Given the function $f(x)=(1+x)^n$ Show that $L(x)=1$+nx is the linearization of $f$ at $0$ … The 2019 Stack Overflow Developer Survey Results Are InPower series of a function about a non zero pointhow to determine delta epsilon figures for an equation involving the constant eProblem with differentials and relative rates of changeabsolute extrema problem…Show that the limit as x approaches zero for $frac2^1/x - 2^-1/x2^1/x + 2^-1/x$ does not existDerivatives and Average VelocityGraphing function input output relationships on the coordinate planeIntegrating $intfrac1cos(theta)dtheta$ by following certain stepsEvaluating the integral as a power series Tips and tricksverifying given linear approx. @ a=0 , then determine x values for which linear approx. is accurate to within 0.1
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Given the function $f(x)=(1+x)^n$ Show that $L(x)=1$+nx is the linearization of $f$ at $0$ …
The 2019 Stack Overflow Developer Survey Results Are InPower series of a function about a non zero pointhow to determine delta epsilon figures for an equation involving the constant eProblem with differentials and relative rates of changeabsolute extrema problem…Show that the limit as x approaches zero for $frac2^1/x - 2^-1/x2^1/x + 2^-1/x$ does not existDerivatives and Average VelocityGraphing function input output relationships on the coordinate planeIntegrating $intfrac1cos(theta)dtheta$ by following certain stepsEvaluating the integral as a power series Tips and tricksverifying given linear approx. @ a=0 , then determine x values for which linear approx. is accurate to within 0.1
$begingroup$
So this is how the question goes.
1. Given the function $f(x)=(1+x)^n$.$$$$
a. Show that $L(x)=1$+nx is the linearization of $f$ at $0$.$$$$
b. A friend claims that the cube root of 1.1 is about 1.03. Without using a calculator, do you think the $$$$
c. Without using a calculator do you think $(1.1)^frac13$ is larger or smaller than 1.03? Explain. (Hint graphing may help)
So for a I did$$L(x)=f'(a)(x-a)+f(a)$$ $$a= 0$$
$$f(a)=(1+0)^n=1$$
$$f'(x) = n(1+x)^n-1$$
$$f'(a)=n(1+0)^n-1=n$$
so $$L(x)=n(x-0)+1=1+nx$$
So for b I used $(1+.1)^frac13$ and plugged into $L(x)$ and got $1.033...$. But I have no Idea on how to go about solving c. Do I just use my answer from b and say that's why? I think there's an specific process towards solving that precisely but I'm not sure how. Thanks in advance!
calculus
edited Jul 8 '14 at 3:46
Jeel Shah
5,296115599
asked Jul 8 '14 at 3:40
KenshinKenshin
629714
$endgroup$
add a comment |
$begingroup$
So this is how the question goes.
1. Given the function $f(x)=(1+x)^n$.$$$$
a. Show that $L(x)=1$+nx is the linearization of $f$ at $0$.$$$$
b. A friend claims that the cube root of 1.1 is about 1.03. Without using a calculator, do you think the $$$$
c. Without using a calculator do you think $(1.1)^frac13$ is larger or smaller than 1.03? Explain. (Hint graphing may help)
So for a I did$$L(x)=f'(a)(x-a)+f(a)$$ $$a= 0$$
$$f(a)=(1+0)^n=1$$
$$f'(x) = n(1+x)^n-1$$
$$f'(a)=n(1+0)^n-1=n$$
so $$L(x)=n(x-0)+1=1+nx$$
So for b I used $(1+.1)^frac13$ and plugged into $L(x)$ and got $1.033...$. But I have no Idea on how to go about solving c. Do I just use my answer from b and say that's why? I think there's an specific process towards solving that precisely but I'm not sure how. Thanks in advance!
calculus
edited Jul 8 '14 at 3:46
Jeel Shah
5,296115599
asked Jul 8 '14 at 3:40
KenshinKenshin
629714
$endgroup$
$begingroup$
Hint: solve an equation. and find out for what values of $x$; $(1+.1)^frac1x$ is larger , smaller or equal to $1.03$. that will tell you about $(1+.1)^frac13$
$endgroup$
– toufik_kh.17
Jul 8 '14 at 4:13
$begingroup$
An other idea would be to look at the sign of the remaining terms that you remove when you do the linerarization. You can explicitely do the math for n=2, n=3 and recursively prove that for every n, the results holds.
$endgroup$
– Martigan
Jul 8 '14 at 14:50
$begingroup$
Hiko Seijuro didn't teach you this?
$endgroup$
– IAmNoOne
Jul 14 '14 at 4:53
add a comment |
$begingroup$
So this is how the question goes.
1. Given the function $f(x)=(1+x)^n$.$$$$
a. Show that $L(x)=1$+nx is the linearization of $f$ at $0$.$$$$
b. A friend claims that the cube root of 1.1 is about 1.03. Without using a calculator, do you think the $$$$
c. Without using a calculator do you think $(1.1)^frac13$ is larger or smaller than 1.03? Explain. (Hint graphing may help)
So for a I did$$L(x)=f'(a)(x-a)+f(a)$$ $$a= 0$$
$$f(a)=(1+0)^n=1$$
$$f'(x) = n(1+x)^n-1$$
$$f'(a)=n(1+0)^n-1=n$$
so $$L(x)=n(x-0)+1=1+nx$$
So for b I used $(1+.1)^frac13$ and plugged into $L(x)$ and got $1.033...$. But I have no Idea on how to go about solving c. Do I just use my answer from b and say that's why? I think there's an specific process towards solving that precisely but I'm not sure how. Thanks in advance!
calculus
edited Jul 8 '14 at 3:46
Jeel Shah
5,296115599
asked Jul 8 '14 at 3:40
KenshinKenshin
629714
$endgroup$
So this is how the question goes.
