MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Jinquan Lin RUID: 187007046 EMAIL: jl2339@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]=-(8*y^7)/(7*z^6) Answer[ 2 ]= Answer[ 3 ]=12 Answer[ 4 ]=(0,7) Answer[ 5 ]= Answer[ 6 ]=DOES NOT EXIST Answer[ 7 ]=k(t)=112/(sqrt(64+196*t^2))^3 Answer[ 8 ]=[0, -8] Answer[ 9 ]= Answer[ 10 ]=(0, 0, 7) TOO MANY ERROR, PLEAE TAKE 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]=8 , a[3]=7 , a[4]=0 , a[5]=0 , a[6]=7 , a[7]=0 , a[8]=4 , a[9]=6 -------------------------------------------- --------------------------------------------- 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 Find dz/dy at the point (1,1,1) if z(x,y) is given implicitly by the equation x^1+y^8+z^7+0*x*y*z^2 = 3+0 Here is how I do it (Explain everything) x+y^8+z^7= 3 differentiate with respect to y (x+y^8+z^7)'= 3 0+8*y^7+7*z^6*(dz/dy)=0 7*z^6*(dz/dy)=-8*y^7 solve for dz/dy using algebra dz/dy=-(8*y^7)/7*z^6 Ans.:dz/dy=-(8*y^7)/(7*z^6) --------------------------------------------- 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)=<1, 0, 2>. Is f increasing or decreasing at the direction <1, 7, 0>? Here is how I do it (Explain everything) Ans.: --------------------------------------------- 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 Find the directional derivative of the function f(x,y,z) x^3*7+y^3*7+z^3*4 At the point P=(1,-1,1) in the direction pointing to Q=(1,-1,3) Here is how I do it (Explain everything) v=PQ=<1-1, -1-(-1), 3-1>=<0, 0, 2> Find a vector v in the direction of PQ ||v||=sqrt(0^2+0^2+2^2)=2 Find unit vector u in the direction of v u=v/||v||=k=<0, 0, 1> fx(x,y,z)=21*x^2 Find the partial dierivates fy(x,y,z)=21*y^2 fz(x,y,z)=12*z^2 grad(f)=<21*x^2, 21*y^2, 12*z^2> Plug-in P(1,-1,1) for gradient grad(f)(1,-1,1)=<21, 21, 12> grad(f)(1,-1,1).u=<21, 21, 12>.<0, 0, 1>=12 Take the dot product Ans.: 12 --------------------------------------------- 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 Find a saddle point of the function f(x,y)= exp(x-0)-(x-0)*exp(y-7) exp(x)-(x)*exp(y-7) Here is how I do it (Explain everything) fx=exp(x)-exp(y-7) fy=-x*exp(7-y) fxx=exp(x) fxy=-exp(7-y) fyy=-x*exp(7-y) exp(x)-exp(y-7)=0, -x*exp(7-y)=0 x=0, y=7 =>(0,7) fxx(0,7)=1, fxy(0,7)=-1, fyy(0,7)=0 D=1*0-(-1)^2=-1 Therefore f(x,y) exists saddle point at (0,7) Ans.:(0,7) --------------------------------------------- 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 0*x + 0*y + 8 Find the ABSOLUTE MINIMUM VALUE of f(x,y) INSIDE the TRIANGLE whose VERTICES ARE A = [1, 8], B = [7, 0], C = [0, 7] Here is how I do it (Explain everything) Since the function f(x,y)=8 is a constant there is no minimum value of f(x,y) Ans.:No Solution --------------------------------------------- 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*0^2-y^2*0^2)/(x*0-y*0) Find the LIMIT of f(x,y) as (x,y) goes to the point [0,0], or show that it does not exist Here is how I do it (Explain everything) f(x,y)=(x^2*0^2-y^2*0^2)/(x*0-y*0) f(x,y)=(0-0)/(1-1)=0 The function does not exist Therefore there is no solution for this question with my RUID --------------------------------------------- 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, 8*t, 7*t^2] At the point (1,0,0) Here is how I do it (Explain everything) r(t)=i+8*t*j+7*t^2*k Compute r'(t) and r''(t) r'(t)=0+8*j+14*t*k r''(t)=14*k r'(t)xr''(t)= 112i Compute the cross product of r'(t) and r''(t) |r'(t)xr''(t)|=sqrt(112^2)=112 Find the magnitude of vector and r'(t) |r'(t)|=sqrt(8^2+(14*t)^2)=sqrt(64+196*t^2) k(t)=|r'(t)xr''(t)|/|r'(t)|^3 Use the formula for curvature k(t)=112/(sqrt(64+196*t^2))^3 Ans.:k(t)=112/(sqrt(64+196*t^2))^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 [-1*sin(t), -8*cos(t)] At time t=0 its position is , [0, 8] and its velocity is , [1, 0] Where is it located at time , t = Pi Here is how I do it (Explain everything) a(t)=-1*sin(t)*i-8*cos(t)*j int(a(t))=v(t)=cos(t)*i-8*sin(t)*j+C v(0)=cos(0)-8*sin(0)+C=[1,0] C=0 v(t)=(cos(t))*i-(8*sin(t))*j int(v(t))=p(t) p(t)=sin(t)+8*cos(t)+C p(0)=0*i+8*j+C=[0,8] C=0 p(t)=sin(t)*i+8*cos(t)*j p(pi)=-8*j =>[0, -8] --------------------------------------------- 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) --------------------------------------------- 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 Find the point of intersection of the three planes x = 0, y = 0, z = 7 Here is how I do it (Explain everything) The point of intersection is (0, 0, 7)