When integrating by parts with one factor as a variable and the other with a trig function, generally the variable will be set to u and the trig function set to dv

integral udv = uv - integral vdu

let's take integral x^2*sinx

u = x^2

du = 2x dx

dv = sinx

v = -cosx

this gives us -x^2(cosx) - integral -2xsinx

now we have to integral by parts again

u = -2x

du = -2 dx

dv = sinx

v = -cosx

2xcosx - integral 2cosx = 2xcosx -(1/2)sinx

answer is -x^2(cosx) - 2xcosx + (1/2)sinx + C

## Tuesday, December 29, 2015

## Tuesday, December 22, 2015

## Monday, December 14, 2015

## Saturday, December 12, 2015

Suppose x,y are as follows

x 1 1 1 2 2 2 3 3 3 1

y 8 9 10 8 9 10 8 9 10 10

This has a relatively strong negative correlation, but if you remove the data point (1,10), all you have left is a block of data with correlation coefficient of 0.

## Wednesday, December 9, 2015

This is what is all means

sum(x) = all x's added together

sum(y) = all y's added together

sum(x^2) = square each x, then sum the total

sum(y^2) = square each y, then sum the total

sum(xy) = muitiply each xy value , then sum the total

(sumx)^2 = sum of the x's then square

(sumy)^2 = sum of the y's, then square

r^2 is just taking r and squaring it

y-bar is the average of the y's, which is sum(y)/n

x-bar is the average of the x's, which is sum(x)/n

## Friday, December 4, 2015

1. state the Ho and Ha

2. find test statistic z or t, for proportions we can use z if np > 5 and n(1-p) or nq > 5

3. find critical value for z or t using the appropriate chart

4. make the decision on the test, if using p-value, if p-value > significance level then do not reject. If p-value < significance level of the test, reject.

5. state the conclusion

Type I error is when you reject Ho when you should not reject Ho

Type II error is when you do not reject Ho when you should reject Ho

The power of the test is the probability of not committing a type two error, which is basically the probability of making the correct decision of rejecting Ho.

## Friday, November 27, 2015

left tailed test (if test statistic < critical value, then reject Ho), if not, then do not reject Ho

right tailed test (if test statistic >= critical value, then reject Ho), if not, then do not reject Ho

two-tailed test, (if test statistic falls in between the critical values, do not reject Ho, if not then reject Ho

for p-value if p-value is > = alpha level of the test, then reject Ho, if not then do not reject Ho

## Saturday, November 21, 2015

a.

x -bar =____

Sx=

n=___

n-1 = ____

b. Define the random variables X and X (with a line over top of it)

c. Which distribution should use you for this problem?

d. Construct a 95% confidence interval for the population mean time wasted. State the confidence interval

x-bar is the sample mean which is 8

Sx is the standard deviation of x which is 4

n = sample size of 81

n-1 is the degrees of freedom which is 80

part b, x is the time an individual waited to be called for jury duty and x-bar is the sample mean, so that is the mean waiting time

c) this is t-distribution since population standard deviation is not known

part d, 95% CI, for 80 df, t value is 1.99

8 +/- 1.99(4/sqrt(81))

8 +/- 0.88 = (7.12, 8.88)

the error bound is also known as the margin of error which is the value added and subtract from the mean in the interval which is 0.8

## Monday, November 16, 2015

## Wednesday, November 11, 2015

Ho: Mu = 10

Ha: Mu > 10

now get the test statistic t, since sample size is small and population standard deviation is not known.

t = (x-bar - Mu)/(standard deviation/square root(n))

t = (9.5 - 10)/(2.5/square root(16))

t = -0.8

We get the critical value for the test,

look up t at n-1 df for one tailed area of .05

t, 15df, .05 = 1.753

Since -0.8 < 1.753, we do not reject Ho. There is not enough evidence to support the claim that mean is greater than 10

For part b, the CI is x-bar+/- t(standard deviation/square root(n))

t for 95% interval, 15 df is 2.131

CI = 9.5 +/- 2.131(2.5/sqrt(16)) = 9.5 +/- 1.332

(8.168, 10.832)