1. Given the function $f(x)=(1+x)^n$.$$$$
a. Show that $L(x)=1$+nx is the linearization of $f$ at $0$.$$$$
b. A friend claims that the cube root of 1.1 is about 1.03. Without using a calculator, do you think the $$$$
c. Without using a calculator do you think $(1.1)^frac13$ is larger or smaller than 1.03? Explain. (Hint graphing may help)
So for a I did$$L(x)=f'(a)(x-a)+f(a)$$ $$a= 0$$
$$f(a)=(1+0)^n=1$$
$$f'(x) = n(1+x)^n-1$$
$$f'(a)=n(1+0)^n-1=n$$
so $$L(x)=n(x-0)+1=1+nx$$
So for b I used $(1+.1)^frac13$ and plugged into $L(x)$ and got $1.033...$. But I have no Idea on how to go about solving c. Do I just use my answer from b and say that's why? I think there's an specific process towards solving that precisely but I'm not sure how. Thanks in advance!
calculus
calculus
edited Jul 8 '14 at 3:46
Jeel Shah
5,296115599
asked Jul 8 '14 at 3:40
KenshinKenshin
629714
edited Jul 8 '14 at 3:46
Jeel Shah
5,296115599
asked Jul 8 '14 at 3:40
KenshinKenshin
629714
edited Jul 8 '14 at 3:46
Jeel Shah
5,296115599
edited Jul 8 '14 at 3:46
Jeel Shah
5,296115599
edited Jul 8 '14 at 3:46
Jeel Shah
5,296115599
5,296115599
asked Jul 8 '14 at 3:40
KenshinKenshin
629714
asked Jul 8 '14 at 3:40
KenshinKenshin
629714
asked Jul 8 '14 at 3:40
KenshinKenshin
629714
629714
$begingroup$
Hint: solve an equation. and find out for what values of $x$; $(1+.1)^frac1x$ is larger , smaller or equal to $1.03$. that will tell you about $(1+.1)^frac13$
$endgroup$
– toufik_kh.17
Jul 8 '14 at 4:13
$begingroup$
An other idea would be to look at the sign of the remaining terms that you remove when you do the linerarization. You can explicitely do the math for n=2, n=3 and recursively prove that for every n, the results holds.
$endgroup$
– Martigan
Jul 8 '14 at 14:50
$begingroup$
Hiko Seijuro didn't teach you this?
$endgroup$
– IAmNoOne
Jul 14 '14 at 4:53
add a comment |
$begingroup$
Hint: solve an equation. and find out for what values of $x$; $(1+.1)^frac1x$ is larger , smaller or equal to $1.03$. that will tell you about $(1+.1)^frac13$
$endgroup$
– toufik_kh.17
Jul 8 '14 at 4:13
$begingroup$
An other idea would be to look at the sign of the remaining terms that you remove when you do the linerarization. You can explicitely do the math for n=2, n=3 and recursively prove that for every n, the results holds.
$endgroup$
– Martigan
Jul 8 '14 at 14:50
$begingroup$
Hiko Seijuro didn't teach you this?
$endgroup$
– IAmNoOne
Jul 14 '14 at 4:53
$begingroup$
Hint: solve an equation. and find out for what values of $x$; $(1+.1)^frac1x$ is larger , smaller or equal to $1.03$. that will tell you about $(1+.1)^frac13$
$endgroup$
– toufik_kh.17
Jul 8 '14 at 4:13
$begingroup$
Hint: solve an equation. and find out for what values of $x$; $(1+.1)^frac1x$ is larger , smaller or equal to $1.03$. that will tell you about $(1+.1)^frac13$
$endgroup$
– toufik_kh.17
Jul 8 '14 at 4:13
$begingroup$
An other idea would be to look at the sign of the remaining terms that you remove when you do the linerarization. You can explicitely do the math for n=2, n=3 and recursively prove that for every n, the results holds.
$endgroup$
– Martigan
Jul 8 '14 at 14:50
$begingroup$
An other idea would be to look at the sign of the remaining terms that you remove when you do the linerarization. You can explicitely do the math for n=2, n=3 and recursively prove that for every n, the results holds.
$endgroup$
– Martigan
Jul 8 '14 at 14:50
$begingroup$
Hiko Seijuro didn't teach you this?
$endgroup$
– IAmNoOne
Jul 14 '14 at 4:53
$begingroup$
Hiko Seijuro didn't teach you this?
$endgroup$
– IAmNoOne
Jul 14 '14 at 4:53
add a comment |
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$begingroup$
Hint: solve an equation. and find out for what values of $x$; $(1+.1)^frac1x$ is larger , smaller or equal to $1.03$. that will tell you about $(1+.1)^frac13$
$endgroup$
– toufik_kh.17
Jul 8 '14 at 4:13
$begingroup$
An other idea would be to look at the sign of the remaining terms that you remove when you do the linerarization. You can explicitely do the math for n=2, n=3 and recursively prove that for every n, the results holds.
$endgroup$
– Martigan
Jul 8 '14 at 14:50
$begingroup$
Hiko Seijuro didn't teach you this?
$endgroup$
– IAmNoOne
Jul 14 '14 at 4:53