MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Daniel Gameiro RUID: 195000275 EMAIL: dtg62@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= -10/7 Answer[ 2 ]= Decreasing Answer[ 3 ]= 21 Answer[ 4 ]= Does not exist (WRONG ANS., RIGHT WAY, MESSED UP CALCULATIONS, -5 POINTS) Answer[ 5 ]= 12 Answer[ 6 ]= 2 Answer[ 7 ]= 90/729 Answer[ 8 ]= [0,-9] Answer[ 9 ]= (NO ANS. -10 POINTS) Answer[ 10 ]= (1,2,5) 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]= 5, a[4]= 1 , a[5]= 1, a[6]= 1, a[7]= 2, a[8]= 7, a[9]= 5 -------------------------------------------- --------------------------------------------- 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 Find dz/dy at the point (1,1,1) if z(x,y) is given implicitly by the equation x^1+y^9+z^5+1*x*y*z^2 = 3+1 Here is how I do it (Explain everything) x^1+y^9+z^5+1*x*y*z^2 = 4 9*y^8 + 5*z^4*z' + x*z^2 + 2*x*y*z*z' = 0 5*z^4*z' + 2*x*y*z*z' = -(9*y^8 + x*z^2) z' = - (9*y^8 + x*z^2) / (5*z^4 + 2*x*y*z) z' at (1,1,1) = -10/7 Ans.: -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,-1,4>. Is f increasing or decreasing at the direction <1,5,-1>? Here is how I do it (Explain everything) Find direction unit vector: <1,5,-1>/|<1,5,-1>| = <1,5,-1>/sqrt(27) = <1/sqrt(27),5/sqrt(27),-1/sqrt(27)> Take dot product with grad(f): <1/sqrt(27),5/sqrt(27),-1/sqrt(27)> . <1,-1,4> = 1/sqrt(27) - 5/sqrt(27) - 4/sqrt(27) = -8/sqrt(27) Directional derivative is negative so it's 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 Find the directional derivative of the function f(x,y,z) x^3*1 + y^3*5 + z^3*7 At the point P=(1,-1,1) in the direction pointing to Q=(1,-1,3) Here is how I do it (Explain everything) Find grad(f)(P): grad(f) = <3*x^2, 15*y^2, 21*z^2> grad(f)(P) = <3,15,21> Find unit direction vector: PQ = <1-1, -1+1, 3-1> = <0,0,2> u_PQ = <0,0,1> Directional derivative: grad(f)(P) . u_PQ = <3,15,21> . <0,0,1> = 21 Ans.: 21 --------------------------------------------- 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 Find a saddle point of the function f(x,y)= exp(x-1)-(x-1)*exp(y-1) If there is no saddle point, write in the Answers: "Does Not exist". Explain what you are doing Here is how I do it (Explain everything) f(x,y) = exp(x-1) - x*exp(y-1) + exp(y-1) Find partial derivatives: f_x = exp(x-1) - exp(y-1) f_y = exp(y-1) - x*exp(y-1) f_xx = exp(x-1) f_xy = -exp(y-1) f_yy = exp(y-1) Find critical points: exp(x-1) - exp(y-1) = 0, exp(y-1) - x*exp(y-1) = 0 P = (1,1) Find the discriminant at P: D = f_xx * f_yy - f_xy^2 D(P) = exp(0) * exp(0) - (-exp(0))^2 = 0 Ans.: 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 Let f(x,y) be the function 1*x + 2*y + 9 Find the ABSOLUTE MINIMUM VALUE of f(x,y) INSIDE the TRIANGLE whose VERTICES ARE A = [1, 9], B = [5, 1], C = [1, 1] Here is how I do it (Explain everything) f_x = 1, f_y = 2 –> No critical points Contestants: (1,1), (1,9), (5,1) f(1,1) = 12, f(1,9) = 28, f(5,1) = 16 Min = 12 Ans.: 12 --------------------------------------------- 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 Let f(x,y) be the function (x^2*1 - y^2*1)/(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) f can be written as: (x+y)*(x-y)/(x-y) (x-y) cancels on top and bottom we have: lim(x,y –> 1,1) (x+y) = 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 Find the curvature of the curve r(t) = [1, 9*t, 5*t^2] At the point (1,0,0) Here is how I do it (Explain everything) Find t_0: t=0 r'(t) = [0, 9, 10*t] r''(t) = [0,0,10] k(t) = |r'(t) x r''(t)| / |r'(t)|^3 = |[90,0,0]|/|[0,9,10*t]|^3 k(0) = 90/9^3 = 90/729 --------------------------------------------- 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 A particle is moving in the plane with ACCELERATION given by [-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) a(t) = [-1*sin(t), -9*cos(t)] v(t) = [cos(t), -9*sin(t)] r(t) = [sin(t), 9*cos(t)] r(pi) = [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) --------------------------------------------- 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 Find the point of intersection of the three planes x = 1, y = 2, z = 5 Here is how I do it (Explain everything) These 3 lines intersect at (1,2,5)