MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Suhayb Islam RUID: 194005384 EMAIL: suhayb.islam@rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]=10 Answer[ 2 ]=decreasing Answer[ 3 ]==[15, -12, 72] Answer[ 4 ]=(1,5) Answer[ 5 ]=(1,9) Answer[ 6 ]=2 Answer[ 7 ]=72*sqrt(145)/21025 Answer[ 8 ]== [-sin(pi) 9cos(pi)] Answer[ 9 ]= Answer[ 10 ]= (1 3 4) ----------------------------------------------------------------- WRONG ANSWERS #1 (WRONG WAY, -10 POINTS) #3 (WRONG WAY AND A BIG CONCEPTUAL ERROR, THE ANSWER IS A NUMBER, -10 POINTS) #5 (RIGHT ANSWER TO THE WRONG QUESTION, I ASKED FOR THE MINUIMUM VALUE, NOT LOCATION, -5 POINTS) #7 (STARTED THE RIGHT WAY, BUT A GROSS ERROR IN MAGNITUTE TAKING, -5 POINTS) #9 (NO ANSWER, -10 POINTS) COMMENT: The answer to #8 IS CORRECT, BUT NOT SIMPLIFIED (I GAVE YOU FULL CREDIT, THOUGH) SCORE: 60 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]= 1, a[2]= 9, a[3]= 4, a[4]=1 , a[5]= 1, a[6]= 5, a[7]= 3, a[8]= 8, a[9]= 4 -------------------------------------------- --------------------------------------------- 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^9+z^4+1*x*y*z^2 = 3+1 Here is how I do it (Explain everything) Ans.: Write out equation - x+y^9+z^4+x*y*z^2=4 Bring values over to one side and simplify - x+y^9+z^4+x*y*z^2-4=0 Partial differentiation of function z(x,y) w/ respect to y. This means we keep all other variables BUT y constant since we measure on y-axis - dz/dy= xz^2+9y^8 Substitute the point (1,1,1) in for the differentiated expression to get differentiation at that point in the graph - (1)(1)^2+9(1)^8=10 --------------------------------------------- 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, -1, 3+2> Is f increasing or decreasing at direction <1, 4, -1>? Here is how I do it (Explain everything) Ans.: grad(f)(P) = <1, -1, 5> Is f increasing or decreasing at direction <1, 4, -1>? To project this grad in direction of the vector [1 4 -1], we use dot product [1 -1 5].[1 4 -1]=1-1(4)-1(-1)= -2 Gradient is decreasing in directio of this vector --------------------------------------------- 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*5+y^3*4+z^3*8 Here is how I do it (Explain everything) Ans.: Rewrite expression: x^3*5+y^3*4+z^3*8 Simplify: 5x^3+4y^3+8z^3 Find Gradient of f: grad(f)=[df/dx, df/dy, df/dz] So.... grad(f) = [15x^2, 12y^2, 24z^2] Now evaluate grad(f) at P=(1,-1,1): [15(1)^2, 12(-1)^2, 24(1)^2] [15, 12, 24] Since we are looking for the directional derivative, dot gradient with the vector Q [15, 12, 24].[1, -1, 3] =[15, -12, 72] --------------------------------------------- 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-5) e^(x-1)-(x-1)*e^(y-5) Here is how I do it (Explain everything) Ans.: e^(x-1)-(x-1)*e^(y-5) To find saddle points, we need to first find grad(f) and see at what points it is equal to 0 to look for tangent planes: grad(f) = [df/dx, df/dy] = [e^(x-1)-e^(y-5), -e^(y-5)*(x-1)] e^(x-1)-e^(y-5) = 0 -e^(y-5)*(x-1) = 0 x=1,y-5 How that we have proven tangent planes, find second partial derivatives and apply second partial derivatives test: d^f/dx^2= e^(x-1) -> a d^f/dy^2= -e^(y-5)*(x-1) -> b d^f/dxdy= -e^(y-5) -> c We assigne the different partial derivitives a, b, and c. Evaluate at crit pts. If (a*b)-c^2 < 0, there's a saddle point at that point. (e^(1-1))*(-e^(5-5)*(1-1))-(-e^(5-5))^2=-1 Saddle Point at (1,5). YAY!!!! --------------------------------------------- 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 4x+3y+9 A=[1,9] B=[4,1] C=[1,5] Here is how I do it (Explain everything) Ans.: Find crit points where tangent plane is flat: df/dx = 4x = 0 df/dy = 3y = 0 x,y = (0,0) f(x,0)= 4x+9 1<=x<=4 min val = 1 f(0,y) = 3y+9 1<=y<=9 min val = 1 (1,9) --------------------------------------------- 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-y^2*1^2)/(x*1-y*1) (x^2-y^2)/(x-y). Find limit as (x,y) goes to [1,1] Here is how I do it (Explain everything) Rewrite expression: (x^2-y^2)/(x-y) Simplify Expression by factoring out (x+y) and cancel out (x-y): (x+y)(x-y)/(x-y) = (x+y) Substitute value (1,1) for expression: (1+1) = 2 = Answer --------------------------------------------- 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(r)= [1, 9t, 4t^2] at point (1,0,0) Here is how I do it (Explain everything) Curvature formula: K(t) = |r'(t) x r''(t)|/|r'(t)|^(3) First find r'(t), and r''(t): r'(t) = [0 9 8t] r''(t) = [0 0 8] r'(t) x r''(t) |i j k| |0 9 8t| |0 0 8| (9(8)-8t(0))-(0-0)+(0-0)= = [72 0 0] magnitude of this vector is sqrt(72) Now, find magnitude of just r'(t): sqrt(9^2+(8t^2)^2)=sqrt(64t^2+81) Cube "64t^2+81" since it is already squared: (64t^2+81)^(3/2)= This makes the curvature = 72/((64t^2+81)^(3/2)). At (1,0,0), it is - 72*sqrt(145)/21025 --------------------------------------------- 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(t) = [-sin(t), -9cos(t)] t = 0, pos [0 9] velocity [1, 0] Here is how I do it (Explain everything) v(t) = [cos(t) -9sin(t)] x(t) = [-sin(t) 9cos(t)] = [-sin(pi) 9cos(pi)] --------------------------------------------- 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 x=1 y=3 z=4 Here is how I do it (Explain everything) [1 0 0] [0 3 0] [0 0 4] Since these are straight planes with no variation, they intersect at (1 3 4)