MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Fady Besada RUID: 192005137 EMAIL: frb37@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= (-5)/2 Answer[ 2 ]= f is decreasing Answer[ 3 ]= 9 Answer[ 4 ]= Does Not exist Answer[ 5 ]= 12 Answer[ 6 ]= 0 Answer[ 7 ]= 4/81 Answer[ 8 ]= <0,-9> Answer[ 9 ]= 8 Answer[ 10 ]= (1,1,2) ----------------------------------------------------------------- WRONG ANSWERS: #4 (f_xx is irrelevant for saddle points, started correctly, -7 points) #6 (COMPLETELY WRONG WAY (see answer key) but shown that you know about limits -8 points) SCORE: 85 (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]= 2, a[4]= 0, a[5]= 0, a[6]= 5, a[7]= 1, a[8]= 3, 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^2+x*y*z^2=4 Here is how I do it (Explain everything) # x+y^9+z^2+x*y*z^2=4 # (dz/dy) = -Fy/Fz # Fy = 9*y^8+x*z^2 # Fz = 2*z+2*x*y*z # At point (1,1,1), it evaluates to (-5)/2. Ans.: (-5)/2 --------------------------------------------- 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,-1,3> #Direction <1,2,-1> Here is how I do it (Explain everything) #Dot product of the gradient and direction vector is -4 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)=5*x^3+2*y^3+3*z^3 Here is how I do it (Explain everything) #f(x,y,z)=5*x^3+2*y^3+3*z^3 #f_x=15*x^2, f_y=6*y^2, f_z=9*z^2 #PQ = Q-P = <0,0,2> (which is the direction vector) #Unit direction vector: (1/2)*<0,0,2> = <0,0,1> #grad(f) at (1,-1,1) = <15,6,9> #<15,6,9>.<0,0,1> = 9 Ans.: 9 --------------------------------------------- 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 #f(x,y)=e^(x-1)-(x-1)*e^(y-5) Here is how I do it (Explain everything) #Find f_x, f_y, f_xx, f_xy, f_yy #Critical point (1,5) #No saddle points because second order partial derivative is 0 so 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 #f(x,y)=x+y+9 #A=(1,9), B=(2,1), C=(1,5) Here is how I do it (Explain everything) #f_x=1 #f_y=1 #Since there are no critical points, the absolute minimum value of f(x,y) will be the lowest of the y values of A,B,C. #Plugging B into f(x,y), we get: f(x,y)=2+1+9 #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 #f(x,y)=(x^2-y^2)/(x-y) Here is how I do it (Explain everything) #y=cx #lim(x,y)->(0,0) of (x^2-c^2*x^2)/(x-c*x) = x^2*(1-c^2)/x*(1-c) = x(1-c^2)/(1-c) = 0 #Since limit is 0, it may exist and equal 0 but we must use polar coordinates to further investigate. #lim(r->0) of (r^2*cos^2(theta) - r^2*sin^2(theta))/(r*cos(theta)-r*sin(theta)) = r*(cos^2(theta)-sin^2(theta)/(cos(theta)-sin(theta)) = 0 #The limit exists and is equal to 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> at (1,0,0) Here is how I do it (Explain everything) #r(t) = <1,9*t,2*t^2> at (1,0,0) #r'(t)=<0,9,4t> #r''(t)=<0,0,4> #t=0 #r'(0) x r''(0) = <36,0,0> #|<36,0,0>| = 36 #4/81 --------------------------------------------- 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) = r''(t) = <-sin(t), -9*cos(t)> #t=0, <0,9> [position] #t=0, <1,0> [velocity] Here is how I do it (Explain everything) #a(t) = r''(t) = <-sin(t), -9*cos(t)> #t=0, <0,9> [position] #t=0, <1,0> [velocity] #v(t)=r'(t)= + C #C = 0 in this scenario #v(t)=r'(t)= #r(t) = + C #C = 0 in this scenario #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 #a[5]=1 #a[1]=1 #a[7]=1 #a[9]=7 Here is how I do it (Explain everything) #Set up chain rule based off of the problem #a[5]*a[1]+a[7]*a[9] #1*1+1*7=1+7=8 --------------------------------------------- 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 = 1, z = 2 Here is how I do it (Explain everything) #These three planes intersect at point (1,1,2).