QUESTION ONE

1.1  Use any appropriate method to integrate ∫2x^4(x^2-5)^50 dx [8]

1.2. Differentiate f(x) = loga x from the first principle .[9]

1.3 Find the binomial expansion for sqrt(x^2 - 2x) up to 3 terms for which values of x is the expansion valid? [10]

1.4 Given that z = x + jy, express z=2x - jy in terms of z and or z modulus in a simplest form. [6]

solve using variation of perameters

y'''-16y' = 2

Use Laplace transformations to solve the following ODE for x(t):

x¨(t) + 2x(t) = u˙(t) + 3u(t)

u(t) = e^−t

Initial conditions

x(0) = 1, x˙(0) = 0, u(0) = 0

Given the integral 1/x dx upper bound 2 lower bound 1

(a) use simpson's rule to approximate the answer with n=4

Formula:f(x)=1/3[f(x0)+4f(x1)+2f(x2)+...+f(xn)]Δx(keep answer to 6 decimals)

b)how large is n in order for the error of Simpsons rule for the given integral is no more than 0.000001

Formula: |Es|=(k)(b-a)^5/(180 n^4), where |f^4(x)≤k|

please show all work and steps

The following observations were obtained when conducting a two-way ANOVA experiment with no interaction.

a. Calculate SST, SSA, SSB, and SSE. (Round intermediate calculations to at least 4 decimal places. Round your answers to 2 decimal places.)

b. Calculate MSA, MSB, and MSE. (Round intermediate calculations to at least 4 decimal places. Round your answers to 2 decimal places.)

c. Construct an ANOVA table. (Round intermediate calculations to at least 4 decimal places. Round "SS", "MS" to 2 decimal places, "F" 3 decimal places.)

d. At the 1% significance level, do the levels of Factor B differ?

• Yes, since we reject the null hypothesis.

• No , since we reject the null hypothesis.

• Yes, since we do not reject the null hypothesis.

• No, since we do not reject the null hypothesis.

e. At the 1% significance level, do the levels of Factor A differ?

• Yes, since we reject the null hypothesis.

• No, since we reject the null hypothesis.

• Yes, since we do not reject the null hypothesis.

• No, since we do not reject the null hypothesis.

For the function, supply a valid technology formula.

r(x) = 30 (1 + 1/3.8)^4x

30*(1 + 1/3.8)^(−4*x)

30*(1 + 1/3.8)^(4*x)   

30*(1 + 1/3.8)*(4*x)

30*(1 + 3.8)*(4*x)

30*(1 + 3.8)^(−4*x)

Then use technology to compute the missing values in the table accurate to four decimal places.

Solve the following IVPs and determine the interval of validity of the solution.

(i)y′ = √1+x2 , y(0)=1

(ii)y′ =e−y(x−2), y(4)=0

Question 12.20 – Use these data - incidents of reports of underage drinking – to perform the following: “Dry” campus, state school: 47, 52, 27, 50 “Dry” campus, private school: 25, 33, 31 “Wet” campus, state school: 77, 61, 55, 48 “Wet” campus, private school: 52, 68, 60

a.) Calculate the cell and marginal means. Notice the unequal Ns b.) Draw a bar graph. c.) Calculate the five different degrees of freedom, and indicate the critical F value based on each set of degrees of freedom, assuming the p level is 0.05. d.) Calculate the total sum of squares. e.) Calculate the between-groups sum of squares of the independent variable campus. f.) Calculate the between-groups sum of squares for the independent variable school. g.) Calculate the within-groups sum of squares. h.) Calculate the sum of squares for the interaction. i.) Create a source table.

Prove the following by induction: 2 + 4 + 6 + …+ 2n = n(n+1) for all integers n

Show all work

Find the intersection point (if any) of the lines r1(t)=(17,54,−22)+t(−3,−8,3)r1(t)=(17,54,−22)+t(−3,−8,3) and r2(s)=(−67,−50,37)+s(12,8,−7)r2(s)=(−67,−50,37)+s(12,8,−7).

Please show full working/steps to help with learning

Which is the most accurate method of the decomposition methods used for the following data set.
Additive with seasonal only, Additive with trend plus seasonal , Multiplicative with seasonal only ,Multiplicative with trend plus seasonal

Sum An a series and |An| cnverges to 0. If the partial sum Sn (A1+A2+...+An) is bounded, is the partial sum Sn' of all absolute value of An (|A1|+|A2|+...+|An|) also bounded?