Stewart 9e Section 2.9: Linear Approximations and Differentials

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Stewart 9e Section 2.9: Linear Approximations and Differentials 0/54
1 Linearization · Level 1
Find the linearization \(L(x)\) of the function at \(a\): \(f(x) = x^3 - x^2 + 3\), \(a = -2\).
2 Linearization · Level 2
Find the linearization \(L(x)\) of the function at \(a\): \(f(x) = \cos(2 x)\), \(a = \dfrac{\pi}{6}\).
3 Linearization · Level 1
Find the linearization \(L(x)\) of the function at \(a\): \(f(x) = \sqrt[3]{x}\), \(a = 8\).
4 Linearization · Level 2
Find the linearization \(L(x)\) of the function at \(a\): \(f(x) = \dfrac{2}{\sqrt{x^2 - 5}}\), \(a = 3\).
5 Linear Approximation · Level 2
Find the linear approximation of the function \(f(x) = \sqrt{1 - x}\) at \(a = 0\) and use it to approximate the numbers \(\sqrt{0.9}\) and \(\sqrt{0.99}\). Illustrate by graphing \(f\) and the tangent line.
6 Linear Approximation · Level 2
Find the linear approximation of the function \(g(x) = \sqrt[3]{1 + x}\) at \(a = 0\) and use it to approximate the numbers \(\sqrt[3]{0.95}\) and \(\sqrt[3]{1.1}\). Illustrate by graphing \(g\) and the tangent line.
7 Linear Approximation Accuracy · Level 3
Verify the given linear approximation at \(a = 0\): \(\sqrt[4]{1 + 2 x} \approx 1 + \left(\dfrac{1}{2}\right) x\). Then determine the values of \(x\) for which the linear approximation is accurate to within \(0.1\).
8 Linear Approximation Accuracy · Level 3
Verify the given linear approximation at \(a = 0\): \((1 + x)^{-3} \approx 1 - 3 x\). Then determine the values of \(x\) for which the linear approximation is accurate to within \(0.1\).
9 Linear Approximation Accuracy · Level 3
Verify the given linear approximation at \(a = 0\): \(1/(1 + 2 x)^4 \approx 1 - 8 x\). Then determine the values of \(x\) for which the linear approximation is accurate to within \(0.1\).
10 Linear Approximation Accuracy · Level 2
Verify the given linear approximation at \(a = 0\): \(\tan x \approx x\). Then determine the values of \(x\) for which the linear approximation is accurate to within \(0.1\).
11 Differentials · Level 2
Find the differential of the function: \(y = (x^2 - 3)^{-2}\).
12 Differentials · Level 2
Find the differential of the function: \(y = \sqrt{1 - t^4}\).
13 Differentials · Level 2
Find the differential of the function: \(y = \dfrac{1 + 2 u}{1 + 3 u}\).
14 Differentials · Level 2
Find the differential of the function: \(y = \theta^2 \sin(2 \theta)\).
15 Differentials · Level 2
Find the differential of the function: \(y = 1/(x^2 - 3 x)\).
16 Differentials · Level 2
Find the differential of the function: \(y = \sqrt{1 + \cos \theta}\).
17 Differentials · Level 2
Find the differential of the function: \(y = \sqrt{t \cos t}\).
18 Differentials · Level 2
Find the differential of the function: \(y = \left(\dfrac{1}{x}\right) \sin x\).
19 Differentials - Evaluation · Level 2
Let \(y = \tan x\). (a) Find the differential \(d y\). (b) Evaluate \(d y\) for \(x = \dfrac{\pi}{4}\) and \(d x = -0.1\).
20 Differentials - Evaluation · Level 2
Let \(y = \cos(\pi x)\). (a) Find the differential \(d y\). (b) Evaluate \(d y\) for \(x = \dfrac{1}{2}\) and \(d x = -0.02\).
21 Differentials - Evaluation · Level 2
Let \(y = \sqrt{3 + x^2}\). (a) Find the differential \(d y\). (b) Evaluate \(d y\) for \(x = 1\) and \(d x = -0.1\).
22 Differentials - Evaluation · Level 2
Let \(y = \dfrac{x + 1}{x - 1}\). (a) Find the differential \(d y\). (b) Evaluate \(d y\) for \(x = 2\) and \(d x = 0.05\).
