Which choice is the solution to the system? x + 2y = 1 y = − 1 2 x + 4
1 answer:
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Answer:
a) d(sinh(f(x)))/dx = cosh(f(x))·df(x)/dx
b) d(cosh(f(x))/dx = sinh(f(x))·df(x)/dx
c) d(tanh(f(x))/dx = sech(f(x))²·df(x)/dx
d) d(sech(4x+2))/dx = -4sech(4x+2)tanh(4x+2)
Step-by-step explanation:
To do these, you need to be familiar with the derivatives of hyperbolic functions and with the chain rule.
The chain rule tells you that ...
(f(g(x)))' = f'(g(x))g'(x) . . . . where the prime indicates the derivative
The attached table tells you the derivatives of the hyperbolic trig functions, so you can answer the first three easily.
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a) sinh(u)' = sinh'(u)·u' = cosh(u)·u'
For u = f(x), this becomes ...
sinh(f(x))' = cosh(f(x))·f'(x)
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b) After the same pattern as in (a), ...
cosh(f(x))' = sinh(f(x))·f'(x)
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c) Similarly, ...
tanh(f(x))' = sech(f(x))²·f'(x)
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d) For this one, we need the derivative of sech(x) = 1/cosh(x). The power rule applies, so we have ...
sech(x)' = (cosh(x)^-1)' = -1/cosh(x)²·cosh'(x) = -sinh(x)/cosh(x)²
sech(x)' = -sech(x)·tanh(x) . . . . . basic formula
Now, we will use this as above.
sech(4x+2)' = -sech(4x+2)·tanh(4x+2)·(4x+2)'
sech(4x+2)' = -4·sech(4x+2)·tanh(4x+2)
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Here we have used the "prime" notation rather than d( )/dx to indicate the derivative with respect to x. You need to use the notation expected by your grader.
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<em>Additional comment on notation</em>
Some places we have used fun(x)' and others we have used fun'(x). These are essentially interchangeable when the argument is x. When the argument is some function of x, we mean fun(u)' to be the derivative of the function after it has been evaluated with u as an argument. We mean fun'(u) to be the derivative of the function, which is then evaluated with u as an argument. This distinction makes it possible to write the chain rule as ...
f(u)' = f'(u)u'
without getting involved in infinite recursion.
Answer:
Step-by-step explanation:
From the given information:
A sandwich shop offers eight types of sandwiches, and Jamey likes all of them equally.
The probability that Jamey picks any one of them is 1/8
Suppose
X represents the number of times he chooses salami
Y represents the number of times he chooses falafel
Z represents the number of times he chooses veggie
Then X+Y+Z ≤ 5 and;
5-X-Y-Z represents the no. of time he chooses the remaining
8 - 3 = 5 sandwiches
However, the objective is to determine the P[X=x,Y=y,Z=z] such that 0≤x,y,z≤5
So, since he chooses x no. of salami sandwiches with probability (1/8)x
and y number of falafel with probability (1/8)y
and for z (1/8)z
Therefore, the remaining sandwiches are chosen with probability
So. these x days, y days and z days can be arranged within five days in
Thus;
since 0 ≤ x, y, z ≤ 5 and x + y + z ≤ 5.
The distribution is said to be Multinomial distribution.