MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME:SHUBIN XIE RUID:203002353 EMAIL:sx111@rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]=(dz/dy=-2/5) Answer[ 2 ]=decreasing Answer[ 3 ]=(directional derivative=15) Answer[ 4 ]=(1,2) is saddle point. Answer[ 5 ]=6 Answer[ 6 ]=the limit does not exist.(WRONG ANS., WRONG WAY -10 POINTS) Answer[ 7 ]=6 Answer[ 8 ]=r(pi)=[0,-1] Answer[ 9 ]=3 (WRONG ANS., NO WAY INDICATED, -10 POINTS) Answer[ 10 ]=(9,3,3) (WRONG ANS., WRONG WAY, - 5 POINTS) [Comment: This question was so easy, too bad you missed it (you misunderstood the question) I only took 5 points off] SCORE: 75 POINTS (out of 100) ----------------------------------------------------------------- Instructions: Download this file with its original name, mt1.txt, then rename it, in your computer mt1FirstLast.txt Edit it with your answers and solutions USING COMPUTEREZE: e.g.: x times y IS x*y, x to the power y is x^y and Email DrZcalc3@gmail.com, 80 minutes (or sooner) after starting (for most people 10:00am, Oct. 15) Subject: mt1 with an attachment. YOU MUST NAME IT EXACTLY mt1FirstLast.txt --------------------------------------------------------------------------- For each of the questions you MUST first figure, YOUR version, with the following convention For i=1,2,3,4,5,6,7,8,9 , a[i]:= The i-th digit of your RUID, BUT of it is zero make it 1 Example: RUID=413200125; a[1] = 4, a[2] = 1, a[3] = 3, a[4] = 2, a[5] = 1, a[6] = 1, a[7] = 1, a[8] = 2, a[9] = 5 ----------------------------------------------------------------------------------------------------------------------------------------------- HERE WRITE THE ACTUAL a[i] (my RUID:203002353) a[1]= 2 , a[2]=1 , a[3]=3 , a[4]=1 , a[5]=1 , a[6]=2 , a[7]= 3, a[8]=5 , a[9]= 3 -------------------------------------------- --------------------------------------------- Problem 1: Find dz/dy at the point (1,1,1) if z(x,y) is given implicitly by the equation x^a[1]+y^a[2]+z^a[3]+a[5]*x*y*z^2 = 3+a[5] With my RUID data the question is x^2+y^1+z^3+1*x*y*z^2=3+1 Here is how I do it (Explain everything) Step1:x^2+y+z^3+x*y*z^2=4(Simplify equation) Step2: derivative y and z( I should use implicit way) (X is stupid constant because we want to get dz/dy) 1+3*z^2*z'+x*(z^2+y*2*z*z')=0 (2*x*y*z+3*z^2)*z'=-x*z^2-1 dz/dy=z'=(-x*z^2-1)/(2*x*y*z+3*z^2)=-(x*z^2+1)/(2*x*y*z+3*z^2) plug in(1,1,1) dz/dy=-(1*1^2+1)/(2*1*1*1+3*1^2)=-2/5 Ans.:dz/dy=-2/5 --------------------------------------------- Problem 2: Suppose that grad(f)(P)=. Is f increasing or decreasing at the direction ? With my RUID data the question is grad(f)(P)=<2,-1,5> direction<2,3,-1> Here is how I do it (Explain everything) grad(f)(P). unit direction <0 Ans.:decreasing --------------------------------------------- Problem 3: Find the directional derivative of the function f(x,y,z) x^3*a[6]+y^3*a[3]+z^3*a[8] At the point P=(1,-1,1) in the direction pointing to Q=(1,-1,3) With my RUID data the question is x^3*2+y^3*3+z^3*5 Here is how I do it (Explain everything) Find gradient for the function: 6*x^2+9*y^2+15*z^2 plug in (1,-1,1) grad<6,9,15> direction: PQ=<1-1,-1-(-1),3-1>=<0,0,2> find unit direction: u=<0,0,2>/2=<0,0,1> Directional derivative= u.grad(p)=<0,0,1>.