MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Yeram Sarah Jung RUID: 192004589 EMAIL: ysj7@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= -2.5 (CORRECT, 10 POINTS) Answer[ 2 ]= Answer[ 3 ]= 78/√11 (WRONG ANS. BUT STARTED CORRECTLY, 5 POINTS) Answer[ 4 ]= (1,4) (CORRECT, 10 POINTS) Answer[ 5 ]= Answer[ 6 ]= 2 (CORRECT, 10 POINTS) Answer[ 7 ]= Answer[ 8 ]= Answer[ 9 ]= Answer[ 10 ]= (1,5,2) (CORRECT, 10 POINTS) ----------------------------------------------------------------- SCORE: 45 POINTS (out of 100). You are welcome to take the make-up miderm. What you did you did well) 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]= 2 , a[4]= 1 , a[5]= 1 , a[6]= 4 , a[7]= 5 , a[8]= 8 , a[9]= 9 -------------------------------------------- --------------------------------------------- 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^9+z^2+1*x*y*z^2 = 4 Here is how I do it (Explain everything) dz/dy is equal to -fy/fz so we take the partial derivative fy from the equation which is 9y^8+xz^2 and divide it by the partial derivative fz which is 2z+2xyz and multiple it by -1. Since the question asks for dz/dy at the point (1,1,1) we plug in our values of x, y, and z which in this case is all 1 and it gives us -10/4 which is equal to -2.5. Ans.: -2.5 --------------------------------------------- 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,-1,7>. Is f increasing or decreasing at the direction <1,2,-1>? 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) 4x^3+2y^3+8z^3 at the point P=(1,-1,1) in the direction pointing to Q=(1,-1,3). Here is how I do it (Explain everything) In order to find the directional derivative given the parameters, I first found the partial derivatives fx, fy, and fz. It came out to 12x^2, 6y^2, and 24z^2 respectively giving grad(f) = <12x^2, 6y^2, 24z^2>. Then I calculate the unit vector |<1,-1,3>| which is equal to <1/√11, -1/√11, 3/√11>. Then I find the gradient vector at point P = (1,-1,1) and find that it is <12,6,24> and calculate the dot product of grad(f) and the unit vector which gives 78/√11, the directional derivative. Ans.: 78/√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-4). 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 first find the partial derivatives fx, fy, fxx, fxy, and fyy and set the derivatives I find for fx and fy equal to 0 to find the critical point. fx = exp(x-1)-exp(y-4) and fy = -xexp(y-4) + exp(y-4) so x = 1 and y = 4 making the critical point (1,4). I then find the discriminant at this point getting D = (1)(0) - (-1)^2 and since it equals -1 a negative value, (1,4) is a saddle point. Ans.: (1,4) --------------------------------------------- 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 + 5y + 9. Find the ABSOLUTE MINIMUM VALUE of f(x,y) INSIDE the TRIANGLE whose VERTICES ARE A = [1,9], B = [2,1], C = [1,4] 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 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) To find the limit I first factorized (x^2-y^2) to (x+y)(x-y) which allowed me to cancel out (x-y) from both the numerator and denominator leaving only (x+y) as the function. So from there I simply plug in [1,1] since it is the point (x,y) approach and I get 2 as the limit. 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 Find the curvature of the curve r(t) = [1, 9t, 2t^2] at the point (1,0,0). 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 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 = π? 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 A certain function depends on variables x and y. Right now the rate of change of the function with respect to x is 1 and the rate of change of the function with respect to y is 5. Both x and y depend on time. Right now the rate of change of x with respect to time is 1 and the rate of change of y with respect to time is 9. How fast is the function changing right now? 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 = 1, y = 5, z = 2. Here is how I do it (Explain everything) Since all three planes are constant values, the point of intersection will simply be (1,5,2) Ans.: (1,5,2)