MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: RUID: EMAIL: BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]=-2 Answer[ 2 ]=decreasing Answer[ 3 ]=21 Answer[ 4 ]=(1,6) Answer[ 5 ]=-10 Answer[ 6 ]=2 Answer[ 7 ]=6 Answer[ 8 ]=(0,1) Answer[ 9 ]=6 Answer[ 10 ]=(1,1,3) WRONG ANSWERS #5 (AND WRONG WAY) : 10 points off #7 (DIVIDED BY r' insted of r'''): 5 points off #8 (ALSO A SERIOUS CONCEPTUAL ERROR) : 10 points off SCORE: 75 (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]=3 , a[4]=0 , a[5]=0 , a[6]=6 , a[7]=0 , a[8]=7 , 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 Equation is x^(1)+y^(9)+z^(3)+1*x*y*z^2=3+1 which is equal to x+y^9+z^3+zyx=4 Here is how I do it (Explain everything) 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 Ans.:-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)=<1,0,2> u=<1,3,0> Here is how I do it (Explain everything) 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 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 Function is 6x^3+3y^3+7z^3 Here is how I do it (Explain everything) Directional derivative=<18x^2,9y^2,21z^2> At point (1,-1,1), directional derivative =<18,-9,21> Pointing in Q-P, <0,0,2> directional derivative =<18,-9,21>.<0,0,2>/2 =21 Ans.:21 --------------------------------------------- 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-6) Here is how I do it (Explain everything) f_x=exp(x-1)-exp(y-6) f_y=(1-x)exp(y-6) The critical point after placing the derivative equal to zero, the critical point is (1,6) f_xx=exp(x-1)---->1 f_yy=(1-x)*exp(y-6)---->0 f_xy=-exp(y-6)---->-1 D=0-1=-1, which is less than 0. Local min=1 Local max=6 Ans.:(1,6) --------------------------------------------- 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=[1,9], B=[3,1], c=[1,6] f(x,y)=x+y+9 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, 13 f(1,y)=y+10 Points are, -10, 11, 19 f(x,x)=2x+9 So, points are x=-9/2, 11, 1 Ans.:-10 --------------------------------------------- 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 Function is (x^2-y^2)/(x-y) Limit goes from (1,1) Here is how I do it (Explain everything) function=(x+y)---->after simplyfying limit=1+1=2 --------Ans:= 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, 9t, 3t^2] at point (1,0,0) Here is how I do it (Explain everything) r'(t)=[0,9,6t]--->[0,9,0], magnitude of which is 9 r''(t)=[0,0,6] Curvature is (|r'(t)xr''(t)|)/|r'(t)^3| 54/9=6 --------------------------------------------- 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 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+c Position is (0,-1) --------------------------------------------- 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/dt=df/dx*dx/dt+df/dy*dy/dt df/dt=1*1+1*5=6 --------------------------------------------- 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=3 Here is how I do it (Explain everything) Intersection is <1,0,0>x<0,1,0>=<0,0,0> Point of interection is (1,1,3)