## Saturday, October 31, 2015

For the first part we want to prove the claim that mean life of time of battery exceeds 400 hours, so Ho would be that the mean is 400 and Ha is that the mean is greater than 400

step 1: Ho: Mu = 400

Ha: Mu > 400

for the second step, recall that since sample size is small and standard deviation is not known, have to use the t-distribution, so test statistic is t

t = (sample mean - population mean)/(sample standard deviation/square root (n))

x-bar = 473.46

s = 210.77 as done on the calculator

n = 13

t = (473.46 - 400)/(210.77/square root(13))

t = 1.26

For the next step,we know using the p-value approach that we reject if p-value is less than alpha level of the test. So we reject Ho if p-value < .025

For the next step, I obtained the p-value from this site..

http://www.socscistatistics.com/pvalues/tdistribution.aspx

Putting in the values for t-statistic and one-tailed test, p-value is .11581

Last step, since .115811 > .025, we do not reject Ho since there is not significant evidence to conclude mean battery life is more than 400 hours.

## Friday, October 23, 2015

What are the expected value and the standard deviation of the sample mean ?

Describe the probability distribution of .

What is the probability that the sample mean is greater than 32?

What is the probability that the sample mean falls between 27 and 29?

What is the probability that the sample mean will be within ±3 of the population mean?

The distribution is the normal distribution which is symmetrical about the mean. Symmetric means it looks the same on both sides of the mean.

P(sample mean > 32)

We need to find a Z-score, which shows how many standard deviations away from the mean a value is.

Z = (sample mean - population mean)/(standard deviation/square root(n))

Z = (32 - 31)/(12/square root (49))

Z = 1/(12/7)

Z = 0.58

Now find Z(0.58) on the standard normal distribution chart. Since the chart shows probabilities less than, we need to take 1- Z(0.58)

Here is the chat I used for all these. http://www.regentsprep.org/regents/math/algtrig/ats7/zchart.htm

Z(0.58) = .7190

1- .7190 = .2010

The probability it falls between 27 and 29 ... P(27 < X < 29)

We need 2 Z-scores now, one for 27 and one for 29

Z = (27 -31)/(12/sqrt(49))

Z = -4/(12/7)

Z = -2.33

and

Z = (29 - 31)/(12/sqrt(49))

Z = -2/(12/7)

Z = -1.17

Now we get Z(-1.17) and Z(-2.33) and subtract them

.1210 - .0099 = .1111

Probability sample mean is within +/- 3standard deviations we simply find Z(3) and Z(-3) and subtract

So we get .9987 - .0013 = .9974

## Sunday, October 18, 2015

he 2003 Statistical Abstract of the United States reported the percentage of people 18 years of age and older who smoke. Suppose that a study designed to collect new data on smokers and nonsmokers uses a preliminary estimate of the proportion who smoke of .26.

a. How large a sample should be taken to estimate the proportion of smokers in the population with a margin of error of .02 (to the nearest whole number)? Use 95% confidence.

a. How large a sample should be taken to estimate the proportion of smokers in the population with a margin of error of .02 (to the nearest whole number)? Use 95% confidence.

b. Assume that the study uses your sample size recommendation in part (a) and finds 520 smokers. What is the point estimate of the proportion of smokers in the population (to 4 decimals)?

c. What is the 95% confidence interval for the proportion of smokers in the population (to 4 decimals)?

