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Chapter 3

Differentiation Rules

Educators

CW
+ 1 more educators

Problem 1

Find the linearization $ L(x) $ of the function at $ a. $
$ f(x) = x^3 - x^2 + 3, a = -2 $

Amrita B.
Numerade Educator

Problem 2

Find the linearization $ L(x) $ of the function at $ a. $
$ f(x) = \sin x, a = \pi/6 $

Amrita B.
Numerade Educator

Problem 3

Find the linearization $ L(x) $ of the function at $ a. $
$ f(x) = \sqrt x, a = 4 $

Amrita B.
Numerade Educator

Problem 4

Find the linearization $ L(x) $ of the function at $ a. $
$ f(x) = 2^x, a = 0 $

Amrita B.
Numerade Educator

Problem 5

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.

Oswaldo J.
Numerade Educator

Problem 6

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.

Amrita B.
Numerade Educator

Problem 7

Verify the given linear approximation at $ a = 0. $ Then determine the values of $ x $ for which the linear approximation is accurate to within 0.1.
$ \ln (1 + x) \approx x $

Amrita B.
Numerade Educator

Problem 8

Verify the given linear approximation at $ a = 0. $ Then determine the values of $ x $ for which the linear approximation is accurate to within 0.1.
$ (1 + x)^{-3} \approx -3x $

Amrita B.
Numerade Educator

Problem 9

Verify the given linear approximation at $ a = 0. $ Then determine the values of $ x $ for which the linear approximation is accurate to within 0.1.
$ \sqrt [4]{1 + 2x} \approx 1 + \frac {1}{2}x $

Amrita B.
Numerade Educator

Problem 10

Verify the given linear approximation at $ a = 0. $ Then determine the values of $ x $ for which the linear approximation is accurate to within 0.1.
$ e^x \cos x \approx 1 + x $

Amrita B.
Numerade Educator

Problem 11

Find the differential of each function.
(a) $ y = xe^{4x} $
(b) $ y = \sqrt {1 -t^4} $

Amrita B.
Numerade Educator

Problem 12

Find the differential of each function.
(a) $ y = \frac {1 + 2u}{1 + 3u} $
(b) $ y = \theta^2 \sin 2\theta $

Amrita B.
Numerade Educator

Problem 13

Find the differential of each function.
(a) $ y = \tan \sqrt t $
(b) $ y = \frac {1 - v^2}{1 + v^2} $

Amrita B.
Numerade Educator

Problem 14

Find the differential of each function.
(a) $ y = \ln (\sin \theta) $
(b) $ y = \frac {e^x}{1 - e^x} $

Amrita B.
Numerade Educator

Problem 15

(a) Find the differential $ dy $ and (b) evaluate $ dy $ for the given values of $ x $ and $ dx. $
$ y = e^{x/10}, x = 0, dx = 0.1 $

Leon D.
Numerade Educator

Problem 16

(a) Find the differential $ dy $ and (b) evaluate $ dy $ for the given values of $ x $ and $ dx. $
$ y = \cos \pi x, x = \frac {1}{3}, dx = -0.02 $

Amrita B.
Numerade Educator

Problem 17

(a) Find the differential $ dy $ and (b) evaluate $ dy $ for the given values of $ x $ and $ dx. $
$ y = \sqrt {3 + x^2}, x = 1, dx = -0.1 $

Amrita B.
Numerade Educator

Problem 18

(a) Find the differential $ dy $ and (b) evaluate $ dy $ for the given values of $ x $ and $ dx. $
$ y = \frac {x + 1}{x - 1}, x = 2, dx = 0.05 $

Amrita B.
Numerade Educator

Problem 19

Compute $ \Delta y $ and $ dy $ for the given values of $ x $ and $ dx = \Delta x. $ Then sketch a diagram like Figure 5 showing the line segments with lengths $ dx, dy, $ and $ \Delta y. $
$ y = x^2 - 4x, x = 3, \Delta x = 0.5 $

Linda H.
Numerade Educator

Problem 20

Compute $ \Delta y $ and $ dy $ for the given values of $ x $ and $ dx = \Delta x. $ Then sketch a diagram like Figure 5 showing the line segments with lengths $ dx, dy, $ and $ \Delta y. $
$ y = x - x^3, x = 0, \Delta x = -0.3 $

Amrita B.
Numerade Educator

Problem 21

Compute $ \Delta y $ and $ dy $ for the given values of $ x $ and $ dx = \Delta x. $ Then sketch a diagram like Figure 5 showing the line segments with lengths $ dx, dy, $ and $ \Delta y. $
$ y = \sqrt {x -2}, x = 3, \Delta x = 0.8 $

