MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Joe Barr RUID: 194007666 EMAIL: jlb661@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= 0.6 Answer[ 2 ]= decreasing Answer[ 3 ]= 36 Answer[ 4 ]= DNE Answer[ 5 ]= 19 (CORRECT) Answer[ 6 ]= 1 Answer[ 7 ]= 8 Answer[ 8 ]= <-pi,-9> Answer[ 9 ]= 6 Answer[ 10 ]= (1,6,4) ALL WRONG EXCEPT FOR #5; PLEASE TAKE 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 a[i] a[1]= 1 , a[2]= 9 , a[3]= 4, a[4]=1 , a[5]= 1 , a[6]= 7, a[7]= 6, a[8]= 6, 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: 194007666 Here is how I do it (Explain everything) Ans.: First I created a f(x,y,z) function then I took the partial derivative of z and y. Then I found the quotient of the partial derivatives and plugged in the point. --------------------------------------------- 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.: I said decreasing because I calculated the directional derivative and it was negative. --------------------------------------------- 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.: I first found the vector u which is PQ. Then I found the gradient of the function and evaluated it at P. Then I took the dot product of the function and the direction. --------------------------------------------- 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.: DNE because there are no critical points. --------------------------------------------- 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.: There were no critical points, so I plugged in the points A-C and at Point B I got the smallest value --------------------------------------------- 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) I found the limit of the y and x axis separately. Since they both were equivalent at L =1 that was the limit --------------------------------------------- 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) I found the first and second derivative of r(t) and evaluated them at t = 0. Then I found the magnitude of the cross product of the first and second derivative evaluated at t and divided it by the magnitude of the first deviated evaluated at the t squared. --------------------------------------------- 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): I first took the integral of acceleration then found the unknown constants using the initial velocity. Then I took the integral once again to get the position function and found the constants by using the initial position. Then I used my general equation for position to find the position at t = 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) Dy/dx = 6. i took the ratio of change of y over change of x. --------------------------------------------- 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) I used parametric equations and they all have the same t.