The margin of error is the 1.96(sqrt(pq/n) and that has to equal .02. We know p =.26, so q = .74

So we have 1.96(sqrt(.26*.74/n) = .02

3.8416(.1924)/n = .0004

0.73912384/n = .0004

n = 1848

b) point estimate is 524/1848 = .2835

c) .26 +/- .02 = (.24, .28)

## Wednesday, October 7, 2015

what we have to do is notice that the ring sizes or 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9 and 9.5 are a certain number of standard deviations from the mean. This is how we will solve this

4 is -2 st dev obtained by taking (4 - 6)/1 = -2

4.5 is -1.5 st dev by taking (4.5 - 6)/1 = -1.5 etc

5 is -1 st dev

5.5 is -0.5 st dev

6 is the mean , so 0 st dev

6.5 is 0.5 st dev

7 is 1 st dev

7.5 is 1.5 st dev

8 is 2 st dev

8.5 is 2.5 st dev

9 is 3 st dev

9.5 is 3.5 st dev.

This is how we will figure out what percentage of each ring should be bought. We can't just say we look at standard normal distribution probabilities and take P(Z = -2) for size 4.0 because remember in a continuous distribution, there are no probabilities for exact values. So we take P(-2.5 < Z < -2.0) = .0166 (i did half standard deviations because each ring size is half st dev apart) . Now take .0166(5000) = 83

The same method is applied to each size

size 4.5 = P(-2 < Z < -1.5) = .044, multiplied by 5000 = 220

size 5 = P(-1.5 < Z < -1) = .0919, multiplied by 5000 = 460

size 5.5 = P(-1 < Z < -0.5) = .1498, multiplied by 5000 = 749

size 6 = P(-0.5 < Z < 0) = .1915, multiplied by 5000 = 958

Since this is the normal distribution, this is symmetrical, so size 6.5 = 958, size 7 = 749, size 7.5 = 460, size 8 = 220, size 8.5 = 83

For size 9, same idea = P(2.5 < Z < 3) = .0049 multiplied by 5000 = 25

size p.5 = P(3 < Z < 3.5) = .0011 times 5000 = 6

Now that gives a total of 4971, the rest are either bigger than 9.5 or less than 4, since no rings will be produced bigger than size 9.5, the other 29 should be of size 4, making size 4 total 112

## Friday, October 2, 2015

What is the probability that an elementary school student in southern Kenya is greater than or equal to 41.3 inches tall? (Round to the nearest percent.)

z score is (x - mean)/standard deviation

so (41.3 - 40.1)/2.2

- and look up that on a standard normal distribution chart
- z-chart
- to that value for z (.55) is .7088
- that is the probability of LESS than z(.55)

## Thursday, September 24, 2015

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

## Monday, September 14, 2015

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.

## Sunday, September 6, 2015

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.

## Wednesday, September 2, 2015

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

## Thursday, August 27, 2015

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%.

## Thursday, August 20, 2015

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

## Saturday, August 15, 2015

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.

## Sunday, August 9, 2015

NY NJ CT

Male 100 60 50

Female 80 50 80

1) Probability person is female and lives in NY

P(F and NY)

80 females from New York and (100 + 60 + 50 + 80 + 50 + 80) = 420 people total in the company from those three states

P(F and NY) = 80/420 = 4/21

2) Probability person is male or lives in NJ

P(M or NJ) = (100 + 60 + 50 + 50)/420

= 260/420

= 13/21

3) P(CT if M)

Number of male and living in Connecticut = 50

Number of males (100 + 60 + 50) = 210

P(CT if M) = 50/210 = 5/21

## Thursday, August 6, 2015

Ho: Mean1 = Mean2

Ha: Mean 1 does not equal Mean2

x-bar1 (south of Center Street) = 145,500 s1 = 4775, n1 = 28

x-bar2 (north of Center Street) = 148,300 s2 = 5900, n2 = 25

test statistic for the test is (x1-bar - x2-bar)/sqrt(s1^2/n1 + s2^2/n2)

If you put the values into the formula and calculate correctly, you should get

t= -1.919

Make sure you understand how to get that

P-value was obtained from this site. http://www.socscistatistics.com/pvalues/tdistribution.aspx

Just enter the t-score of -1.919, 24 df two sides test and alpha = .05 and you get

.0670

b) the conclusion, since .0670 > .05, we have to accept Ho. So you fail to reject Ho since there is not significant evidence of a difference of means.

Remember when p-value > alpha, accept Ho and when p-value < alpha, reject Ho.