23 Increment vs Differential · Level 2
Compute \(\Delta y\) and \(d y\) for \(y = x^2 - 4 x\), \(x = 3\), \(\Delta x = 0.5\). Then sketch a diagram showing the line segments with lengths \(d x\), \(d y\), and \(\Delta y\).
24 Increment vs Differential · Level 2
Compute \(\Delta y\) and \(d y\) for \(y = x - x^3\), \(x = 0\), \(\Delta x = -0.3\). Then sketch a diagram showing the line segments with lengths \(d x\), \(d y\), and \(\Delta y\).
25 Increment vs Differential · Level 2
Compute \(\Delta y\) and \(d y\) for \(y = \sqrt{x - 2}\), \(x = 3\), \(\Delta x = 0.8\). Then sketch a diagram showing the line segments with lengths \(d x\), \(d y\), and \(\Delta y\).
26 Increment vs Differential · Level 2
Compute \(\Delta y\) and \(d y\) for \(y = x^3\), \(x = 1\), \(\Delta x = 0.5\). Then sketch a diagram showing the line segments with lengths \(d x\), \(d y\), and \(\Delta y\).
27 Comparing Delta y and dy · Level 2
Compare the values of \(\Delta y\) and \(d y\) for \(f(x) = x^4 - x + 1\) if \(x\) changes from 1 to 1.05. What if \(x\) changes from 1 to 1.01? Does the approximation \(\Delta y \approx d y\) become better as \(\Delta x\) gets smaller?
28 Comparing Delta y and dy · Level 2
Compare the values of \(\Delta y\) and \(d y\) for \(f(x) = (x^3 + 3)^2\) if \(x\) changes from 1 to 1.05. What if \(x\) changes from 1 to 1.01? Does the approximation \(\Delta y \approx d y\) become better as \(\Delta x\) gets smaller?
29 Comparing Delta y and dy · Level 2
Compare the values of \(\Delta y\) and \(d y\) for \(f(x) = \sqrt{5 - x}\) if \(x\) changes from 1 to 1.05. What if \(x\) changes from 1 to 1.01? Does the approximation \(\Delta y \approx d y\) become better as \(\Delta x\) gets smaller?
30 Comparing Delta y and dy · Level 2
Compare the values of \(\Delta y\) and \(d y\) for \(f(x) = 1/(x^2 + 1)\) if \(x\) changes from 1 to 1.05. What if \(x\) changes from 1 to 1.01? Does the approximation \(\Delta y \approx d y\) become better as \(\Delta x\) gets smaller?
31 Estimation - Linear Approximation · Level 2
Use a linear approximation (or differentials) to estimate the given number: \((1.999)^4\).
32 Estimation - Linear Approximation · Level 2
Use a linear approximation (or differentials) to estimate the given number: \(\dfrac{1}{4}.002\).
33 Estimation - Linear Approximation · Level 2
Use a linear approximation (or differentials) to estimate the given number: \(\sqrt[3]{1001}\).
34 Estimation - Linear Approximation · Level 2
Use a linear approximation (or differentials) to estimate the given number: \(\sqrt{100.5}\).
35 Estimation - Linear Approximation · Level 2
Use a linear approximation (or differentials) to estimate the given number: \(\tan 2^{\circ}\).
36 Estimation - Linear Approximation · Level 2
Use a linear approximation (or differentials) to estimate the given number: \(\cos 29^{\circ}\).
37 Explanation - Linear Approximation · Level 1
Explain, in terms of linear approximations or differentials, why the approximation \(\sec 0.08 \approx 1\) is reasonable.
38 Explanation - Linear Approximation · Level 1
Explain, in terms of linear approximations or differentials, why the approximation \(\sqrt{4.02} \approx 2.005\) is reasonable.
39 Error Estimation - Cube · Level 3
The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing (a) the volume of the cube and (b) the surface area of the cube.
40 Error Estimation - Disk · Level 3
The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. (a) Use differentials to estimate the maximum error in the calculated area of the disk. (b) What is the relative error and the percentage error?
41 Error Estimation - Sphere · Level 3
The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. What is the relative error? (b) Use differentials to estimate the maximum error in the calculated volume. What is the relative error?
42 Error Estimation - Paint Volume · Level 2
Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m.
