MIDTERM 1 FOR  Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z.                                                                                NAME: Brianna Patnaude                                                                                RUID: 196001752                                                                                EMAIL: briannapatnaude@gmail.com                                                                 BELOW WRITE THE LIST OF THE ANSWERS                                                                           Answer[ 1  ]= dz/dy = (9*y^8 + x*z^2)/ (-6*z^5 - 2*x*y*z) (WRONG ANS., IT SHOULD BE A NUMBER,CONCEPTUAL ERROR, -10 POINTS)                                                                           Answer[ 2  ]= increasing  (WRONG ANS., WRONG WAY -10 POINTS)                                                                           Answer[ 3  ]= 5 (WRONG ANS., WRONG WAY -10 POINTS)                                                                           Answer[ 4  ]= (1,1)                                                                           Answer[ 5  ]= (1,7)  (WRONG ANS., YOU HAD TO FIND THE VALUE, CONCEPTUAL ERROR, -10 POINTS)                                                                           Answer[ 6  ]= 2                                                                           Answer[ 7  ]=36/93 (WRONG ANS., BUT RIGHT WAY, -1 POINT)                                                                           Answer[ 8  ]= (0,-9)                                                                           Answer[ 9  ]= 15                                                                           Answer[ 10  ]= (1,7,6) SCORE: 59 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]= 6,  a[4]=1  , a[5]=1 , a[6]= 1, a[7]= 7,  a[8]= 5,  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 z(x,y) = x^1 + y^9 + z^6 +(1)*x*y*z^2= 3 + 1                                                               Here is how I do it (Explain everything) z(x,y) = 9y^8 + 6z^5 (dz/dy) + xz^2 + 2xyz(dz/dy) = 0                                                                                 Ans.: dz/dy = (9*y^8 + x*z^2)/ (-6*z^5 - 2*x*y*z)                                                             ---------------------------------------------                                                                              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,-1,9> Direction: <1,6,-1>                                                               Here is how I do it (Explain everything) f_x= 1 f_y= -1 f_z= 9                                                                                Ans.: increasing                                                             ---------------------------------------------                                                                              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 f(x,y,z) = x^3 + 6*y^3 +5*z^3                                                               Here is how I do it (Explain everything) PQ= <1-1,-1+1,3-1>= <0,0,2> U = PQ/ |PQ| u= <0,0,2>/ (2)= <0,0,1> = Directional Derivative = f_x*a+f_y*b+f_z*c = 0 + 0 + 5z^3 Evaluate at P=(1,-1,1) Directional Derivative at P = 5(1)^3 = 5                                                                                Ans.:5                                                             ---------------------------------------------                                                                              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) f_x= exp(x-1) - exp(e^y-1) f_xx= exp(x-1) f_y= -x*exp(y-1) = exp(y-1) f_yy= -x*exp(y-1) + exp(y-1) Set equal to zero and solve equations for x to obtain x=y  Plug x=y into equation to find x=1 then if x=1 and x=y then y=1 Saddle point (1,1)                                                               Here is how I do it (Explain everything)                                                                                Ans.: (1,1)                                                             ---------------------------------------------                                                                              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 + 7*y + 9 A = [1, 9] B = [6, 1] C = [1, 1]                                                               Here is how I do it (Explain everything) f_x= 1 f_y=7  Critical point at (1,7)                                                                                 Ans.: (1,7)                                                              ---------------------------------------------                                                                              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 (x^2-y^2)/(x-y)                                                               Here is how I do it (Explain everything) Simplify to x+y and take the limit by plugging in (1,1), 1+1 = 2 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) = (1,9*t,6*t^2) at point (1,0,0)                                                               Here is how I do it (Explain everything) K= | r’(t) x r’’(t)|/ |r’(t)|^3 r’(t)= <0,9,12t> r’’(t)- <0,0,12> | r’(t) x r’’(t)| = 108 |r’(t)|^3 = (3*sqrt(16*t^2 +9))^3 Solve fo rt: 9t=0 → t=0 Plug in t=0 into k equation =108/729=36/93 Answer: 36/93                                                             ---------------------------------------------                                                                              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(t) = (-sin(t) + 2t, 9cost(t) ) position= (0,9) velocity= (1,0)                                                               Here is how I do it (Explain everything) Take integral of a(t) and solve for constants: v(t)= (-cos(t) +2, 9*sint(t)) Take integral of v(t) and solve for constants: r(t)= (sin(t) + 2*t, 9*cos(t)) Plug in pi to this equation to get (0,-9) Answer: (0,-9)                                                             ---------------------------------------------                                                                              Problem 9:  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                                                    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,  7                                                                      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, 2                                                             How fast is the function changing right now? 15                                                               Here is how I do it (Explain everything) (dx)(dx/dt)+(dy)(dy/dt) (1)(1) +(7)(2) =15                                                             ---------------------------------------------                                                                             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=7 z=6                                                               Here is how I do it (Explain everything) Point of intersection is at (1,7,6) because since they are all constants they all intersect that that point.  Answer: (1,7,6)