## Tuesday, July 28, 2015

To get p^ you take x/n, so the first two are quite simple

p1^ = 25.176 = .142

p2^ = 32/143 = .224

for 90% CI you use 1.645 for zcritical

formula is p1^ - p2^ +/- 1.645(sqrt(p1^q1^/n1 + p2^q2^/n2))

if you do this correctly for the data we have you get

-.082 +/- .071850965

(-.154, -.010)

## Thursday, July 23, 2015

A type II error is the probability of not rejecting the null hypothesis when we should reject it, which is Beta.

By intuition, the greater the departure from Ho, the less likely that departure will be detected, therefore there is a less chance of rejecting the null hypothesis, which decreases the chance of Type II error, which means Beta is smaller

## Sunday, July 19, 2015

For example. The difference between hours for a person 20 years old and 30 years old, using this equation is 1.86+ 20(.02) = 2.26 hours and 1.86 + 30(.02) = 2.46 hours That comes out to just a 12 minute difference. Even for a wide difference in age from 20 to 70 it only adds 1 extra hour.. for a 50 year difference. Each increase of 1 year in age adds only 1.2 extra minutes of tv time, showing no practical significance.

## Wednesday, July 15, 2015

American Continental Delta United Total

Yes 48 69 68 25 210

No 52 41 62 35 190

Total 100 110 130 60 400

Now what we have to do is get the expected number in each spot. For instance we need the expected number from American and YES. We get the expected number by taking the row total times the column total divided by the total number in all.

For American and YES that would be 210(100)/400 (row total of 210)(column total of 100)/(total of 400), which equals 52.5

You have to do that for ALL of the 8 different spots on the table. I'll show you one more .. For Continental and YES it's (210)(110)/400 = 57.75

If you do that correctly this is what you will have and i'll show on another table

American YES NO

Observed = 48 52

Expected = 52.5 47.5

Continental YES NO

Observed = 69 41

Expected = 57.5 52.25

Delta YES NO

Observed 68 62

Expected 60.25 61.75

United YES NO

Observed 25 35

Expected 31.5 28.5

Now to get the test statistic you have to take the (observed - expected)^2 . Divided that by the expected and sum all the values.

For American and YES that is (48-52.5)^2/52.5

For Continental and YES that is (69 - 57.75)^2/57.75

and so on....

If you do that for all eight values you should get 8.251 (you could have a slightly different number depending on how you round)

Now you need the critical value for the test. To get the df you take (rows - 1)(columns - 1)

We have 2 rows in the table and 4 columns, so df = (4-1)(2-1) = 3

Look in any chart for Chi-square distribution for 3 df and alpha = .05 and you get 9.488

We compare our test statistic to 9.488

Remember we reject Ho and conclude there is a preference to which airline we choose if the test statistic is greater than the critical value, but 8.251 < 9.488 so we do what? We do not reject Ho, so we have no preference as to what airline to choose

## Friday, July 10, 2015

For multiple linear regression, it's best to use Excel or use this online calculator for multiple regression, simply put in the values for the x variables and the y variable and you will the result you need

http://www.xuru.org/rt/MLR.asp#Manually

Remember that a multiple linear regression equation will take on the form y^ = a + b0x1 + b1x2 and so on.

## Friday, July 3, 2015

by choosing another sample at random from the population based on data from your sample.we sample instead from an artificial population constructed on our computer and that embodies everything we know

about the population of interest. In many, but not all, of the examples

that follow, this artificial population is the very data set from which

we seek to draw inferences. Since the data set is itself a sample of the

whole population, we are taking a sample from the sample: resampling.

This doesn’t, of course, provide more information about the population,

but it does provide us with a way of understanding the consequences of

sampling variability for drawing inferences about the population based

on our data.

