MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Khush Tated RUID: 192002457 EMAIL:kt514@scarletmail.rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= -2.5 Answer[ 2 ]= Decreasing Answer[ 3 ]= 15 Answer[ 4 ]= Does not exist (WRONG ANS., BUT OTHERWISE PERFECT, NEGATIVE D MEANS IT IS A SADDLE POINT, -3 POINTS) Answer[ 5 ]= 18 (WRONG ANS., RIGHT WAY! CARELESS ERROR, f(2,1)=15, -3 POINTS) Answer[ 6 ]= 0 (WRONG ANS. WRONG WAY, -10 POINTS) Answer[ 7 ]= 4/81 Answer[ 8 ]= <0, -9> Answer[ 9 ]= 29 Answer[ 10 ]= (1, 4, 2) SCORE: 84 POINTS (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] = 2, a[4] = 1, a[5] = 1, a[6] = 2, a[7] = 4, a[8] = 5, a[9] = 7 -------------------------------------------- --------------------------------------------- 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 x+y^9+z^2+1*x*y*z^2 = 3+1 Here is how I do it (Explain everything) We need to differentiate everything with respect to y. This becomes x + y^9 + x^2 + 1 + x*y*z^2 = 4 When differentiated with respect to y, this becomes 9y^8 + 2z * dz/dy + 2*x*y*z*dz/dy + xz^2 = 0 Now solve for dz/dy -(9y^8 + x*z^2)/(2z + 2*x*y*z) =-10/4 = -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 grad(f)(P)=<1,-1,6> Direction: <1, 2, -1> Here is how I do it (Explain everything) Take the dot product of the 2 vectors 1 -2 -6 = -1 -6 = -7 Due to a negative dot product, this is 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 x^3*2+y^3*2+z^3*5 At the point P=(1,-1,1) in the direction pointing to Q=(1,-1,3) Here is how I do it (Explain everything) Gradient is <6x^2, 6y^2, 15z^2> PQ is <0, 0, 2> Unit direction is <0, 0, 1> Gradient at point P is <6, 6, 15> Dot product of gradient and unit direction is 15. Ans.:15 --------------------------------------------- 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-a1)*exp(y-2) Here is how I do it (Explain everything) Find Fx, Fy, Fxy Fx = (exp)(x-1) - exp(y-2) Fy = (x-1)(exp(y-2)) Fxy = -exp(y-2) Critical point after setting Fx and Fy to 0 is 1,2 Fx(1,2) = 0 Fy(1,2) =0 Fxy(1, 2) = -1 Because the discriminant is -1, there is no saddle point. Ans.: Does not exist --------------------------------------------- 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 Function: x + 4y + 9 =f(x,y) A = (1, 9) B = (2, 1) C = (1, 2) Here is how I do it (Explain everything) Fx = 1 Fy = 4 Fxy = 0 There are no critical points. We can look at the points located on the triangles The absolute minimum value in this triangle is at (1, 2) 1 + 4(2) + 9. Ans.: 18 --------------------------------------------- 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 (x^2*1-y^2*1)/(x-y) as it goes to (1,1) Here is how I do it (Explain everything) Subsitute (1, 1) = 0/0. Indeterminate Let y = cx (x^2 - c^2*x^2) / (x - c*x) / (x- c*x) Simplify x*(x-c) / (1-c). This limit depends on x. Let's go to polar lim r goes to 0 of r^2 cos^2(theta) - r^2*sin^2(theta) / (r*costheta - r*isntehta) = 0 Ans : 0 --------------------------------------------- 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, 9*t, 2*t^2] at (1, 0, 0) Here is how I do it (Explain everything) Find r'(t) = <0, 9, 4t> = <0, 9, 0> at relevant point r''(t) = <0, 0, 4> Cross product magnitude between r'(t) x (r'(t)) = <36, 0, 0> Magnitude of r'(t) = 9 36 / 9^3 = 4/81 Ans: 4/81 --------------------------------------------- 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 ACCELERATION = <-1*sin(t), -9*cos(t)> velocity = <1, 0> time = pi Here is how I do it (Explain everything) Integrate ACCELERATION with respect to t. velocity = + c = < 1, 0>. C = 0 Integreate velocity with respect to t. Position = + c = <0, 9> So at t = pi, position is <0, -9> Ans: <0, -9> --------------------------------------------- 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 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, 4 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, 7 Here is how I do it (Explain everything) This is a chain rule question. Multiply dz/dx *dx/dt = 1*1 Multiply dz/dy * dy/dt = 4*7 Add them together 29. 1 + 28 = 29 Ans: 29 --------------------------------------------- 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 = 4 z = 2 Here is how I do it (Explain everything) Because the x plane spans out in every direction from x = 2, y plane spans out from y = 4, and z spans out from z = 2, the point of intersection if (1, 4, 2) Ans: (1, 4, 2)