MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Zixin Qu RUID: 200005776 EMAIL: zq84@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= -2/3 Answer[ 2 ]= decreasing Answer[ 3 ]= 3*sqrt(11) (WRONG ANS., STARTED CORRECLY, BUT TOOK WRONG DIRECTION, -5 POINTS) Answer[ 4 ]= DOES NOT EXIST (WRONG ANS., RIGHT WAY. D<0 MEANS SADDLE POINT! -5 POINTS) Answer[ 5 ]= 1 (WRONG ANS., WRONG WAY (AND WAY TOO COMPLICATED), -10 POINTS) Answer[ 6 ]= 2 Answer[ 7 ]= 2 Answer[ 8 ]= -1 (WRONG ANSWER< BUT STARTED THE RIGHT WAY, ANSWER SHOULD BE A VECTOR, -5 POINTS) Answer[ 9 ]= 44 Answer[ 10 ]=(1,7,1) 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] a[1]=2 , a[2]=1 , a[3]=1 , a[4]=1 , a[5]=1 , a[6]=5 , a[7]=7 , a[8]=7 , a[9]=6 -------------------------------------------- --------------------------------------------- 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 Find dz/dy at the point (1,1,1) if z(x,y) is given implicitly by the equation x^2+y^1+z^1+1*x*y*z^2 = 3+1 Here is how I do it (Explain everything) #dz/dy= 1 + dz/dy + x*z^2 + x*y*2*z*(dz/dy)=0 #dz/dy*(1+2*x*y*z)=-x*z^2-1 #dz/dy=(-x*z^2-1)/(1+2*z*x*y) #dz/dy at point (1,1,1)= -2/3 Ans.: -2/3 --------------------------------------------- Problem 2: Suppose that grad(f)(P)=. Is f increasing or decreasing at the direction ? With my RUID data the question is Suppose that grad(f)(P)=<2,-1,9>. Is f increasing or decreasing at the direction <2,1,-1>? Here is how I do it (Explain everything) #|(2,1,-1)|= sqrt(6) #u=(2,1,-1)/sqrt(6) #grad(f).u=4/sqrt(6)+(-1)/sqrt(6)+(-9)/sqrt(6)=-sqrt(6) #it is negative #so it is decreasing Ans.: --------------------------------------------- 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 Find the directional derivative of the function f(x,y,z) x^3*5+y^3*1+z^3*7 At the point P=(1,-1,1) in the direction pointing to Q=(1,-1,3) Here is how I do it (Explain everything) #fx=15*x^2 #fy=3*y^2 #fz=21*z^2 #grad(f)=(15*x^2, 3*y^2, 21*z^2) #plug in P(1,-1,1) #grad(f)=(15,3,21) #|(1,-1,3)|=sqrt(11) #u=(1,-1,3)/(sqrt (11)) #grad(f).(u)=15*(1/sqrt(11))+3*(-1/sqrt(11))+21*(1/sqrt(11)) # =33/(sqrt(11))= 3*sqrt(11) Ans.:3*sqrt(11) --------------------------------------------- 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 Find a saddle point of the function f(x,y)= exp(x-1)-(x-1)*exp(y-5) If there is no saddle point, write in the Answers: "Does Not exist". Explain what you are doing Here is how I do it (Explain everything) #fx=exp(x-1)-exp(y-5) #fy=-(x-1)*exp(y-5) #fxx=exp(x-1) #fxy=-exp(y-5) #fyy=-1*(x-1)*exp(y-5) #fx=0 x=1,y=5 #fy=0 x=1, y=5 #the critical point is (1,5) #D=-1 Therefore, D<0, (1,5)is neither max nor min but a saddle point. The answer does not exist. Ans.:DOES NOT EXIST --------------------------------------------- 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 Let f(x,y) be the function 1*x + 7*y + 1 Find the ABSOLUTE MINIMUM VALUE of f(x,y) INSIDE the TRIANGLE whose VERTICES ARE A = [2,1], B = [1,1], C = [1,5] Here is how I do it (Explain everything) #fx=1, fy=7, critical point is (0,0) # 1<=x<=2, 1<=y<=5 #f(1,y)=1+7*y, f(1,0)=1 f(1,1)=8, f(1,5)=36 #f(2,y)=2+7*y, f(2,0)=2 f(2,1)=9, f(2,5)=37 the absolute minimum is 1 #f(x,1)=x+7, f(0,1)=7 f(1,1)=8 f(2,1)=9 #f(x,5)=x+35, f(0,1)=35 f(1,1)=36 f(2,1)=37 the absolute minimum is 7 Therefore, the absolute minimum is 1 Ans.: --------------------------------------------- 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 Let f(x,y) be the function (x^2*1^2-y^2*1^2)/(x*1-y*1) Find the LIMIT of f(x,y) as (x,y) goes to the point [1,1], or show that it does not exist Here is how I do it (Explain everything) #let y=cx #((x^2)-(c*x)^2)/(x-c*x)=x*(1-c)*(1+c)/(1-c)=x*(1+c) #The point is (1,1) #so plug in the x*(1+c)=x*(1+y/x) # =1*(1+1)=2 Answer is 2 --------------------------------------------- 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 Find the curvature of the curve r(t) = [2, 1*t, 1*t^2] At the point (2,0,0) Here is how I do it (Explain everything) #r'(t)=j+2*t*k #r"(t)=2*k #r'(t) x r"(t) = 2i #|r'(t) x r"(t) |= 2 #|r'(t)|= sqrt(1+4*t^2) #because point is (2,0,0), this means t = 0 #|r'(t)|=1 #k(t)=2/1=2 --------------------------------------------- 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 particle is moving in the plane with ACCELERATION given by [-2*sin(t), -1*cos(t)] At time t=0 its position is , [0, 1] and its velocity is , [2, 0] Where is it located at time , t = Pi Here is how I do it (Explain everything) #v(t)=∫(a(t))dt=2*cos(t)*i -sin(t)*j +C #2*cos(0)=2, -sin(0)=0 #C=0 #v(t)=2*cos(t)*i -sin(t)*j #r(t)=∫(v(t))dt=2*sin(t)*i+cos(t)*j+C #2*sin(0)=0, cos(0)=1 #C=0 #r(t)=2*sin(t)*i+cos(t)*j #r(pi)=2*0+(-1)=-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 certain function depends on variables x and y Right now the rate of change of the function with respect to x is, 1 and the rate of change of the function with respect to y is, 7 Both x and y depend on time Right now the rate of change of x with respect to time is, 2 and the rate of change of y with respect to time is, 6 How fast is the function changing right now? Here is how I do it (Explain everything) #Let the certain function be z #dz/dt= dz/dx*dx/dt+dz/dy*dy/dt #=1*2+7*6 #=44 #The function now is changing at 44 --------------------------------------------- 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 Find the point of intersection of the three planes x = 1, y = 7, z = 1 Here is how I do it (Explain everything) #ax+by+cz=d #the point of intersection should be (1,7,1) # # #