## Thursday, June 25, 2015

For example, suppose the function if f(x) = x^3 + 2x^2 - 5x

f'(x) = 3x^2 +4x - 5

f''(x) = 6x + 4

6x + 4 = 0

6x = -4

x = -2/3

Now test a point on both sides of -2/3, we'll use -1 and 0

f''(-2) = 6(-2) + 4 = -8

f''(0) = 6(0) + 4 = 4

that means we have concave down from (-infinity, -2/3) and concave up from (-2/3, infinity)

## Sunday, June 21, 2015

For the exponential distribution Mean (Mu) = B and variance (sigma squared) = B^2

Examine these and you can see the difference.. The means are basically the same except gamma has a and exponential does not. Variance for gamma also includes a, while exponential does not.

If a = 0 and B = 0 then Mu for exponential is 0 and sigma^2 = 0.

If a = 0 and B = B then Mu for exponential is B and sigma^2 = B^2

If a = 1 and B = B then Mu for exponential is B and sigma^2 = B^2, but note that the Gamma Distribution will also have Mu = B and sigma^2 = B^2.

That's the key, the exponential distribution is EQUAL to the Gamma distribution when a = 1 and B = B.

## Thursday, June 18, 2015

For example, if the population standard deviation is 10 and we have a sample of size 50 from the population, the standard deviation of the sample mean is 10/sqrt(50)

## Wednesday, June 10, 2015

## Wednesday, June 3, 2015

The way to figure out the highest reasonable outcome, take the 1.96(which is 1.96 standard deviations above the mean) and multiply by the standard deviation and add to the mean.

70,000 + 1.96(10,000) = highest maximum reasonable value.

For part c, we need to take (80,000 - 70,000)/(5,000/sqrt(100)) to get the z value. The formula is (value - mean)/(standard deviation/square root of n)

If this falls between -1.96 and 1.96, it is a reasonable value. It does not, as you will see, so it's not reasonable.

d) For this, we know the z-value that will give the maximum reasonable mean salary is 1.96

so use the formula

1.96 = (x - 70,000)/(5,000/sqrt(100))

The x is the value we are solving for, which is the mean salary for a random sample.

If you do this correctly, you will get 70,980

## Monday, June 1, 2015

## Monday, May 18, 2015

Volume = lwh

l = 48 - 2x

w = 32 - 2x

h = x

V= x(48 - 2x)(32 - 2x)

V= x(1536 - 96x - 64x + 4x^2)

V = 1536x - 160x^2 + 4x^3

V' = 1536 - 320x + 12x^2

set V' = 0 and solve for x.

From quadratic formula, x = 6.28 inches

## Thursday, May 14, 2015

## Monday, May 11, 2015

## Thursday, May 7, 2015

## Wednesday, May 6, 2015

unknown quantities are generally combined with mathematical operations (addition, subtraction, multiplication,division, square root, cube root, exponents, etc) to form statements that describe the relationship of things that change over a period of time. These statements are expressed using equations, expressions and terms. Problems can be solved by translating words into algebraic equations. The description of a problem using an equation and other mathematical concepts is known as a mathematical model. A mathematical model can be used to solve numerous types of problems in every day life

## Sunday, May 3, 2015

## Thursday, April 30, 2015

## Friday, April 24, 2015

## Wednesday, April 22, 2015

## Tuesday, April 21, 2015

## Monday, April 20, 2015

## Sunday, April 19, 2015

## Thursday, April 16, 2015

## Tuesday, April 7, 2015

## Saturday, April 4, 2015

You can solve this system by substituting 2y+3 for x in the second equation. The second equation becomes 3(2y+3) + 4y = 16. Now we can solve for y.

6y + 9 + 4y = 16

10y = 7

y = 7/10

To solve for x, substitute 7/10 for y in the first equation. That gives us

x = 2(7/10) + 3

x = 14/10 + 3

x = 4.4

## Tuesday, March 31, 2015

We find the points of intersection by setting f(x) = g(x). Then we determine which curve is the upper curve in the integral and which is the lower curve in the integral. Suppose the points of intersection are x=1 and x=3.

