MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME:Fayed Raza RUID: 192002683 EMAIL: fr247@rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= -9y^8/2z Answer[ 2 ]= Increasing Answer[ 3 ]= 84/sqrt(96) Answer[ 4 ]= Does Not exist Answer[ 5 ]= 9 Answer[ 6 ]= 0 Answer[ 7 ]= 36/81 Answer[ 8 ]= [2t - 2, 0] Answer[ 9 ]= 4 Answer[ 10 ]= (0,6,2) ----------------------------------------------------------------- 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]= 2, a[4]=0 , a[5]= 0, a[6]= 2, a[7]= 6, a[8]=8 , 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 Find dz/dy at the point (1,1,1) if z(x,y) is given implicitly by the equation x^1+y^9+z^2 = 3 Here is how I do it (Explain everything) Ans.: Fz = 2z Fy = 9y^8 dz/dy = -Fy/Fz = -9y^8/2z --------------------------------------------- Problem 2: Suppose that grad(f)(P)=. Is f increasing or decreasing at the direction ? With my RUID data the question is Here is how I do it (Explain everything) Suppose that grad(f)(P)=<1,0,8>. Is f increasing or decreasing at the direction <1,2,0>? Ans.: <1,2,0> * <1,0,8> = 1 + 0 + 0 = 1 Increasing since 1 > 0 --------------------------------------------- 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^6+y^6+z^24 Here is how I do it (Explain everything) 1/sqrt(11) <1,-1,3> <6x^5,6y^5, 24z^23> -> <6,-6,24> * 1/sqrt(96) <1,-1,3> = 84/sqrt(96) Ans.: 11/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)-(x)*exp(y-2) Here is how I do it (Explain everything) fx= exp(x)-exp(y-2) x=y-2 exp(x)=exp(2-2) exp(x)=1 x = ln(1) Fy = -(y-2)*exp(y-2) y=2 (0,2) fxx= exp(x) = 1 fxy = 0 Fyy = -(x)*exp(y-2) = 0 Fxxfyy - fxy^2 = 0-0 = 0 No saddle point since d = 0 Does Not exist Ans.: --------------------------------------------- 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 6*y + 9 Find the ABSOLUTE MINIMUM VALUE of f(x,y) INSIDE the TRIANGLE whose VERTICES ARE A = [9, 9], B = [2, 0], C = [0, 2]] Here is how I do it (Explain everything) Fx = 0 Fxx =0 Fyy =0 Fxy = 0 Fy = 6 (0,6) 0-0 D=0 9 9 = 54 + 9 = 63 2 0 = 9 0 2 = 21 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 (-y^2*1^2)/(x*2-y) Find the LIMIT of f(x,y) as (x,y) goes to the point [0,0], or show that it does not exist Here is how I do it (Explain everything) r^2cos0^2/(2rsin0 - rcos0)= lim r-> 0 rcos0^2 /(2sin0-cos0) = 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 Find the curvature of the curve r(t) = [1, 9*t, 2*t^2] At the point (1,0,0) Here is how I do it (Explain everything) R'(t) = [0,9,4t] R''t = [0,0,4] I j k 0 9 4t 0 0 4 36i-0j-0k 36/(16 (0)^2 + 81) = 36/81 --------------------------------------------- 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 [-*sin(t), -9*cos(t)] At time t=0 its position is , [0, 9] and its velocity is , [1, 0] Where is it located at time , t = Pi Here is how I do it (Explain everything) 1 = -cos(0) + c c= 2 0 = -9sin(0) + C vt(t) = [-sin(t) + 2, -9*cos(t)] 0 = cos(t) + 2t + C 0 = 1 + C C =-1 v(t) = -9sint(t) + C 0 = C [cos(t) + 2t - 1, -9sint(t)] [cos(pi) + 2t - 1, -9sint(pi)] [2t - 2, 0] --------------------------------------------- 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, 0 and the rate of change of the function with respect to y is, 1 Both x and y depend on time Right now the rate of change of x with respect to time is, 4 and the rate of change of y with respect to time is, 5 How fast is the function changing right now? Here is how I do it (Explain everything) (4-0) = 4 (5-1) = 4 --------------------------------------------- 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 = 0, y = 6, z = 2 Here is how I do it (Explain everything) (0,6,2) All plane will touch at (0,6,2)