MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Aayushi Kasera RUID: 193008020 EMAIL: amk408@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]=-2 Answer[ 2 ]= decreasing Answer[ 3 ]= 6 Answer[ 4 ]=does not exist Answer[ 5 ]=-10 Answer[ 6 ]=2 Answer[ 7 ]=54/729 Answer[ 8 ]=(0,-1) Answer[ 9 ]=10 Answer[ 10 ]=(1,1,1) ----------------------------------------------------------------- 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]= , a[2]= , a[3]= , a[4]=0 , a[5]= , a[6]= , a[7]= , a[8]= , a[9]= 1,9,3,1,1,8,1,2,1 -------------------------------------------- --------------------------------------------- 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^3+xyz^2=4 Here is how I do it (Explain everything) Ans.: x+y^9+z^3+xyz^2=4 x+y^9+z^3+xyz^2=3 9y^8+3z^2(dz/dy)+2xyz(dz/dx)+xz^2=0 Dz/dy=-9y^8-xz^2 Dz/dy=(-9y^8-xz^2)/(3z^2+2xyz) =(-9-1)/(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> , v=<1,3,-1> Here is how I do it (Explain everything) Ans.: grad(f)(P)=<1,-1,3> , v=<1,3,-1> |v|=sqrt(11) u=<1/sqrt(11),3/sqrt(11),-1/sqrt(11)> Dot product of the grad(F) and direction vector <1,-1,3>.<1/sqrt(11),3/sqrt(11),-1/sqrt(11)> =1/sqrt(11)-3/sqrt(11)-3/sqrt(11) -8/sqrt(11) Since the dot product is negative, it 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^24+y^9+z^6 Here is how I do it (Explain everything) Ans.: F_x= 24x^23 F_y=9y^8 f_z=6z^5 grad_f=<24x^23,9y^8,6z^5> grad_f(1,-1,1)=<24,-9,6> v=Q-P=<0,0,2> |v|=2 u=<0,0,1> Take the dot product of grad_f.u 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 f(x,y)=exp(x-1)-(x-1)*exp(y-8)) Here is how I do it (Explain everything) Ans.: F_x=exp(x-1)-exp(y-8) f_y=-(x-1)exp(y-8) Find critical points f_x=0 x-1=y-8 x-y=-7 x+7=y y=8 f_y=0 x=1 CP=(1,8) f_xx=exp(x-1) =1 f_yy=-(x-1)exp(y-8) =0 f_xy=-exp(y-8) =-1 D=1 Since D>0,f_xx>0 (1,8) is a local minimum --------------------------------------------- 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 Let f(x,y) be the function x + y + 9 Find the ABSOLUTE MINIMUM VALUE of f(x,y) INSIDE the TRIANGLE whose VERTICES ARE A = [1,9], B = [3,1], C = [1,8] Here is how I do it (Explain everything) f_x=y f_y=x f_xy=1 f(x,1)=x+10 Points can be -10, 11, 18 f(1,y)=y+10 Points are, -10, 11, 18 f(x,x)=2x+9 So, points are x=-9/2, 11, 25 Ans.: --------------------------------------------- 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 Let f(x,y) be the function (x^2-y^2)/(x-y) Find the LIMIT of f(x,y) as (x,y) goes to the point [1,1], or show that it does not exist Here is how I do it (Explain everything) f(x,y)=(x^2-y^2)/(x-y) f(x,y)=x+y lim(x,y)->(1,1)(x+y) =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 Find the curvature of the curve r(t) = [1, 9t, 3t^2] At the point (1,0,0) Here is how I do it (Explain everything) R'(t)=<0,9,0> R"(t)=<0,0,0> R'(t)xR"(t) <0,9,0>x<0,0,6> =<54,0,0> |R'(t)xR"(t)|=54 |R'(t)|=9 |R'(t)|^3=9^3 Curvature kuppa= 54/9^3 --------------------------------------------- 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 particle is moving in the plane with ACCELERATION given by [-sin(t), -9cos(t)] At time t=0 its position is , [0, 9] and its velocity is , [1, 0] Where is it located at time , t = Pi Here is how I do it (Explain everything) a(t)=-sin(t)i-9cos(t)j Integrate a(t)=v(t)=cos(t)i-9sin(t)j+c v(0)=i v(0)=cos(0)i-9sin(0)j+c i=I+c c=0 v(t)=cos(t)i-9sin(t)j x(t)= integrate v(t) integrate v(t)= x(t)=sin(t)i+9cos(t)j+c x(0)=9j x(0)=j+c c=8j x(t)=sin(t)i+(9cos(t)+8)j+c sin(pi)I+(9cos(pi)+8)j =-1j --------------------------------------------- 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) Df/dx=1 Dy/dt=1 Df/dy=1 Dy/dt=9 Ans:1+9 10 --------------------------------------------- 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 = 1 Here is how I do it (Explain everything) a=<1,0,0> b=<0,1,0> c=<0,0,1> aXb=0