43 Differentials - Cylindrical Shell · Level 3
(a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height \(h\), inner radius \(r\), and thickness \(\Delta r\). (b) What is the error involved in using the formula from part (a)?
44 Error Estimation - Right Triangle · Level 3
One side of a right triangle is known to be 20 cm long and the opposite angle is measured as \(30^{\circ}\), with a possible error of \(\pm 1^{\circ}\). (a) Use differentials to estimate the error in computing the length of the hypotenuse. (b) What is the percentage error?
45 Application - Ohm's Law · Level 3
If a current \(I\) passes through a resistor with resistance \(R\), Ohm's Law states that the voltage drop is \(V = R I\). If \(V\) is constant and \(R\) is measured with a certain error, use differentials to show that the relative error in calculating \(I\) is approximately the same (in magnitude) as the relative error in \(R\).
46 Application - Poiseuille's Law · Level 3
When blood flows along a blood vessel, the flux \(F\) (the volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius \(R\) of the blood vessel: \(F = k R^4\). Show that the relative change in \(F\) is about four times the relative change in \(R\). How will a \(5%\) increase in the radius affect the flow of blood?
47 Differential Rules - Proof · Level 2
Establish the following rules for working with differentials (where \(c\) denotes a constant and \(u\) and \(v\) are functions of \(x\)): (a) \(d c = 0\), (b) \(d(c u) = c d u\), (c) \(d(u + v) = d u + d v\), (d) \(d(u v) = u d v + v d u\), (e) \(d\left(\dfrac{u}{v}\right) = (v d u - u d v)/v^2\), (f) \(d(x^n) = n x^{n-1} d x\).
48 Application - Pendulum · Level 3
In physics textbooks, the period \(T\) of a pendulum of length \(L\) is often given as \(T \approx 2 \pi \sqrt{\dfrac{L}{g}}\), provided that the pendulum swings through a relatively small arc. In the course of deriving this formula, the equation \(a_T = -g \sin \theta\) for the tangential acceleration of the bob of the pendulum is obtained, and then \(\sin \theta\) is replaced by \(\theta\) with the remark that for small angles, \(\theta\) (in radians) is very close to \(\sin \theta\). (a) Verify the linear approximation at 0 for the sine function: \(\sin \theta \approx \theta\). (b) If \(\theta = \dfrac{\pi}{18}\) (equivalent to \(10^{\circ}\)) and we approximate \(\sin \theta\) by \(\theta\), what is the percentage error? (c) Use a graph to determine the values of \(\theta\) for which \(\sin \theta\) and \(\theta\) differ by less than \(2%\). What are the values in degrees?
49 Linear Approximation from Graph · Level 2
Suppose that the only information we have about a function \(f\) is that \(f(1) = 5\) and the graph of its derivative is as shown. (a) Use a linear approximation to estimate \(f(0.9)\) and \(f(1.1)\). (b) Are your estimates in part (a) too large or too small? Explain.
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50 Linear Approximation - Concavity Check · Level 2
Suppose that we don't have a formula for \(g(x)\) but we know that \(g(2) = -4\) and \(g'(x) = \sqrt{x^2 + 5}\) for all \(x\). (a) Use a linear approximation to estimate \(g(1.95)\) and \(g(2.05)\). (b) Are your estimates in part (a) too large or too small? Explain.
51 Example - Linearization and Linear Approximation · Level 2
Find the linearization \(L(x)\) of the function \(f(x) = \sqrt{x + 3}\) at \(a = 1\) and use it to approximate the numbers \(\sqrt{3.98}\) and \(\sqrt{4.05}\). Are these approximations overestimates or underestimates?
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52 Example - Accuracy of Linear Approximation · Level 3
For what values of \(x\) is the linear approximation \(\sqrt{x + 3} \approx \dfrac{7}{4} + \dfrac{x}{4}\) accurate to within \(0.5\)? What about accuracy to within \(0.1\)?
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53 Example - Differentials vs Increment · Level 2
Compare the values of \(\Delta y\) and \(d y\) if \(y = f(x) = x^3 + x^2 - 2x + 1\) and \(x\) changes (a) from 2 to 2.05 and (b) from 2 to 2.01.
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54 Example - Error Estimation with Differentials · Level 3
The radius of a sphere was measured and found to be 21 cm with a possible error in measurement of at most 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere?

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