MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Krithika Patrachari RUID: 200006996 EMAIL: kp951@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= -1 dz/dy = ------------- (1 + 4*x*z) Answer[ 2 ]= Decreasing Answer[ 3 ]= 51/sqrt(11) Answer[ 4 ]= dne Answer[ 5 ]= 1 Answer[ 6 ]= Answer[ 7 ]= Answer[ 8 ]= Answer[ 9 ]= Answer[ 10 ]= ----------------------------------------------------------------- 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]= 1, a[3]= 1, a[4]= 1 , a[5]= 1, a[6]= 6, a[7]= 9, a[8]= 9, 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^2+y^1+z^1+2*x*y*z^2 = 3+1 Here is how I do it (Explain everything) First, I differentiate with respect to x and make sure that I use dz/dy when taking the derivative of a z variable I treat all x variables as a constant and I get 1 + dz/dy + 4*x*z*dz/dy =0 Next, I isolate dz/dx on one side of the equation dz/dy + 4*x*z*dz/dx = -1 I solve for dz/dx by factoring the dz/dx out of the left side and dividing the right side by the numbers that are left when dz/dx is factored dz/dy * (1 + 4*x*z) = -1 -1 dz/dy = ------------- (1 + 4*x*z) Ans.: -1 dz/dy = ------------- (1 + 4*x*z) --------------------------------------------- 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,-1,9+2>. Is f increasing or decreasing at the direction <2,1,-1>? Here is how I do it (Explain everything) First, I find the magnitude of the vector <2,1,-1> by taking the square root of the components squared and added sqrt(2^2 + 1^2 + (-1)^2) sqrt(4 + 1 + 1) = sqrt(6) I find the unit vector by dividing the direction vector by the magnitude/direction u= <2/sqrt(6) , 1/sqrt(6), -1/sqrt(6)> Next, I find the dot product of grad(f)(P) and u <2,-1,11> . <2/sqrt(6) , 1/sqrt(6), -1/sqrt(6)> = (2*2/sqrt(6)) + (-1*1/sqrt(6)) + (11*-1/sqrt(6)) = (4/sqrt(6)) + (-1/sqrt(6)) + (-11/sqrt(6)) = -8/sqrt(6) Since the result is negative, the it indicates that the directional derivative is in the negative direction and is there for decreasing. Ans.: 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 Find the directional derivative of the function f(x,y,z) x^3*6+y^3*1+z^3*9 At the point P=(1,-1,1) in the direction pointing to Q=(1,-1,3) Here is how I do it (Explain everything) First, to find the gradient of the function f(x,y,z), I find the partial derivatives of the function d/dx = 18*x^2 d/dy = 3*y^2 d/dz = 27*z^2 The grad(f) = <18*x^2, 3*y^2, 12*z^2> Next I find the magnitude of the directional vector Q sqrt (1^2 + -1^2 + 3^2) = sqrt(11) Now, I divide the vector Q by its magnitude to get the unit vector u = <1/sqrt(11) , -1/sqrt(11) , 3/sqrt(11)> I find the grad(f)(P) or grad(f)(1,-1,1) grad(f)(1,-1,1) = <18*1^2, 3*-1^2, 12*1^2> = <18, 3, 12> I take the dot product of u and grad(f)(1,-1,1) <1/sqrt(11) , -1/sqrt(11) , 3/sqrt(11)> . <18, 3, 12> (18*1/sqrt(11)) + (3*-1/sqrt(11)) + (12*3/sqrt(11)) = 51/sqrt(11) Ans.: 51/sqrt(11) --------------------------------------------- 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-1)-(x-1)*exp(y-6) 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) I find the partial derivatives d/dx = exp(x-1) d/dy = -x*exp(y-6) + exp(y-6) I find the second partial derivatives f_xx = exp(x-1) f_xy = 0 f_yy = -x*exp(y-6) + exp(y-6) I set d/dx and d/dy =0 d/dx = exp(x-1) = 0 ---> x= dne d/dy = -x*exp(y-6) + exp(y-6) ---> dne Ans.: dne --------------------------------------------- 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 1*x + 9*y + 1 Find the ABSOLUTE MINIMUM VALUE of f(x,y) INSIDE the TRIANGLE whose VERTICES ARE A = [2, 1], B = [1, 1], C = [1, 6] Here is how I do it (Explain everything) 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 Here is how I do it (Explain everything) --------------------------------------------- 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 Here is how I do it (Explain everything) --------------------------------------------- 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) --------------------------------------------- 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 Here is how I do it (Explain everything)