MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Daniel Lim RUID: 201000735 EMAIL: d.lim@rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= -(1+xz^2/1+2xyz) (WRONG ANSWER, ANSWER SHOULD BE A NUMBER, -10 POINTS) Answer[ 2 ]= f is decreasing Answer[ 3 ]= 27/sqrt11 (WRONG ANS. STARTED CORRECTLY, BUT WRONG DIRECTION VECTOR, -5 POINTS) Answer[ 4 ]= Does Not Exist (WRONG ANS., STARTED CORRECTLY BUT THEN MESSED UP , -7 POINTS) Answer[ 5 ]= Absolute Min Does Not Exist (WRONG ANS., THERE IS ALWAYS ABSOLUTE MIN (and MAX) in a FINITE REGION, -10 POINTS) Answer[ 6 ]= Does Not Exist (WRONG ANS. WRONG WAY, -10 POINTS) Answer[ 7 ]= 2 Answer[ 8 ]= (2sin(Pi) - Pi)I + (cos(Pi))j (WRONG ANS., RIGHT WAY, -5 POINTS) Answer[ 9 ]= 37 Answer[ 10 ]= (1,7,1) ----------------------------------------------------------------- SCORE: 53 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]= 2 , a[2]= 1 , a[3]= 1 , a[4]= 1 , a[5]= 1 , a[6]= 1 , a[7]= 7 , a[8]= 3 , 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 x^2 + y + z + x*y*z^2 = 3 Here is how I do it (Explain everything) Dz/dy = -dy/dz. Dy = 1 + x*z^2. Dz = 1 + 2xyz Ans.: -(1+xz^2/1+2xyz) --------------------------------------------- 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)=<2,-1,9>. Is f increasing or decreasing at the direction <2,1,-1>? Here is how I do it (Explain everything) Find D_u = (2)(2) + (-1)(1) + (9)(-1) = -6 Ans.: f is 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 f(x,y,z) = x^3 + y^3 + 3z^3 Here is how I do it (Explain everything) Find the gradient vector = take derivative of f(x,y,z) = <3x^2,3y^2,9z^2> Find unit vector from the given direction = Q = (1,-1,3) = <1/sqrt11,-1/sqrt11,3/sqrt11> D_u = (3x^2)(1/sqrt11) + (3y^2)(-1/sqrt11) + (9z^2)(3/sqrt11) Plug in P=(1,-1,1) for (x,y,z). Ans.: D_u = 27/sqrt11 --------------------------------------------- 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) Find f_x, f_y, f_xx, f_yy, f_xy f_x = exp(x-1) - exp(y-1) f_y = exp(x-1) - exp(y-1)(x-1) f_xx = exp(x-1) f_yy = exp(y-1)(x-1) f_xy = 0 Find the value of D. D = f_xx * f_yy - f_xy^2 D = 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 A = [2,1] B = [1,1] C = [1,1]. f(x,y) = x + 7y + 1 Here is how I do it (Explain everything) First find f_x, f_xx, f_y, f_ty, and f_xy f_x = 1 f_xx = 0 f_y = 7 f_yy = 0 f_xy = 0 The three vertices of then triangle will be bounds for where the absolute minimum value will be found. However my RUID makes the vertices B and C the same point. Ans.: f_x and f_y does not yield critical points, therefore absolute min DNE --------------------------------------------- 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) at point (1,1) Here is how I do it (Explain everything) Limit at given equation equals 0/0 let y = cx where c is some random value F(x,y) = (x^2 - c^2*x^2)/(x - cx) Factor out the x and simplify to (x-x*c^2)/(1-c) Limit may exist, but is dependent on the slope c Answer: Therefore limit Does Not Exist --------------------------------------------- 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) = [2,t,t^2]. At point (2,0,0) Here is how I do it (Explain everything) Find r'(t) = [0,1,2t] Find r''(t) = [0,0,2] Take the cross product of r'(t) and r''(t) = (2,0,0) Magnitude of value above = 2 Magnitude of r'(t) = sqrt(4t^2 + 1). -> (4t^2 + 1)^3/2 Plug in 0 for t Answer = 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 A = [-2sint,-cost] Here is how I do it (Explain everything) Acceleration = Derivative of Velocity so Int(A) = 2cost - sint + c At t=0 velocity = 1 so c = -1 Velocity = (2cost - 1)I - (sint)j Take int of Velocity to find position Position = (2sint - t)i + (cost)j + c At t=0 position is (0,1) so c = 0 Plug in Pi for t in position equation. --------------------------------------------- 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) a[5] = 1, a[7] = 7, a[1] = 2, a[9] = 5 x and y = some function of t df/dt = (df/dx)(dx/dt) + (df/dy)(dy/dt) df/dt = (1)(2) + (7)(5) = 37 --------------------------------------------- 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 = 1 Here is how I do it (Explain everything) The planes are all a singular value, so the point of intersection is (1,7,1)