MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME:Jiahe Li RUID:200005714 EMAIL:jl2669@roseprogram.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]=Does not exist (WONG ANS. and WAY, -10 POINTS) Answer[ 2 ]=Increasing Answer[ 3 ]=3 Answer[ 4 ]=(0,5) Answer[ 5 ]=0 Answer[ 6 ]=Does not exist Answer[ 7 ]=Does not exist Answer[ 8 ]=(0,0) Answer[ 9 ]=28 Answer[ 10 ]=(0,7,0) ----------------------------------------------------------------- Dear Jiahe, You forgot to change the 0's into 1's, but even then most of the answers are wrong. Please take the make-up. I will not give you a grade. 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]= 0 , a[3]= 0 , a[4]= 0 , a[5]= 0, a[6]= 5, a[7]= 7, a[8]= 1, a[9]= 4 -------------------------------------------- --------------------------------------------- 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^2+y^0+z^0+0*x*y*z^2 = 3+0 Here is how I do it (Explain everything) The equation is x^2-1=0, x=+-1, it doesn't contain z and y. So dz/dy doesn't exist. Ans.: doesn't exist. --------------------------------------------- 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)=<2,0,9>. Is f increasing or decreasing at the direction <2,0,0>? Here is how I do it (Explain everything) sqrt(2^2)=2 so the unit vector is <2/2,0,0>=<1,0,0> <2,0,9>.<1,0,0>=2>0 so f increasing at this direction 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 Find the directional derivative of the function f(x,y,z) x^3*5+y^3*0+z^3*1 At the point P=(1,-1,1) in the direction pointing to Q=(1,-1,3) Here is how I do it (Explain everything) The function is 5*x^3+z^3, <5*x^3,0, z^3> The gradient is <15*x^2,0,3*z^2> When P=(1,-1,1),the gradient is <15,0,3> The vector is <1-1, -1-(-1),3-1>=<0,0,2> sqrt(2^2)=2 The unit vector is <0,0,2/2>=<0,0,1> The directional derivative is <0,0,1>.<15,0,3>=3 Ans.:3 --------------------------------------------- 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)-x*exp(y-5) If there is no saddle point, write in the Answers: "Does Not exist". Explain what you are doing Here is how I do it (Explain everything) f_x(x,y)=exp(x)-exp(y-5) f_y(x,y)=-x*exp(y-5) If both of them are 0, x=y-5, x=0, y=5 So the critical point is (0,5) f_xx(x,y)=exp(x) f_yy(x,y)=-x*exp(y-5) f_xy(x,y)=-exp(y-5) For (0,5) D=1*0-1=0 So the test fails. By graphing, we found that this is a saddle point. Ans.:(0,5) --------------------------------------------- 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 7*y Find the ABSOLUTE MINIMUM VALUE of f(x,y) INSIDE the TRIANGLE whose VERTICES ARE A = [2, 0], B = [0, 0], C = [0, 5] Here is how I do it (Explain everything) Because there are only one variable, we only need to minimize it. When it's on AB, y minimize, it's 0 Ans.:0 --------------------------------------------- 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 [a[5],a[4]], or show that it does not exist Here is how I do it (Explain everything) The function is 0/0 doesn't exist. Ans: 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 Find the curvature of the curve r(t) = [2, 0, 0] At the point (2,0,0) Here is how I do it (Explain everything) Because this is not a curve, it's just a point. So there are no curvature. Ans: Does not exist --------------------------------------------- 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 [-2*sin(t), 0] At time t=0 its position is , [0, 0] and its velocity is , [2, 0] Where is it located at time , t = Pi Here is how I do it (Explain everything) int([-2*sin(t),0])=[2*cos(t),0]+C When t=0, [2,0]+C=[2,0],C=0 So the speed is [2*cos(t),0] int([2*cos(t),0])=[2*sin(t),0]+C When t=0, [0,0]+C=[0,0], C=0 So the position is [2*sin(t),0] When t=pi The position is (0,0) Ans: (0,0) --------------------------------------------- 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 certain function depends on variables x and y Right now the rate of change of the function with respect to x is, 0 and the rate of change of the function with respect to y is, 7 Both x and y depend on time Right now the rate of change of x with respect to time is, 2 and the rate of change of y with respect to time is, 4 How fast is the function changing right now? Here is how I do it (Explain everything) df/dx=0, df/dy=7, dx/dt=2, dy/dt=4 df/dt= df/dx*dx/dt+ df/dy*dy/dt=7*4=28 Ans:28 --------------------------------------------- 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 = 7, z = 0 Here is how I do it (Explain everything) The point that follows x = 0, y = 7, z = 0 is (0,7,0) Ans:(0,7,0)