MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Niharika Kompella RUID: 192001812 EMAIL: nlk39@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= z' = (-9y^8 - z^2) / (2z + 2y) (WRONG ANS., AND TYPE OF ANS. -10 POINTS) Answer[ 2 ]= decreasing (CORRECT) Answer[ 3 ]= 5sqrt(2) (WRONG ANS. BUT RIGHT WAY (MESSED UP CALCULATIONS, -3 POINTS) Answer[ 4 ]= Does not exist (WRONG ANS. BUT STARTED CORRECTLY, -5 POINTS) Answer[ 5 ]= no abs min (WRONG ANS., CONCEPTUAL ERROR THERE IS ALWAYS A MIN VALUE (and MAX VALUE) ON A FINITE (CLOSED) REGION , -10 POINTS) Answer[ 6 ]= 0 (WRONG ANS., WRONG WAY, -10 POINTS) Answer[ 7 ]= 6 / (sqrt(9^2 + 4t^2)) (WRONG ANS. , NONSENSE ANS. -10 POINTS) Answer[ 8 ]= (NO ANS. -10 POINTS) Answer[ 9 ]= (NO ANS. -10 POINTS) Answer[ 10 ]= (1, 8, 2) SCORE: 32 POINTS (OUT of 100). You are welcome to take the 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]= 2 , a[4]= 1 , a[5]= 1 , a[6]= 1 , a[7]= 8 , a[8]= 1 , a[9]= 2 -------------------------------------------- --------------------------------------------- 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) 1. Differentiate both sides wrt y : xz^2+9y^8 = 0 2. (xz^2+9y^8)' = 0 3. 9y^8+2zz'+[y'z^2+y(z^2)'] 4. 9y^8+z^2+2zz'+2yz' 5. Factor z' : 9y^8+z^2+z'(2z+2y) 6. Solve for z' : z' = (-9y^8 - z^2) / (2z + 2y) Ans.: z' = (-9y^8 - z^2) / (2z + 2y) --------------------------------------------- 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,-1,8+2> ; <1,2,-a[5]> Here is how I do it (Explain everything) Ans.: 1. PQ is <0, 3, -18> 2. Mag is 1^2 +2^2+ -8^2 = sqrt21 3. 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*1+y^3*2+z^3*1 Here is how I do it (Explain everything) 1. Find PQ--> (vector) : <0, -2, 2> 2. Find all partial derivatives : grad f = <3x^2 , 6y^5 , 4z^3> 3. Find unit vector : sqrt(0^2 + (-2)^2 + 2^2) = sqrt(8, u = <0/sqrt8 , -2/sqrt8 , 2/sqrt8> 4. Plug x0 y0 and z0 into grad f : 3(1)^2 , 6(-1)^5 , 4(1)^3 5. Dot product of grad f and u : 5sqrt(2 Ans.: 5sqrt(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-1) Here is how I do it (Explain everything) 1. Find all partial derivatives: f_x = e^(x+y-2)x f_y = e^(x+y-2)(x-1) f_xx = e^(x+y-2)x+e^(x+y-2) f_xy = e^(x+y-2)x f_yy = e^(x+y-2)(x-1) 2. Set f_x and f_y to zero, solve sys of eq. : e^(x+y-2)x = 0 e^(x+y-2)(x-1) = 0 (0, -2) 3. Plug in points f_xx, f_xy and f_yy to then use D = f_xxf_yy - (f_xy)^2 -1 -1 0 D = -1*-1*(0^2) D = 1 > 0, f_xx< 0, Does not exist 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 1*x + 8*y + 2 ; A = [1, 9] B = [2, 1] C = [1, 1] Here is how I do it (Explain everything) 1. Find all partial derivatives f_x = 1 f_y = 8 2. Plug into f : 1*(1) + 8(8) + 2 = 68 3. Because when we look on all three sides, the values of f_x and f_y are a constant value, there is no abs. min Ans.: no abs min :( --------------------------------------------- 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) ; (1, 1) Here is how I do it (Explain everything) 1. lim(x,y)->(1,1) of (x^2-y^2)/(x-y) 2. Because just plugging in gives us 0/0, we can use l'hopitals and find that 2x-2y and plug in to get 2-2 which is 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, 9*t, 2*t^2] ; (1,0,0) Here is how I do it (Explain everything) 1. Find r'(t) and r''(t) : 0i + 9j + 4tj and 0i +0j + 4j 2. Find cross product of r'(t) and r''(t) = 36 3. Find mag using k -> 6 / (sqrt(9^2 + 4t^2)) --------------------------------------------- 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)] ; [0, 9] ; [1, 0] Here is how I do it (Explain everything) 1. Find --------------------------------------------- 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 1; 8; 1; 2 Here is how I do it (Explain everything) 1. --------------------------------------------- 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 = 8, z = 2 Here is how I do it (Explain everything) 1. Set up a system of eq : 1x + 0y + 0z = 1 0x + 1y + 0z = 8 0x + 0y + 1z = 2 2.solve for answer: (1, 8, 2)