MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: RUID: EMAIL: BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= -5/2 Answer[ 2 ]= decreasing Answer[ 3 ]= 15 Answer[ 4 ]= Does Not Exist (WRONG ANS., STARTED CORRECTLY, -5 POINTS) Answer[ 5 ]= 18 Answer[ 6 ]= 0 (WRONG ANS., WRONG WAY, -10 POINTS) Answer[ 7 ]= 4/81 Answer[ 8 ]= <0, -9> Answer[ 9 ]= 50 Answer[ 10 ]= (1,7,9) (WRONG ANS. BUT CARELESS ERROR, DID NOT TAKE POINTS OFF) ----------------------------------------------------------------- SCORE: 85 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]= 2, a[4]= 0, a[5]= 0, a[6]= 5, a[7]= 7, a[8]= 5, a[9]= 7 -------------------------------------------- --------------------------------------------- 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 Here is how I do it (Explain everything) Ans.: Equation: x + y^9 + z^2 + xyz^2 = 4 Find the partial derivatives: Fy = 9y^8 + xz^2, Fz = 2z + 2xyz This is the implicit diff rule: Dz/dy = -Fy/Fz The answer: -Fy/Fz = -10/4 @ (1,1,1) --------------------------------------------- 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.: Take dot product of both vectors and you get -2 it is 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 Here is how I do it (Explain everything) Ans.: Equation: 5x^3 + 2y^3 + 5z^3 Find the partial derivs: Fx = 15x^2, Fy = 6y^2, Fz = 15z^2 grad(f) = <15x^2, 6y^2, 15z^2> Find unit vector: <0,0,1> Find dot product of grad and unit = 15 --------------------------------------------- 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.: Equation: e^(x-1) - (x-1)e^(y-5) Find partial derivs and second derive Find critical point: (1,5) Second derivative test yields 0, so no saddle points Does Not exist --------------------------------------------- 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.: No critical points so answer is lowest y value which is at point b (2,1) plug into equation Abs min is 18 --------------------------------------------- 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) Equation: (x^2 - y) / (x-y) Plug in y = cx and limit = 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 Here is how I do it (Explain everything) You get t = 0 if u plug in the point with the curve. Get cross r't and r''t and divide by r't^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 Here is how I do it (Explain everything) Take integral of a(t) and take the integral of that and you should get <0,-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) Do the chain rule to get dz/dt = 1*1 + 7*7 Dz/dt = 50 --------------------------------------------- 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) Just plug in points <1,7,9> And you get (1,7,9)