MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Matthew Sternesky RUID: 186004140 EMAIL:mjs754@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= 4/3 Answer[ 2 ]= Increasing Answer[ 3 ]= 9.045 Answer[ 4 ]= Inconclusive Answer[ 5 ]= Answer[ 6 ]= 0 Answer[ 7 ]= 96/(sqrt(64*144t^2))^3 Answer[ 8 ]= (0, -8) Answer[ 9 ]= Answer[ 10 ]= All Answers are wrong except for 8, PLEASE TAKE THE MAKE-UP ----------------------------------------------------------------- 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]= 1, a[2]= 8, a[3]= 6, a[4]=0 , a[5]= 0, a[6]= 4, a[7]= 1, a[8]= 4, a[9]= 0 -------------------------------------------- --------------------------------------------- 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^1+y^8+z^6+0*x*y*z^2=3 Here is how I do it (Explain everything) Treat x as constant Derive with respect to y dz/dy=0+8y^7+6z^5*dz/dy=0 8(1)^7+6(1)^5*dz/dy=0 8+6*dz/dy=0 dz/dy=8/6 Ans.: 4/3 --------------------------------------------- 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)=<1,0,3> at <1, 6, 0> Here is how I do it (Explain everything) Take the unit vector of <1, 6, 0> P/|P|=u ux=1/sqrt(7), uy=6/sqrt(7), uz=0 grad(f)(P).u (1)(1/sqrt(7))+(0)(6/sqrt(7))+(3)(0) 1/sqrt(7)+0+0 = 1/sqrt(7) Ans.: Increasing --------------------------------------------- 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 4*x^3+6*y^3+4z^3 at P=(1, 1, 1) point to Q=(1,-1,3) Here is how I do it (Explain everything) fx=12x^2 fy=18y^2 fz=12z^2 grad(f)=<12x^2,18y^2,12z^2> grad(f)(1, 1, 1)=<12, 18, 12> P/|P|=u |P|=sqrt(1^2+-1^2+3^2)=sqrt(11) ux=1/sqrt(11) uy=-1/sqrt(11) uz=3/sqrt(11) grad(f).u =>(12)(1/sqrt(11))+(18)(-1/sqrt(11))+(12)(3/sqrt(11)) Ans.: 9.045 --------------------------------------------- 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 f(x, y) = e^(x-0)-xe^(y-4) Here is how I do it (Explain everything) fx=e^x-e^(y-4) fy=-xe^(y-4) fxx=e^x fxy=-e^(y-4) fyy=-xe^(y-4) e^x-e^(y-4)=0 , -xe^(y-4)=0 x=y-4 y=4 x=0 critcal point: (0, 4) fxx(0, 4) = 1 fxy(0, 4) = 0 fyy(0, 4) = 0 D=fxx*fyy-fxy^2=0 Ans.: Inconclusive --------------------------------------------- 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 Here is how I do it (Explain everything) 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 lim(x,y)->(1,1) of (x^2*1^2-y^2*1^2)/(1x-1y) Here is how I do it (Explain everything) lim(x,y)->(1,1) of (x^2-y^2)/(x-y) lim(x,y)->(1,1) of x-y lim(x,y)->(1,1) of 1-1 = 0 Ans.: 0 --------------------------------------------- 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) = [1, 8*t, 6t^2] at (1, 0, 0) Here is how I do it (Explain everything) r'(t) = 0i + 8j + 12tk r''(t) = 12k |i j k| |0 8 12t| |0 0 12| =96i - 0j + 0k |r'(t)xr''(t)| = 96 |r'(t)| = sqrt(8^2 + (12t)^2)=sqrt(64*144t^2) Ans.: 96/(sqrt(64*144t^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]] t = 0 s(t)=[0,8] and its velocity is , [a[1], 0] v=[1,0] Where is it located at time , t = Pi t=Pi With my RUID data the question is [-1*sin(t), -8*cos(t)] Here is how I do it (Explain everything) v(t)=int[-1*sin(t), -8*cos(t)] v(t)=[cos(t), -8sin(t)] s(t)=int(v(t)) s(t)=[sin(t)), 8cos(t)] Ans.: (0, -8) --------------------------------------------- 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 Here is how I do it (Explain everything) --------------------------------------------- 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 Here is how I do it (Explain everything)