MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Jennifer Gonzalez RUID: 187005703 EMAIL: jg1426@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= 8y^7 + 7z^6dz/dy = 0 Answer[ 2 ]= Answer[ 3 ]= -(6√11)/11 Answer[ 4 ]= Answer[ 5 ]= Answer[ 6 ]= Answer[ 7 ]= 7/32 Answer[ 8 ]= Answer[ 9 ]= Answer[ 10 ]= (0,0,7) ONLY #7 IS CORRECT. #10 WOULD HAVE BEEN CORRECT IF YOU WOULD HAVE CHANGED THE 0s in your RUID to 1's SCORE: 20 POINTS (OUT of 100). 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 187005703 a[1]= 1 , a[2]= 8 , a[3]= 7, a[4]=0 , a[5]= 0, a[6]= 5, a[7]= 7, a[8]= 0, 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 x^1+y^8+z^7+a(0)*x*y*z^2 = 3+0 Here is how I do it (Explain everything) Ans.: dz/dy = 8y^7 + 7z^6dz/dy = 0 --------------------------------------------- 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) Ans.: --------------------------------------------- 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^15 + y^21 + z^0 Here is how I do it (Explain everything) First I found the gradient f the function: D/dx = 15x^14 D/dy = 21y^20 D/dz = 0 grad = (15(1)^14, 21(-1)^20, 0) = (15, 21, 0) Next the unit vector in the given direction: ||(1, -1, 3)|| = √(1^2 + (-1)^2 + 3^2) = √(1+1+9) = √(11) Unit vector---->(1/√11, -1/√11, 3/√11) Now the dot product: (15, 21, 0)•(1/√11, -1/√11, 3/√11) = 15-21+0/√11 = -6/√11 or -(6√11)/11 Ans.: -(6√11)/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 Here is how I do it (Explain everything) 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 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 Here is how I do it (Explain everything) --------------------------------------------- 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, 7*t^2] At the point (1,0,0) We get point (1,0,0) when t=0 Here is how I do it (Explain everything) r'(t) = [0, 8, 14t] r''(t) = [0, 0, 14] ||r'(t)|| = √0+64+(14t)^2 = √64+196t^2 Cross product: i(8*14) - j(0) + k(0) = 112i r'(t) x r''(t) = [112, 0, 0] ||r'(t) x r''(t)|| = √112^2+0+0 = 112 k(t)= 112/(64+196t^2)^3/2 k(0)= 112/(64^(3/2)) = 7/32 Curvature at given point is 7/32 --------------------------------------------- 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 Here is how I do it (Explain everything) --------------------------------------------- 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 = 0, y = 7, z = 7 Here is how I do it (Explain everything) Thinking of this geometrically and drawing the planes on paper, the intersection should be (0,7,7)