$ or in surface area $d A .$

A pool has a rectangular base of 10 ft by 20 ft and a depth of 6 ft. What is the change in volume if you only fill it up to 5.5 ft?

Review Exercises 531

True or False? Justify your answer with a proof or a counterexample. Assume that $f (x)$ is continuous and differentiable unless stated otherwise.

Find the linear approximation $L (x)$ to $y = x^{2} + tan (π x)$ near $x = \frac{1}{4} .$

Key terms · 7

differential: the differential $d x$ is an independent variable that can be assigned any nonzero real number; the differential $d y$ is defined to be $d y = f ' (x) d x$
differential form: given a differentiable function $y = f ' (x),$ the equation $d y = f ' (x) d x$ is the differential form of the derivative of $y$ with respect to $x$
linear approximation: the linear function $L (x) = f (a) + f ' (a) (x - a)$ is the linear approximation of $f$ at $x = a$
percentage error: the relative error expressed as a percentage
propagated error: the error that results in a calculated quantity $f (x)$ resulting from a measurement error dx
relative error: given an absolute error $\text{Δ} q$ for a particular quantity, $\frac{\text{Δ} q}{q}$ is the relative error.
tangent line approximation (linearization): since the linear approximation of $f$ at $x = a$ is defined using the equation of the tangent line, the linear approximation of $f$ at $x = a$ is also known as the tangent line approximation to $f$ at $x = a$

Learn 4.2 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 4.2 in the textbook · This section on openstax.org