16BFinalExam(Practice1)sol

Q: How do you solve the constrained optimization problem for y^2 with the given constraint?

Set up the necessary equation using the constraint and solve for stationary points, then evaluate the candidate points to confirm the maximum exists.

Q: What does the integral test require to determine convergence or divergence?

You must verify the integral test hypotheses for the given positive term function and then compare the convergence of the corresponding improper integral.

Q: How are the cumulative distribution function and variance for X computed from f(x)=sin(x) on [0, π/2]?

Compute F(x) by integrating f(t) from 0 to x, then use the definition of variance with E[X] and E[X^2] obtained from integrals over the same interval.

Sep 30, 2026

Uploaded by Fans

Practice exam set covering multivariable optimization with constraints, double integrals over regions, and first-order differential equations including solving with initial conditions. Includes differential equation analysis via graphical solution of y'=cos(y), and convergence/divergence assessment using the integral test plus infinite series tests. Also requests Taylor series expansions with at least five non-zero terms, and probability computations for a continuous density on [0, π/2], including the CDF and variance. Final sections address standard normal and normal probability calculations such as Pr(X≤2).

LanguageEnglish

CategoryExam

FormatPDF

Pages11

Reading Time28 min

AccessFree to download

Questions

Differential equations and solutions

Integral tests and series convergence

Taylor series expansions

Probability and normal distributions

| This exam consists of 10 questions.Answer the questions in the
spaces provided. | | | | --- | --- | --- | | | | | 1.(25 points)Find the points(x,y)which maximise the function y²z,subject to the con-straint y²+4acy=1200.You may assume,withoutjustification,that a maximum exists.Solution: 3y²一12oo=0==16 →入：-5ft i wi2d--2) agpois=A= 5y= 2=>2=40oy=20-2x=,)=2+入十山-120 y=-20→x= (16,20)aud(-10,-2)2o².10 >(-z)²(-10 =)(16,20) Move P 2.(25 points)Let R be the region enclosed by y=1/x²,y=1/x and z =2 in the firstquadrant.Determine the following double integral Solution: (/→x1) 一儿u(一l2d 3.(25 points)Find the general solution to the following differential equation: x²y'=ycos(1/xc)+y Hence,or otherwise,find solutions with the following initial conditions:y(1/π)=0 andy(1/π)=-1. Solution: soltTouy'=yxcm十x-2→y=0的 s x~cmc×=)d= -cmcy) du --SmCu →uly si(一个 ±e⁴auy na-2eo cmAety ={±ee) 七Lu owAa大 soltou 4.(25 points)A company expects to have a constant income rate for the next ten years.They will invest it in a saving account with 50%annual interest rate.If they want tohave $10000000 in the account after ten years,what must their income rate be?You donot need to simplify your answer. Solution: aunudlk= congt autwtevert rate 00、0】eVoluu oevkd七 = 160Accumwatd16 => C个 kes(-2+2)=100000 yoa 5.(25 points)Consider the differential equation y'=cos(y).Graph the solutions with thefollowing initial conditions:y(0)=-π/4,y(0)=π/2 and y(0)=3π/4.Be sure to labeleach solution carefully. 6.(a)(15 points)Using the integral test,determine whether is convergent or divergent.Be sure to check all the hypotheses of the integral test. Solution: couvageCa l to21≠'(v)= 二3<0m CI,∞机 k≥ (b)(10 points)Using this,or otherwise,determine if the infinite series is convergent or divergent. Solution: ≤Sin²(W≤to o 7.(25 points)Determine the Taylor series of the following functions.Write at least thefirst five non-zero terms of the sum to illustrate the general pattern. (a) f(x)=e Solution: (b) f(x)=In(1-2x²) Solution: 4u(1-2z2)=1u(1+(-2²)) 8.(25 points)Let X be a continuous random variable on [0,π/2]with probability densityfunction f(x)=sin(x).Calculate the cumulative distribution function of X.CalculateVar(X). F(x iu(2)x=-(+ F(o)二0=) -cm(0+C0C10 F()=1-c( ≠'()=1G =一ceg Lem 氧x)=x²g(z)=HuL²≠'(2=2z G(2)=-cm (z)个《(x) )个ur 九u) A'()=1G(a)= Sinlx) 《2}\sin(x)dx=-x^{2}\c00(x)+2x\sin(x)+2o\c π/z 9.(25 points)Let Z be the standard normal random variable.Assume that If X is a normal random variable with variance 4 and such that,calculate Pr(X≤2). Solution: Vaw(X)==92= P√(X≥1)=方= Mediam(x)= I=μ= SAaudawA Nnmd.(X2) A三)=P(已≤1=0 → Pr(×≤2)=00·乙

FAQs

How do you solve the constrained optimization problem for y^2 with the given constraint?

Set up the necessary equation using the constraint and solve for stationary points, then evaluate the candidate points to confirm the maximum exists.

What does the integral test require to determine convergence or divergence?

You must verify the integral test hypotheses for the given positive term function and then compare the convergence of the corresponding improper integral.

How are the cumulative distribution function and variance for X computed from f(x)=sin(x) on [0, π/2]?

Compute F(x) by integrating f(t) from 0 to x, then use the definition of variance with E[X] and E[X^2] obtained from integrals over the same interval.

Sep 30, 2026

Uploaded by Fans