It's clear that because of symmetry, the derivatives at -1/2, -1 and -2 will be the negative of those at 1/2, 1 and 2 for the function f(x) = x^2 since that is the graph of a parabola. Therefore I get f'(-1/2) = -1, f'(-1) = -2 and f'(-2) = -4


Applied the definition to get the derivative of x^2. Put in (x + h) for x in f(x) to get (x+ h)^ then subtract off f(x) which is x^2 and divide the whole thing by h. Remember to expand (x+h)^2, which is (x+h)(x+h), then simplify and put in 0 for h to get 2x, which matches the guess in part c

For the first part, suppose we want to know approximately how much data falls within 2 standard deviations from the mean using Chebyshev's theorem. This is how we do it. We square the standard deviations first to get 4. Then we take 1/4. Now simple subtract 1/4 from 1 to get 3/4. Therefore at least 75% of the data will fall within 2 st dev from the mean. In this problem we don't know the standard deviations, but we know it has to be at least 70%. So we basically are working in reverse now.
That means that 1 - .3 = .7........... therefore .3 = 1/(standard deviations)^2
.3(standard deviations)^2 = 1
divide by .3 to get (standard deviations)^2 = 3.33, therefore standard deviations = 1.83
We know that at least 70% of the data falls within 1.83 standard deviations from the mean.
If you don't understand, please let me know!
For the second part, we simply get the mean and standard deviation of the data set. I did it on my calculator and get mean = 5 and st dev = 1.777
Therefore to see the range of values that are within 1.83 st deviations, take 5 +/- 1.83(1.777)
That gives us 1.75 and 8.25. Look at the data and you see the lowest value is 2.22 and the highest is 8.11, so ALL of the data, 100% fall within the range.

The parabola in the form y = (x - h)^2 + k has a vertex of (h,k). The problem we have is y-3 = (x-1)^2, so we have to get the -3 from the left side and move to the right, so add 3 to both sides of the equation. That gives us y = (x-1)^2 + 3, so the vertex is (1,3)

Since the x^2 term is positive, the parabola opens up away from the x-axis, so there are no x-intercepts. If you are unsure of this you can put 0 in for y and solve for x, like i demonstrated and by the quadratic formula you will see there are no real number solutions for x, so no x-intercepts.

For the y-intercept, put 0 in for x and you'll see that y = 4.
The axis of symmetry is simply the line the goes through the vertex, cutting the parabola in half. The domain is all the possible x values, and since there are no restrictions on x, it's all real numbers. The range is all the possible y-values that the function takes on. As you can see on the graph, the lowest y value is at (1,3) so the range is from 3 to infinity.

 Solve: Cos(Sec^-1 u)

Recall that sec is 1/cos
Suppose to make this a little easier to understand that the problem says sec^-1(2), which means u = 2 . So we want the angle which has a sec value equal to 2. That is the same as saying 1/cos = 2 which means cos = 1/2

Cos is 1/2in the first quadrant 60 degrees

That gives us cos(60) which we know is 1/2 and 1/2 = 1/u.

Therefore the answer is simply 1/u. That makes logical sense too since sec = 1/cos and cos = 1/sec. They are inverses.

You can also do this by labeling parts of the right triangle. You know that sec^-1 u means that the adjacent side of the right triangle is 1 and the hypotenuse is u, since sec = hypotenuse/adjacent.

Therefore cos of the angle equal 1/u

Yes the CLT is a bit of a confusion here because as we know, The central limit theorem states that the sampling distribution of any statistic will be normal or nearly normal, if the sample size is large enough. That's where the problem comes in, sample size should be at least 30, some say 40. That would lead you to believe that you cannot use apply CLT here, BUT. the more closely the original population resembles a normal distribution, the fewer sample points will be required.

But what we can do is use the normal approximation to the binomial if this condition holds true. If np > 5 and n(1-p)> 5, then it can be used. We know p = .5 and if you consider 4 trials, then np < 5, as is n(1-p). BUT if you use all 12 tosses, then n = 12 and np = 6 and n(1-p) = 6. But doing it this was we would get the average number of heads expected in 12 tosses to be 6 and the average in all 12 tosses to be 8, instead of 2 in 3 tosses.

npq = (12)(.5)(.5) = 3
sqrt(npq) = 1.732
Formula is z = (x-bar - np)/(sqrt(npq)
z= (8 - 6)/(1.732) = 1.15
z(1.15) = .8749
1- .8749 = .1251
If you notice when you go through a coin tossing experiment tossing three times, there are 8 outcomes and the only way to get greater the 2 heads is if you get HHH, which is 1 out of 8, which is 12.5%.

since population standard deviation is known, we can use Z

Test stat Z = (x-bar - mean)/(standard deviation/sqrt(n))
Z = (16.2 - 15)/(5.6/sqrt(49))

If you calculate this correctly , you should get Z = 1.50

The critical values are Z = 2.43 and Z = -2.43 since this is a two-tailed test. The critical value was found looking up .9925 on the chart. Took .015/2 = .0075 and then 1-. 0075 to get .9925. It was alpha/2 because of the two-tailed test.

Since 1.5 < 2.43, accept Ho

 P-value was .133614. Using the chart it's .1336. In any event, since that value is greater than alpha for the test, accept Ho

The power of the test is probability of Type II error. IN this case it is probability of type II error when x-bar = 23.6. This is the probability of Ho not being rejected when it is false because x-bar = 23.6
B(23.6) = 1 - z(23.6 - 25.1)(14/sqrt(100))
1- z(-1.07)
Finr z(-1.07) on standard normal distribution chart to get .1423
1- .1423 = .8577
b) we know the power of the test is .9
so 1 - z(23.6 - 25.1)/(14/sqrt(n)) = .9
therefore z(-1.5/914/sqrt(n)) = .1
looking at the z-chart we know z(-1.28) = .1
so -1.28 = -1.5(14/sqrt(n))
-17.92/sqrt(n) = -1.5
cross multiply and square both sides and solve for n and you get n = 143.