MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Wenhao Li RUID:201002676 EMAIL:wl545@rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]=z'=-1/(1+2*x*z) Answer[ 2 ]=f is decreasing at the direction <2,1,-1> Answer[ 3 ]=The requested directional derivative is 66/sqrt11 Answer[ 4 ]=It has one saddle point at (1,2) Answer[ 5 ]=the absolute minimum is 1 Answer[ 6 ]=the LIMIT of f(x,y) is 2 Answer[ 7 ]=the curvature of the curve is 2/(sqrt1+4t^2)^3 Answer[ 8 ]=it is locate at [2sinPi,cosPi] at t=Pi Answer[ 9 ]=the function is changing 1 right now Answer[ 10 ]= the point of intersection of the three planes is 6 ----------------------------------------------------------------- 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]=2 , a[7]=6 , 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 x^2+y^1+z^1+1*x*y*z^2=3+1 Here is how I do it (Explain everything) 1+1*z'+x*2z*z'=0 1+z'+2*x*z*z'=0 z'+2*x*z*z'=-1 z'*(1+2*x*z)=-1 z'=-1/(1+2*x*z) Ans.:z'=-y/(1+2*x*y*z) --------------------------------------------- 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) sqrt2^2+1^2+(-1)^2=sqrt6 u=1/sqrt6<2,1,-1>=<2/sqrt6,1/sqrt6,-1/sqrt6> <2,-1,9>.<2/sqrt6,1/sqrt6,-1/sqrt6>=-6/sqrt6<0 f is decreasing at the direction <2,1,-1> Ans.:f is decreasing at the direction <2,1,-1> --------------------------------------------- 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*1+z^3*7 Here is how I do it (Explain everything) fx=6x^2 fy=3y^2 fz=21z^2 grad(f)=<6x^2,3y^2,21z^2> |<1,-1,3>|=sqrt11 u=<1/sqrt11,-1/sqrt11,3/sqrt11> grad(f)(1,-1,1)=<6,3,21> <6,3,21>.<1/sqrt11,-1/sqrt11,3/sqrt11>=66/sqrt11 Ans.:The requested directional derivative is 66/sqrt11 --------------------------------------------- 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) fx=exp(x-1£©-exp(y-2) fy=-(x-1)*exp(y-2) fxx=exp(x-1£© fxy=-exp(y-2) fyy=-(x-1)*exp(y-2) exp(x-1£©-exp(y-2)=0 -(x-1)*exp(y-2)=0 y=2 x=1 fxx£¨1,2)=1 fxy(1,2)=-1 fyy(1,2)=0 D=1*0-1^2=-1 is negative, this is a saddle point. Ans.:It has one saddle point at (1,2) --------------------------------------------- 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 f(x,y)=1*x + 6*y + 1 A = [2, 1], B = [1, 1], C = [1, 2] Here is how I do it (Explain everything) fx=1 fy=6 1<=x<=2 1<=y<=2 f(1,y)=x+6y+1=Fy F'y=6 f(2,y)=2x+6y+1=Fy F'y=6 f(x,1)=x+7=Fx F'x=1 f(x,2)=x+13=Fx F'x=1 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 (x^2*1^2-y^2*1^2)/(x*1-y*1) the point [1,1] Here is how I do it (Explain everything) (x^2-y^2)/(x-y) ((x+y)-(x-y)£©/(x-y) =x+y plug in the point [1,1] 1+1=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 r(t) = [2, 1*t, 1*t^2] At the point (2,0,0) Here is how I do it (Explain everything) r'(t)=<0,1,2t> r''(t)=<0,0,2> r'(t)*r''(t)=2i |r'(t)*r''(t)|=2 |r'(t)|=sqrt1+4t^2 k(t)=2/(sqrt1+4t^2)^3 --------------------------------------------- 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 [-2*sin(t), -1*cos(t)] At time t=0 its position is , [0, 1 ] its velocity is , [2, 0] Here is how I do it (Explain everything) a=[-2*sin(t), -1*cos(t)] v=[2cost,-sint] s=[2sint,cost] t=Pi s=[2sinPi,cosPi] --------------------------------------------- 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 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, 6 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) <1,6> <2,6> <2,6>-<1,6>=<1,0> |<1,0>|=1 --------------------------------------------- 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 = 6, z = 1 Here is how I do it (Explain everything) 1*6*1=6