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Answer:
Opt. D.
Step-by-step explanation:
To solve this problem we need to focus our attention in the abs(x-1) function,
If x>1, then abs|x-1|= x-1, if x<1 then abs|x-1|= 1-x. And in the two intervals, the function opens upwards.
But our function has -3 multiplying abs|x-1|, This means that our function opens downward.
The vertex point can be found when x=1, since abs(x-1)=0. this gives f(1)=12
Answer:
⁵/₃ ln 2
Step-by-step explanation:
∫₀¹ ∫√ₓ¹ 5 / (y³ + 1) dy dx
We want to change the order of integration. To do this, we start by graphing the region contained by the four limits.
x = 0, x = 1
y = √x, y = 1
(Notice that y = √x is the same as x = y²).
Once we've graphed the region, we need to write the domain of x in terms of y. In this case, 0 < x < y².
Then, we find the range: 0 < y < 1.
Now we can rewrite the integral:
∫₀¹ ∫₀ʸ² 5 / (y³ + 1) dx dy
Notice that the integrand itself, 5 / (y³ + 1), does not change. Only the limits have changed.
Solve the integral.
∫₀¹ [ 5 / (y³ + 1) x |₀ʸ² ] dy
∫₀¹ [ 5y² / (y³ + 1) ] dy
⁵/₃ ∫₀¹ [ 3y² / (y³ + 1) ] dy
(⁵/₃ ln|y³ + 1|) |₀¹
⁵/₃ ln|1³ + 1| − ⁵/₃ ln|0³ + 1|
⁵/₃ ln 2
Answers:
(a) Possible values of X are: 0, 1, 2, 3, 4, 5
(b) See attached image below; specifically figure 2
(c) No, it is <u>not</u> symmetric
(d) As a fraction, the answer is 1/6. In decimal form, the answer is approximately 0.1667 (converts to roughly 16.67%)
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Explanations:
(a) The largest possible roll we can get on one die is 6, the smallest is 1, so the largest possible difference is 6-1 = 5. The smallest difference is 0 as we could get doubles (eg: both dice show a "3" at the same time). We can get any difference in between 0 and 5. So X can take on whole numbers from 0 to 5, ie, X can be any number from the set {0, 1, 2, 3, 4, 5}
(b) Consult the attached image. Figure 1 shows a 6 by 6 grid of values that represent all the possible ways to roll two dice. Each value in the interior of the grid is the result of subtracting the outer header values. Figure 2 is where the answer is shown for part b. The idea is that you list out the X values along the top of the table and the probability P(X) values along the bottom row. I show the work on how to compute each probability in the attachment. The probability is equal to the number of occurences over the number total. Example: X = 3 shows up 6 times out of 36, so P(X) is 6/36 = 1/6 when X = 3.
(c) The distribution is not symmetric as figure 3 shows a lop-sided diagram. The mode should be at the center, but this figure shows the mode off to the left side.
(d) Count how many copies of '4' or '5' show up in the interior of the grid shown in figure 1. There are four copies of '4' and two copies of '5'. Overall there are 4+2 = 6 instances of either '4' or '5'. This is out of 36 total, so 6/36 = 1/6 is the fractional form of the answer which approximates to 0.1667 and converts to 16.67% approximately.
