MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME:Elyas Sanzar RUID:193004896 EMAIL:eas348@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]=.5 (WRONG ANS. -10 POINTS) Answer[ 2 ]=decreasing Answer[ 3 ]=<3/sqrt(11),-9/sqrt(11),18/sqrt(11)> Answer[ 4 ]=DNE Answer[ 5 ]=B=(3,1) is minimal value at 20 Answer[ 6 ]=2 Answer[ 7 ]=54/9^3 Answer[ 8 ]=9 Answer[ 9 ]= Answer[ 10 ]=x=1 y=8 z=3, (1,8) (3,8) ----------------------------------------------------------------- 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]=193004896 a[1]= 1 , a[2]=9 , a[3]=3 , a[4]=1 , a[5]=1 , a[6]=4 , a[7]=8 , a[8]=9 , 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+y^9+z^3+x*y*z^2=3+1 Here is how I do it (Explain everything) Ans.:0.5 --------------------------------------------- Problem 2: Suppose that grad(f)(P)=. Is f increasing or decreasing at the direction ? With my RUID data the question is <1,-8,3>, <1,3,-1> Here is how I do it (Explain everything) We take the directional derivative and because it was negative it is decreasing Ans.:decreasing --------------------------------------------- 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+3y^3+2z^3 Here is how I do it (Explain everything) First I found the implicit derivative of the function f(x,y,z) and from there multiplied that by gradient over the u, and got the answer. The ending was found by using the dot product. Ans.:<3/sqrt(11),-9/sqrt(11),18/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 f(x,y)=e^(x-1)-(x-1)*e^(y-4) Here is how I do it (Explain everything) Because the value was a non existing number the answer will be DNE. Work is shown. Ans.:DNE --------------------------------------------- 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)=1x+8y+9 A=[1,9] B=[3,1] C=[1,4] Here is how I do it (Explain everything) We just plug in A,B, and C and see which value is the smallest because it is linear Ans.: B=(3,1) is the minimum value at 20 --------------------------------------------- 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-y^2)/x-y ->(1,1) Here is how I do it (Explain everything) All I did was find the limit by first simplifying the function and then plugging in for x and y. Ans: 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 (1,9t,3t^2). (1,0,0) Here is how I do it (Explain everything) I found the derivative of the the first functions and then the second derivative and then multiplied them by cross product and used the curvature equation to get the answer. Ans: 54/9^3 --------------------------------------------- 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 [-1sin(t)-9cos(t)]. [0,9] Here is how I do it (Explain everything) Work Is shown on the paper Ans: 9 --------------------------------------------- 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=8, z=3 Here is how I do it (Explain everything) Because all the values are linear they will intersect at the same point