MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Angelica Armstrong RUID: 195002383 EMAIL: aa2036@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= (-9-1)/(5+2)=-10/7 Answer[ 2 ]= decreasing Answer[ 3 ]= (-6/sqrt(11)),(-15/sqrt(11)),(72/sqrt(11)) (WRONG ANSWER, WRONG TYPE OF ANSWER, -10 POINT) Answer[ 4 ]= exp(x-1)*-(x-1)*exp(y-2)-(-exp(y-2))^2 (WRONG ANSWER, WRONG TYPE OF ANSWER, -10 POINT) Answer[ 5 ]= (NO ANSWER, -10 POINTS) Answer[ 6 ]= DNE it depends on c (WRONG ANS., WRONG WAY, -10 POINTS) Answer[ 7 ]=90/(9^3) Answer[ 8 ]= 9j (RIGHT ANS. WRONG WAY , -3 POINTS) Answer[ 9 ]= sqrt(10) (WRONG ANS. DOES NOT MAKE SENSE.-10 POINTS) Answer[ 10 ]= <1,3,5> SCORE: 47 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]=5 , a[4]=0 , a[5]=0 , a[6]=2 , a[7]=3 , a[8]=8 , 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^9+z^5+1*x*y*z^2 = 3+1 Here is how I do it (Explain everything) Ans.: (-9-1)/(5+2)=-10/7 --------------------------------------------- Problem 2: Suppose that grad(f)(P)=. Is f increasing or decreasing at the direction ? With my RUID data the question is Suppose that grad(f)(P)=<1,-1,3+2>. Is f increasing or decreasing at the direction <1,5,-1>? 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^3*2+y^3*5+z^3*8 At the point P=(1,-1,1) in the direction pointing to Q=(1,-1,3) Here is how I do it (Explain everything) Ans.: (-6/sqrt(11)),(-15/sqrt(11)),(72/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 exp(x-1)-(x-1)*exp(y-2) Here is how I do it (Explain everything) Ans.: exp(x-1)*-(x-1)*exp(y-2)-(-exp(y-2))^2 --------------------------------------------- Problem 5: Let f(x,y) be the function 1*x + 3*y + 9 Find the ABSOLUTE MINIMUM VALUE of f(x,y) INSIDE the TRIANGLE whose VERTICES ARE A = [1, 9], B = [5, 1], C = [0, 2] 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 (x^2*1^2-y^2*1^2)/(x*1-y*1) Find the LIMIT of f(x,y) as (x,y) goes to the point [1,1], or show that it does not exist 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, 9*t, 5*t^2] At the point (1,0,0) Here is how I do it (Explain everything) --------------------------------------------- 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 [-1*sin(t), -9*cos(t)] At time t=0 its position is , [0, 9] and its velocity is , [1, 0] Where is it located at time , t = Pi 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, 1 and the rate of change of the function with respect to y is, 3 Both x and y depend on time Right now the rate of change of x with respect to time is, 1 and the rate of change of y with respect to time is, 3 How fast is the function changing right now? 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 = 5 Here is how I do it (Explain everything)