MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Jessenia Bello RUID: 192004374 EMAIL: jgb123@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= 4/10 Answer[ 2 ]= Decreasing Answer[ 3 ]= e^x-1*-(x-1)e^y-2+(e^y-2)^2 Answer[ 4 ]= DNE Answer[ 5 ]= c Answer[ 6 ]= 2 Answer[ 7 ]= 54/9^3 Answer[ 8 ]=9 Answer[ 9 ]= Answer[ 10 ]= (1,3,2) ----------------------------------------------------------------- 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]= 4, a[7]= 3, a[8]= 7, a[9]= 4 -------------------------------------------- --------------------------------------------- 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^[9]+z^[2]+[1]*x*y*z^2=3+[1] Here is how I do it (Explain everything) First I would take the impartial derivative of z and y and divide them and then proceed to plug in the points (1,1,1) into x,y,z and solve. Ans.: 4/10 --------------------------------------------- Problem 2: Suppose that grad(f)(P)=. Is f increasing or decreasing at the direction ? With my RUID data the question is grad(f)(p)=<1,-1,3+2> <1,2,-1> Here is how I do it (Explain everything) I take the directional derivative and its negative meaning its 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*[4]+y^3*[2]+z^3*[7] Here is how I do it (Explain everything) I would take the impartial derivative, find the gradient and use U. Giving me 12,6,21 then find fix Ans.: e^x-1*-(x-1)e^y-2+(e^y-2)^2 --------------------------------------------- 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 exp(x-1)-(x-1)*exp(y-4) Here is how I do it (Explain everything) Does not have critical points after finding derivative therefore no saddle point 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 a=1,9 b=2,1 c=1,4. 1*x+3*y+9 Here is how I do it (Explain everything) Plug in points a b c and find the smallest value because it's linear. Ans.: C is the absolute minimum at A=1,9 giving an answer of 7 --------------------------------------------- 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 I take the limit as it approaches [1,1] and simplify by dividing and plugged in my points. Here is how I do it (Explain everything) Answer= 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 r(t)=[1,9t,3t^2] Here is how I do it (Explain everything) I would take the derivative using power rule, then take the derivative a second time then do r' * r" Answer= 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 Here is how I do it (Explain everything) I take the integral and end up with c=1-cost(t)+9sin(t). 1-cos(t)+cos(t)-9sin(t)+0sin(t)=1 s1=t+c 0+9=t+c c=9-t t+9-t and the t cancels out answer= 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=3 z=2 Here is how I do it (Explain everything) Find the parametric equations to find point of intersection and then use x=p0 + value (t) and end up with the same values answer =They intersect at x=1 y=3 z=2