<6,9,15>=15 Ans.:directional derivative=15 --------------------------------------------- Problem 4: Find a saddle point of the function f(x,y)= exp(x-a[4])-(x-a[4])*exp(y-a[6]) If there is no saddle point, write in the Answers: "Does Not exist". Explain what you are doing With my RUID data the question is exp(x-1)-(x-1)*exp(y-2) Here is how I do it (Explain everything) f_x=exp(x-1)-exp(y-2) f_y=(x-1)*exp(y-2) f_xx=exp(x-1) f_xy=exp(y-2) f_yy=(x-1)*exp(y-2) let f_x and f_y equal 0 find critical point: (1,2) f_xx(1,2)=1 f_xy(1,2)=1 f_yy(1,2)=0 D=f_xx*f_yy-f_xy^2=1*0-1^2=-1<0 Therefore(1,2) is saddle point. Ans.:(1,2) is saddle point. --------------------------------------------- Problem 5: Let f(x,y) be the function a[4]*x + a[7]*y + a[2] Find the ABSOLUTE MINIMUM VALUE of f(x,y) INSIDE the TRIANGLE whose VERTICES ARE A = [a[1], a[2]], B = [a[3], a[4]], C = [a[5], a[6]] With my RUID data the question is 1*x+3*y+1 A=[2,1]b=[3,1]C=[1,2] Here is how I do it (Explain everything) f_x=1 f_y=3 we don't have critical point. f(2,1)=1*2+3*1+1=2+3+1=6 f(3,1)=1*3+3*1+1=7 f(1,2)=1*1+3*2+1=8 The absolute minimum value is 6. Ans.:6 --------------------------------------------- Problem 6: Let f(x,y) be the function (x^2*a[4]^2-y^2*a[5]^2)/(x*a[4]-y*a[5]) Find the LIMIT of f(x,y) as (x,y) goes to the point [a[5],a[4]], or show that it does not exist With my RUID data the question is (x^2*1^2-y^2*1^2)/(x*1-y*1) point(1,1) Here is how I do it (Explain everything) (x^2-y^2)/(x-y) plug in (1,1) we find 0/0 we should go next step step 2: (y-1)=c*（x-1） y=c*(x-1)+1 (x^2-(c*(x-1)+1)^2)/(x-(c*(x-1)+1)) when (x,y) goes to the point (1,1) the limit is 0/0 the limit does not exist. --------------------------------------------- Problem 7: Find the curvature of the curve r(t) = [a[1], a[2]*t, a[3]*t^2] At the point (a[1],0,0) With my RUID data the question is r(t)=[2,1*t,3*t^2] point(2,0,0) Here is how I do it (Explain everything) t=0 r'(t)=[0,1,6*t]=[0,1,0] r''(t)=[0,0,6] r'(t)Xr''(t)=[6-0,0,0]=[6,0,0] Ir'(t)Xr''(t)I=6 Ir'(t)I=1 k=6/1^3=6 Ans:6 --------------------------------------------- Problem 8: A particle is moving in the plane with ACCELERATION given by [-a[1]*sin(t), -a[2]*cos(t)] At time t=0 its position is , [0, a[2]] and its velocity is , [a[1], 0] Where is it located at time , t = Pi With my RUID data the question is a=[-2*sin(t),-1*cos(t)] T=0 POSITION[0,1] T=0 velocity[2,0] Here is how I do it (Explain everything) V=2cos(t)*i-sin(t)*j+c c=0 v=2cos(t)*i-sin(t)*j R=2sin(t)*i+cost*j+c C=0 R=2sin(t)*i+cost*j r(pi)=[0,-1] --------------------------------------------- Problem 9: A certain function depends on variables x and y Right now the rate of change of the function with respect to x is, a[5] and the rate of change of the function with respect to y is, a[7] Both x and y depend on time Right now the rate of change of x with respect to time is, a[1] and the rate of change of y with respect to time is, a[9] How fast is the function changing right now? With my RUID data the question is a[5]=1 a[7]=3 a[1]=2 a[9]=3 Here is how I do it (Explain everything) 3 --------------------------------------------- Problem 10: Find the point of intersection of the three planes x = a[5], y = a[7], z = a[3] With my RUID data the question is x=1 y=3 z=3 Here is how I do it (Explain everything) A(1,0,0) B(0,3,0) C(0,0,3) AB=[-1,3,0] AC=[-1,0,3] ABXAC=[9,3,3] (9,3,3)