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Sergeu [11.5K]
1 year ago
7

Find the maclaurin series for f(x) using the definition of a maclaurin series. [assume that f has a power series expansion. do n

ot show that rn(x) → 0. ] f(x) = e−6x
Mathematics
1 answer:
Nady [450]1 year ago
4 0

The equation of f(x) = e^{-6x} by maclaurin series is f(x)=\sum_{i=0}^{\infty} \frac{(-6.x)^{i}  }{i!}.

The maclaurin series for f(x) is defined by the following formula:

f(x) = \sum_{i=0}^{\infty} \frac{f^{(i)} (0)}{i!} .x^{i}--------------(1)

Where f^{i} is the i - th derivative of the function

If f(x) = e^{-6x}, then the formula of the i - th derivative of the function is:

f^{i} =(-6)^{i} .e^{-6x}----------------------(2)

Then,

f^{i}(0) = (-6)^{i}

Lastly, the equation of the trascendental function by Maclaurin series is: f(x)=\sum_{i=0}^{\infty} \frac{(-6)^{i}.x^{i}  }{i!} \\f(x)=\sum_{i=0}^{\infty} \frac{(-6.x)^{i}  }{i!}---------------(3)

Hence,

The equation of f(x) = e^{-6x} by maclaurin series is f(x)=\sum_{i=0}^{\infty} \frac{(-6.x)^{i}  }{i!}.

Find out more information about maclaurin series here

brainly.com/question/24179531

#SPJ4

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Help with these precalculus problems, please.
likoan [24]
Yay, easy peasy, nice refresher


53. evaluate for x=infinity
we get 100/(infiniy^2-5) which is basically 100/infinity which is a very small number so about 0



54.
vertical assemtotes, neato
that is where the denomenator equals 0
but wait, simplify first
ok, fraction is simplified (anything factored out would be a hole)
so set the denom to 0 and solve
x^2-5=0
(x-√5)(x+√5)=0
x=-√5 and √5
vertical assemtotes at x=-√5 and x=√5




55. horizontal assemtotes
for the function p(x)/f(x)

if p(x) has lower degree than f(x) then y=0 is the horizontal assemtote

if p(x) has equal degree to f(x) then divide the leading coefients of p(x) and f(x)

if p(x) has greater degree than f(x) then we got slant assemtotes

100/(x^2-5)
100 has lower degree than x^2
y=0 is the horizontal assemtote



56. evalate for x=-5
we get -150/-10=150/10=15
the limit is 15



57. that negative on the right side means that we must approach it from the left
let's see if we can factor out the denomenator
nope
so go and evaluate values like 4, 4.5, 4.9, etc, approaching x=5 from the left
we keep getting smaller and smaller valuse
this make sense because as x approaches 5, we get 30(c)/(c-5) where c is close to 5, and as c appraoches 5 very close, the top part is appraoching 150 but the bottom approaches a very small negative number, and if we do something like 150/-0.00000000001, we get a very negative number
the limit is negative infinity

alternately, you could figure that there is a vertical assemtote at x=5 so we can see that if we aproximaate closer and closer values, we get smaller values. so from what we know about assemtotes, expecially vertical ones, we say that it approaches negative infinity




58.
evalauating we get infintity/inifinty, not helpful
hmmm, is there a horizontal assemtote to help us?
the degree of top is higher so we got a slant assemtote
the only way is to use long division to divide the top by the bottom (see attachment)
we get
\frac{1}{2}x-\frac{1}{4}+\frac{\frac{1}{4}}{2x+1}
the slant assemtote is y=\frac{1}{2}x-\frac{1}{4}
as you can see, as x approaches infinity, y also will approach infinity











ANSWERS:
53. 0
54. VA is x=-√5 and x=√5
55. HA is y=0
56. the limit is 15
57. the limit is -∞
58. the limit is ∞

6 0
3 years ago
A=8, b=2, and c= 1/2<br><br> ab= ?<br><br> ca=?<br><br> 4a=?
jarptica [38.1K]

Answer:

ab=16

ca=4

4a=32

Step-by-step explanation:

a*b=8*2=16

c*a=1/2*8=4

4*a=4*8=32

hope it helped xx

7 0
3 years ago
Read 2 more answers
chad randomly chooses a card from a standard deck 30 times. what is the probability he gets a diamond exactly 4 times
Margarita [4]

Answer:

From a standard deck of cards, one card is drawn. What is the probability that the card is black and a jack? P(Black and Jack) P(Black) = 26/52 or ½ , P(Jack) is 4/52 or 1/13 so P(Black and Jack) = ½ * 1/13 = 1/26 A standard deck of cards is shuffled and one card is drawn. Find the probability that the card is a queen or an ace.

Step-by-step explanation:

4 0
2 years ago
A random sample of 25 glass sheets is obtained and their thicknesses are measured. The sample mean is x= 3.54 mm and sample stan
lisabon 2012 [21]

Answer: (3.46, 3.62)

Step-by-step explanation:

The formula to find the confidence interval for population mean is given :-

\overline{x}\ \pm\ t_{\alpha/2}\dfrac{s}{\sqrt{n}}

, where n = sample size.

t_{\alpha/2} = Two-tailed t-value for significance level of (\alpha) and degree of freedom df= n-1.

s = sample standard deviation.

As per given , we have

\overline{x}= 3.54 mm

s= 0.20 mm

n= 25

Significance level =\alpha=1-0.95=0.05

Since population standard deviation is not given , it means the given problem has t- distribution.

Two-tailed t-value for significance level of (0.05) and degree of freedom df= 24:

t_{\alpha/2\ ,df}=t_{0.025,\ 24}=2.0639

95% Confidence interval for population mean:

3.54\ \pm\ (2.0639)\dfrac{0.20}{\sqrt{25}}

=3.54\ \pm\ (2.0639)\dfrac{0.20}{5}

=3.54\ \pm\ (2.0639)(0.04)

=3.54\ \pm\ 0.082556

=(3.54- 0.082556,\ 3.54+ 0.082556 )=(3.457444,\ 3.622556)\approx(3.46,\ 3.62)

Hence, the 95% two-sided confidence interval for the mean glass thickness = (3.46, 3.62)

7 0
3 years ago
a jeweler buys gold jewelry and resells the gold to a refinery the jeweler buys gold for 1235.55 per ounce and then resells it f
weqwewe [10]
The jeweler bought it for (1235.55 * 73.5) = 90812.925...expenses
and the jeweler sold it for (1376.44 * 73.5) = 101168.34...revenue

profit = revenue - expenses
profit = 101168.34 - 90812.925 = 10355.415 rounds to 10,355.42 <==
4 0
3 years ago
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