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Sati [7]
2 years ago
8

Find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d^2y/dx

^2 at this point.
x=sin^2t-1, y=csc(t); t=-pi/6
Mathematics
1 answer:
nikdorinn [45]2 years ago
3 0

Answer:

Differentiation will give you the gradient for the tangent at any point, and you use the product rule whenever a function can be thought of as two functions multiplied together.

If

f

(

x

)

=

g

(

x

)

×

h

(

x

)

then

f

'

(

x

)

=

g

'

(

x

)

h

(

x

)

+

g

(

x

)

h

'

(

x

)

so if

y

=

x

×

sin

x

then

d

y

d

x

=

1

×

sin

x

+

x

×

cos

x

=

sin

x

+

x

cos

x

We know that

x

=

π

2

, so the gradient is

m

=

sin

(

π

2

)

+

π

2

cos

(

π

2

)

=

1

+

π

2

×

0

=

1

Therefore, we can say that

y

=

m

x

+

c

y

=

(

1

)

x

+

c

y

=

x

+

c

So all we really need to find now is the value for

c

, the

y

intercept. We do this by working out a point

(

x

,

y

)

on the graph. We are already given that

x

=

π

2

, so

y

=

x

sin

x

=

π

2

sin

(

π

2

)

=

π

2

×

1

=

π

2

∴

(

x

,

y

)

=

(

π

2

,

π

2

)

Now we substitute this into the equation we already have for the tangent,

y

=

x

+

c

,

(

x

,

y

)

=

(

π

2

,

π

2

)

π

2

=

π

2

+

c

c

=

π

2

−

π

2

=

0

∴

y

=

x

+

c

=

x

+

(

0

)

=

x

which means the tangent to the curve

y

=

x

sin

x

at

(

π

2

,

π

2

)

is simply

y

=

x

.

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AlekseyPX

Answer:

a) 0.052 = 5.2% probability that exactly three arrivals occur during a particular hour

b) 0.971 = 97.1% probability that at least three people arrive during a particular hour

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In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

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This is P(X = 3).

P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}

P(X = 3) = \frac{e^{-7}*7^{3}}{(3)!} = 0.052

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(b) What is the probability that at least three people arrive during a particular hour? (Round your answer to three decimal places.)

Either less than three people arrive, or at least three does. The sum of the probabilities of these events is 1. So

P(X < 3) + P(X \geq 3) = 1

We want P(X \geq 3), which is

P(X \geq 3) = 1 - P(X < 3)

In which

P(X \geq 3) = P(X = 0) + P(X = 1) + P(X = 2)

P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}

P(X = 0) = \frac{e^{-7}*7^{0}}{(0)!} = 0.001

P(X = 1) = \frac{e^{-7}*7^{1}}{(1)!} = 0.006

P(X = 2) = \frac{e^{-7}*7^{2}}{(2)!} = 0.022

So

P(X \geq 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.001 + 0.006 + 0.022 + 0.029

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During an hour(60 minutes), 7 people are expected to arrive. So, using proportions:

\frac{45*7}{60} = \frac{3*7}{4} = \frac{21}{4} = 5.25

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=============================================================

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