If f(x) is the upper curve then the integral looks like

Integral (1 to 3) [f(x) - g(x)] dx

If g(x) is the upper curve the the integral looks like

Integral (1 to 3) [g(x) - f(x)] dx

## Saturday, March 28, 2015

https://learnivore.com/instructors/mathematics-help-with-tutor-with-15-years-experience

https://www.fiverr.com/mathtutorkk

## Saturday, March 21, 2015

If we pick two angles from the unit circle that add to 165, we can obtain the exact value for sin(165). The reason is we can substitute those values in for this formula.

sin(a + b) = sin(a)cos(b)+cos(a)sin(b)

Notice that 120 + 45 = 165 and both of those values are on the unit circle

sin(120 + 45)= sin(120)cos(45) + cos(120)sin(45)

= (sqrt(3)/2)(sqrt(2)/2) + (-1/2)(sqrt(2)/2)

= sqrt(6)/4 - sqrt(2)/4

= (sqrt(6) - sqrt(2))/4

## Wednesday, March 18, 2015

The Most Simple Introduction to Hypothesis Testing

## Thursday, March 12, 2015

For example, suppose you want a 95 percent confidence interval for the sample mean. The formula is mean +/- standard error which is Z(alpha/2)standard deviation/square root(n). alpha is 1-.95 = .05. So the Z value is 1.96.

Now if we want an x value that 95% of the data falls below, then we need Z(.05) which is 1.645.

Be careful to understand the difference between the two.

## Friday, March 6, 2015

The first derivative f'(x) = 2x + 3

Second derivative f"(x) = 2

Notice the second derivative is positive for all values of x therefore the function is concave up, therefore opening up.

## Tuesday, March 3, 2015

For example, suppose the radius of a circle is 6 and the angle of the sector is 45 degrees. Therefore the area of the sector is

(45/360)Pi(6)(6) = (1/9)(36)Pi = 14.14

## Sunday, February 22, 2015

Use the formula A + P(1+ r/n)^(nt) where A is the amount, P is the initial investment, r is the interest rate, n is the number of times compounded annually and t is the time in years.

2A = A(1 + .07/4)^(4t)

2 = (1.0175)^4t

log 2 + log (1.0175)^(4t)

log 2 = 4t log(1.0175)

log2/log(1.0175) = 4t

t = 10 years

## Monday, February 16, 2015

Suppose you have 5^x = 33 and you want to solve for x. That is difficult as it is set up. It can be made easier by using logarithms.

We rewrite the exponential equation above in logarithmic form as follow:

log (base b) y = x is the same ax b^x = y.

In our example, the logarthmic equivalent is log (base 5) 33 = x.

Using the change of base formula you get log 33/log 5 = x. This can be solve on a calculator to get 2.1725. We can check this by substituting for x in the equation to get 5^(2.1725) which is approximately 30.

## Wednesday, February 11, 2015

There is a huge difference and one that is often confused.

The "highest 89 percent" is from the 11th percent on upward. In other words, if there are 100 people taking a test and you tare in the highest 89 percent, you have scored better than only 11 people.

The "89th percentile" would be that only 11 people scored higher. So it's the exact opposite.

"89th percentile" = "highest 11 percent"

"highest 89 percent" = "11th percentile"

## Sunday, February 8, 2015

## Tuesday, February 3, 2015

We have to integrate f''(x) first to get f'(x).

f'(x) = x^5 + (1/4)x^4 + (1/2)x^2 + 3x + c, since f'(0) = 4, we get

4 = 0^5 + (1/4)(0)^4 +(1/2)(0)^2 + 3(0) = c, therefore c = 4

f'(x) = x^5 + (1/4)x^4 + (1/2)x^2 + 3x + 4. We get f(x) by integrating f'(x).

Therefore f(x) = (1/6)x^6 + (1/20)x^5 + (1/6)x^3 + (3/2)x^2 + 3x + c. Since we know f(1) = 9.883333, we get

9.883333 = 1/6 + 1/20 + 1/6 + 3/2 + 3 + c. Therefore, c =5 and f(x) is

f(x) = (1/6)x^6 + (1/20)x^5 + (1/6)x^3 + (3/2)x^2 + 3x + 5.

