MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Andrew King RUID: 194003227 EMAIL: Andrew.king@rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= -5/3 Answer[ 2 ]= decreasing Answer[ 3 ]= 6 Answer[ 4 ]= (did not finish) (STARTED CORRECTLY, -5 POINTS) Answer[ 5 ]= 19 (WRONG ANS., PARTIALLT CORRECT WAY, -5 POINTS) Answer[ 6 ]= 0 (WRONG ANS., WRONG WAY, -10 POINTS) Answer[ 7 ]= 72/729 Answer[ 8 ]= [0,-9] Answer[ 9 ]= df/dt = 15 Answer[ 10 ]= (1,2,4) SCORE: 80 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]= 4 , a[4]= 0 (1) , a[5]= 0 (1) , a[6]= 3 , a[7]= 2 , a[8]= 2 , 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 x + y^9 + z^4 + xyz^2 = 4 Here is how I do it (Explain everything): Step 1: I found the deriviatives: f_y and f_z and plugged them into the equation dz/dy = -fy/fx. Step 2: f_y = 4*z^3 + 2*x*y*z and f_y = 9*y^8 + x*z^2 Step 3: When plugged into the equation dz/dy = -fy/fx, I get: -(9*y^8 + x*z^2)/(4*z^3 + 2*x*y*z) and at the point (1,1,1) I get -5/3 Ans.: -5/3 --------------------------------------------- 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) grad(F)(P) = <1,-1,3> direction = <1,4,-1> Step 1: Take dot product and see if it is negative or pos: <1,-1,3> . <1,4,-1> = 1 - 4 - 3 = -6 -6 means it is 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)*3 + (y^3)*4 + (z^3)*2 Here is how I do it (Explain everything) Step 1: I found the gradient(f_x,f_y,f_z) which is <9*x^2, 12*y^3, 6*z^2> and after plugging in the point (1,-1,1) I got <9,-12,6> Step 2: To get the unit vector I found PQ which was (0,0,2), then found the magnitude and divided it by it (which was 2) and got (0,0,1) Step 3: Finally I did dot product with the gradient and the unit vector and got a final answer of 6. Ans.: 6 --------------------------------------------- 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 e^(x-1) - (x-1)*e^(y-3) Here is how I do it (Explain everything) Step 1: I first found all the partial derivatives: f_x = e^(x-1) - e^(y-3) f_y = -(x-1)*e^(y-3) f_xx = e^(x-1) f_xy = -e^(y-3) f_yy = (1-x)*e^(y-3)\ Step 2: critical points = e^(x-1) = 0 and -(x-1)*e^(y-3) = 0 (x,y): x=1, y = 3 (1,3) Ans.: --------------------------------------------- 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 f(x,y) = x + 2y + 9 A = [1,9], B = [4,1], C = [1,3] Here is how I do it (Explain everything) Step 1: f_x = 1 ; f_y = 2 ; NO CRITICAL POINTS because 1=0 and 2=0 DNE. Step 2: The minimum y value is at point B so plugging that into the original function: x + 2y + 9 at (4,1) = 4 + 6 + 9 = 19 Ans.: The minimum value is 19 --------------------------------------------- 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 - y^2)/(x-y) Here is how I do it (Explain everything) Step 1: I plugged y = cx into the limit and solved. lim(x,y) -> (0,0): (x^2 - y^2 / x-y) , y = cx , lim(x,y) -> (0,0) : (x^2 - c^2 * x^2)/(x - cx) = (x^2 * (1-c^2))/(x(1-c)) = 0 Step 2: After getting 0, I have to do polar coords: lim(r -> 0) (r^2 * cos^2(theta) - r^2 * sin^2(theta))/(rcos(theta) - rsin(theta) = r(cos^2(theta) - sin^2(theta))/(cos(theta) - sin(theta)) = 0 Ans.: 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 r(t) = [1, 9t, 4t^2] Here is how I do it (Explain everything) Step 1: r'(t) = <0,9,8t> r"(t) = <0,0,8> ; r'(t) x r"(t) = <72,0,0> ; |72| = 72 ; |r'(t)| = 81 + 64(t) [t = 0] = 81 Step 2: 72 / sqrt81 ^3 = 72/9^3 = 72/729 Ans.: = 72/729 --------------------------------------------- Problem 8: A particle is moving in the plane with ACCELERATION given by [-a[1]*sin(t), -a[2]*cos(t)] [-sin(t), -9cos(t)] At time t=0 its position is , [0, a[2]] [0,9] and its velocity is , [a[1], 0] [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) Step 1: v = [cos(t) + C, -9sin(t) + C] so both Cs are 0 Step 2: x = [sin(t) + C, 9cos(t) + C] C1 = 0, C2 = 0 At t = pi, x(pi) = [sin(pi), 9cos(pi)] = [0, -9] Ans.: [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] [1] and the rate of change of the function with respect to y is, a[7] [2] Both x and y depend on time Right now the rate of change of x with respect to time is, a[1] [1] and the rate of change of y with respect to time is, a[9] [7] How fast is the function changing right now? With my RUID data the question is Here is how I do it (Explain everything) Step 1: The first step is to multiply df/dx by dx/dt and df/dy by dy/dt, and add both. This is 1 * 1 + 2 * 7 = 15 Ans.: df/dt = 15 --------------------------------------------- Problem 10: Find the point of intersection of the three planes x = a[5], y = a[7], z = a[3] x = 1, y = 2, z = 4 With my RUID data the question is Here is how I do it (Explain everything) The point of intersection is just (1,2,4) because each one expands infinitely at each point, eventually meeting at that point. Point of Intersection = (1,2,4)