Problem

Use the same-derivative argument, as was done to …

03:24

Problem 89 Hard Difficulty

The linearization of $\ln (1+x)$ at $x=0$ Instead of approximating ln $x$ near $x=1,$ we approximate $\ln (1+x)$ near $x=0$ .
We get a simpler formula this way.
a. Derive the linearization $\ln (1+x) \approx x$ at $x=0$
b. Estimate to five decimal places the error involved in replacing
$\ln (1+x)$ by $x$ on the interval $[0,0.1] .$
c. Graph $\ln (1+x)$ and $x$ together for $0 \leq x \leq 0.5 .$ Use different colors, if available. At what points does the approximation
of $\ln (1+x)$ seem best? Least good? By reading coordinates
from the graphs, find as good an upper bound for the error as
your grapher will allow.

Answer

a. Use the linearization definition from page 163 .
b. 0.00469
c. Best on $[0,0.1],$ worst near $0.5 .$ Error is 0.0945348919 (to 10 decimal
digits), upper bound depends on calculator precision.

Related Courses

Calculus 1 / AB

Thomas Calculus

Chapter 7

Transcendental Functions

Section 2

Natural Logarithms

Discussion

You must be signed in to discuss.

Top Calculus 1 / AB Educators

Watch More Solved Questions in Chapter 7

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Problem 67

Problem 68

Problem 69

Problem 70

Problem 71

Problem 72

Problem 73

Problem 74

Problem 75

Problem 76

Problem 77

Problem 78

Problem 79

Problem 80

Problem 81

Problem 82

Problem 83

Problem 84

Problem 85

Problem 86

Problem 87

Problem 88

Problem 89

Problem 90

Problem 91

Problem 92

Video Transcript

this problem? We are Lena Rising Function Natural law one plus extra for zero. Now let's in port They performed this immunization being ordered the musician of a function musical function of lit that porn misstated zero plus debuting a lit that same point times distance to that point. Now our function F FX is nation log off plus X So then f 00 as natural lawyer horn which is zero using effects we can find there is a form of access one over one Plus that's anything like that function at a given 10.0 We find it ever to be born now plugging dose and you don't find of the immunization for zero to the zero plus one times X minus zero, which is thanks or in port beat. Now we Well, they were told to take the spacing as points one. So we are trying to find how much difference this spacing going from one point to another, uh, would make Bonaparte baby No, not decorative office function. I will let that given point is born. So, um, this means that going or moving this much would result in a change of a function. 04 point minus F off zero. This is our initial point, or in part, won't be calculated. This to be dear, right? Yeah. So we need to park red F port one, which is natural law off one plus 41. And that is nature lover Horn for one. So using this, we can do for you to be and your law off 1.4 minus warm. This we end up with this because there it is. One, uh, using this week for a change. To be born hero, you're or 69 Important. See, we're on street for the function, error and the approximations off realization. Uh, this is our accesses. Why? Accesses are ex senses. We are reporting this between zero in point by zero. Uh, this is our leader explosions bags or Joe puncher. Well, looks like there's careful backs is natural. And this is how the air order difference between those defections became. As you can see, a CZ, we go from zero point fire air increases so four smaller value. The exit. The immunization gives colter results to the actual function. Then for live your values race and or does the x o point fine we can just like in party. We could find error to be 0.0 during, uh three. Fine. And actually, we could use this as our upper Oh.

We have video lessons for 94.94% of the questions in this textbook