## Saturday, January 31, 2015

x1 = 1, x2= 1.75, x3 = 2.5, x4 = 3.25 and x5 = 4

f(x1) is the height of the first rectangle using the left endpoint.

f(x2) is the height of the second rectangle using the left endpoint and so on.

Now we take f(x1) = 1 + 3 - 4 = 0

f(x2) = 4.3125

f(x3) = 9.75

f(x4) = 16.3125

To get the area we take w*f(x1) + w*f(x2) + w*f(x3) + w*f(x4). Doing so, we get 22.7815.

Now we can estimate the area again using the right endpoint of each triangle. Those values are

f(x2), f(x3), f(x4) and f(x5)

f(x5) = 24

The estimated area is 22.7815 + 24 = 46.7815.

Using integration on the function f(x), we get (1/3)x^3 + (3/2)x^2 - 4x. Substituting 4 for x, we get 29.333333 Substituting 1 for x, we get -2.166666 Subtracting the values we get the area to be 31.5

## Saturday, January 24, 2015

First we integrate the function.

(1/4)x^4 + (2/3)x^3 - 2x^2 + 6x..

Now we evaluate the integral at 4 and then at 1 and subtract.

(1/4)(4)^4 + (2/3)(4)^3 - 2(4)^2 + 6(4) = 98.6666666666

(1/4)(1)^4 + (2/3)(1)^3 - 2(1)^2 + 6(1) = 4.91666666666

Subtract the two to get the area of 93.75

## Tuesday, January 20, 2015

A parameter is a numerical characteristic from a population. According to Wikipedia, a parameter is sometimes taken to be unobservable, and in this case the statistician's task is to infer what they can about the parameter based on observations of random variables distributed according to the probability distribution in question, or, more concretely stated, based on a random sample taken from the population of interest. In other situations, parameters may be fixed by the nature of the sampling procedure used or the kind of statistical procedure being carried out

Source of above paragraph is http://en.wikipedia.org/wiki/Statistical_parameter

So the easy way to distinguish between the two is S-S and P-P. Statistic and sample go together as does parameter and population.

## Sunday, January 11, 2015

Volume of a cone is 1/3(Pi)r^2(h)

We know dv/dt = 5 and we know that r = (1/2)h. Substitute (1/2)h in for r to get

V = (1/3)Pi(1/2 h)^2(h)

V= (1/3)Pi(1/4)(h^3)

V = (1/12)(Pi)h^3

Now take the derivative to get the rate of change of the volume.

V' = (3/12)(Pi)h^2(dh/dt)

V' = (1/4)Pi(h^2)dh/dt

Substitute 5 for V' and 8 for h to get

5 = (1/4)Pi(64)dh/dt

5 = 50.3dh/dt

.1= dh/dt

## Saturday, January 10, 2015

Lim (x --> infinity) (x^3 + 4x^2 - 3)/(4x^3 + 6x)

You can divide each term by x to the highest power in the problem, which is 3. So divide each term by x^3 to get

Lim (x --> infinity) (1 + 4/x - 3/x^3)/(4 + 6/x^2)

as x tends to infinity, the numerator is just 1 because 4/x and 3/x^3 approach zero. Similarly 6/x^2 approaches zero so the denominator is just 4. Therefore the limit is 1/4.

Another way to do this problem is that since the highest power of x is the same in both numerator and denominator, divide the coefficients of those two terms which is just 1/4.

If the highest power in the numerator is less than the highest power in the denominator, the lim as x tends to infinity or negative infinity is 0. If the highest power in the numerator is greater than the highest power in the denominator, the lim as x tends to infinity or negative infinity is undefined

## Tuesday, January 6, 2015

If b^2 - 4ac = 0 there is 1 real root

If b^2 - 4ac > 0 there are 2 distinct real roots

If b^2 - 4ac < 0 there are 2 imaginary roots.

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