Amrita B.
Numerade Educator

Problem 22

Compute $ \Delta y $ and $ dy $ for the given values of $ x $ and $ dx = \Delta x. $ Then sketch a diagram like Figure 5 showing the line segments with lengths $ dx, dy, $ and $ \Delta y. $
$ y = e^x, x = 0, \Delta = 0.5 $

Amrita B.
Numerade Educator

Problem 23

Use a linear approximation (or differentials) to estimate the given number.
$ (1,999)^4 $

Amrita B.
Numerade Educator

Problem 24

Use a linear approximation (or differentials) to estimate the given number.
$ 1/4.002 $

Amrita B.
Numerade Educator

Problem 25

Use a linear approximation (or differentials) to estimate the given number.
$ \sqrt [3]{1001} $

Amrita B.
Numerade Educator

Problem 26

Use a linear approximation (or differentials) to estimate the given number.
$ \sqrt {100.5} $

Oswaldo J.
Numerade Educator

Problem 27

Use a linear approximation (or differentials) to estimate the given number.
$ e^{0.1} $

Amrita B.
Numerade Educator

Problem 28

Use a linear approximation (or differentials) to estimate the given number.
$ \cos 29^o $

Carson M.
Numerade Educator

Problem 29

Explain, in terms of linear approximations or differentials, why the approximation is reasonable.
$ \sec 0.08 \approx 1 $

Amrita B.
Numerade Educator

Problem 30

Explain, in terms of linear approximations or differentials, why the approximation is reasonable.
$ \sqrt {4.02} \approx 2.005 $

Amrita B.
Numerade Educator

Problem 31

Explain, in terms of linear approximations or differentials, why the approximation is reasonable.
$ \frac {1}{9.98} \approx 0.1002 $

Amrita B.
Numerade Educator

Problem 32

Let $ f(x) = (x - 1)^2 g(x) = e^{-2x} $
and $ h(x) = 1 + \ln(1 -2x) $
(a) Find the linearization of $ f, g $ and $ h $ at $ a = 0. $ What do you notice? How do you explain what happened?
(b) Graph $ f, g, $ and $ h $ and their linear approximations. For which function is the linear approximation best? For which is it worst? Explain.

Amrita B.
Numerade Educator

Problem 33

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.

Amrita B.
Numerade Educator

Problem 34

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? What is the percentage error?

Amrita B.
Numerade Educator

Problem 35

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?

Mutahar M.
Numerade Educator

Problem 36

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. $

Carson M.
Numerade Educator

Problem 37

(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)?

Linda H.
Numerade Educator

Problem 38

One side of a right triangle is known to be $ 20 cm $ long and the opposite angle is measured as $ 30^o, $ with a possible error of $ \pm 1^o. $
(a) Use differentials to estimate the error in computing the length of the hypotenuse.
(b) What is the percentage error?

Amrita B.
Numerade Educator

Problem 39

If a current $ I $ passes through a resistor with resistance $ R, $ Ohm's Law states that the voltage drop is $ V = RI. $ 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. $

Amrita B.
Numerade Educator

Problem 40

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 = kR^4 $
(This is known as Poiseuille's Law; we will show why it is true in Section 8.4.) A partially clogged artery can be expanded by an operation called angioplasty, in which a balloon-tipped catheter in inflated inside the artery in order to widen it and restore the normal blood flow.
Show that the relative change in $ F $ is about four times the relative changes in $ R. $ How will a $ 5\% $ increase in the radius affect the flow of blood?

Amrita B.
Numerade Educator

Problem 41

Establish the following rules for working with differentials (where $ c $ denotes as constant and $ u $ and $ v $ are functions of $ x $).
(a) $ dc = 0 $
(b) $ d(cu) = c du $
(c) $ d (u + v) = du + dv $
(d) $ d(uv) = u dv + v du $
(e) $ d ( \frac {u}{v}) = \frac {v du - u dv}{v^2} $
(f) $ d(x") = nx^{n-1} dx $

Clarissa N.
Numerade Educator

Problem 42

On page 431 of $ Physics: Calculus, $ 2d ed., by Eugene Hecht (Pacific Grove, CA: Brooks/ Cole, 2000), in the course of deriving the formula $ T = 2\pi \sqrt {L/g} $ for the period of a pendulum of length $ L, $ the author obtains the equation $ a_T = -g \sin \theta $ for the tangential acceleration of the bob of the pendulum. He then says, "for small angles, the value of $ \theta $ in radians is very nearly the value of $ \sin \theta; $ they differ by less than $ 2\% $ out to about $ 20^o." $
(a) Verify the linear approximation at 0 for the sine function:
$ \sin x \approx x $
(b) Use a graphing device to determine the value of $ x $ for which $ \sin x $ and $ x $ differ by converting from radians to degrees.

Amrita B.
Numerade Educator

Problem 43

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.

Amrita B.
Numerade Educator

Problem 44

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 estimate in part (a) too large or too small? Explain.

Amrita B.
Numerade Educator