MIDTERM 1 FOR Multivariable Calculus, Math 251(22-24), FALL 2020, Dr. Z. NAME: Sai Embar RUID: 193003278 EMAIL: sai.embar@rutgers.edu BELOW WRITE THE LIST OF THE ANSWERS Answer[ 1 ]= -2 Answer[ 2 ]= decreasing Answer[ 3 ]= 60/sqrt(8) (WRONG ANS., RIGHT WAY, CARELESS ERROR, -3 POINTS) Answer[ 4 ]= (1,3) Answer[ 5 ]= 14 Answer[ 6 ]= 2 Answer[ 7 ]= 54/729 Answer[ 8 ]= (0,-9) Answer[ 9 ]= 17 Answer[ 10 ]= (1,2,3) ----------------------------------------------------------------- SCORE: 97 POINTS (out of 100) GOOD JOB! 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]= 1 , a[5]= 1 , a[6]= 3 , a[7]= 2 , a[8]= 7 , a[9]= 8 -------------------------------------------- --------------------------------------------- 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^1+y^9+z^3+1*x*y*z^2 = 3+1 Here is how I do it (Explain everything) The shortcut formula for implicit differentiation for dz/dy = -(F_y)/(F_z) F_y (partial derivative) = 9y^8 + x*z^2 F_z = 3*z^2 + 2*x*y*z dz/dy = -(9y^8 + xz^2)/(3z^2+2xyz) To find dz/dy at a specific point, plug in the point into dz/dy dz/dy at (1,1,1) = (9*1^8 + 1*1^2)/(3*1^2 + 2*1*1*1) = -10/5 = -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 Suppose that grad(f)(P)=<1,-1,4>. Is f increasing or decreasing at the direction <1,3,-1>? Here is how I do it (Explain everything) First we can label the direction vector v = <1,3,-1> Next, we take the magnitude to find the unit vector of v. ||v|| = sqrt(1^2 + 3^2 + (-1)^2) = sqrt(11) u_v (unit vector) = <1/sqrt(11), 3/sqrt(11), -1/sqrt(11)> Next, we find the directional derivative D_u(f)(P) = grad(f)(P) *(dot product) u_v = <1,-1,4> * (dot product) <1/sqrt(11), 3/sqrt(11), -1/sqrt(11)> = -6/sqrt(11) -----> Negative directional derivative means f is decreasing at the direction <1,3,-1> 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*3+y^3*3+z^3*7 Here is how I do it (Explain everything) First find F_x, F_y, and F_z F_x = 9x^2 F_y = 9y^2 F_z = 21z^2 next find grad(f) = <9x^2, 9y^2, 21z^2> next, plug in the point P to find grad(f)(P) = <9, -9, 21> next, find vector v = PQ = <0,-2,2> find magnitude of v = ||v|| = sqrt(0^2 + (-2)^2 + 2^2) = sqrt(8) unit vector of v = u_v = <0, -2/sqrt(8), 2/sqrt(8)> Next, we find the directional derivative D_u(f)(P) = grad(f)(P) *(dot product) u_q = <1,-9,21> * (dot product) <0, -2/sqrt(8), 2/sqrt(8)> = 60/sqrt(8) Ans.: 60/sqrt(8) --------------------------------------------- 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-3) 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) First find F_x and F_y F_x = exp(x-1)-(x-1)*exp(y-3) F_y = -exp(y-3)*(x-1) Set both equations equal to 0. We can then solve for x in the F_y equation and we find that x = 1. Plug x back into the F_x equation to solve for y, which gives y = 3. The critical point is (1,3). Find F_xx, F_xy, and F_yy F_xx = exp(x-1) F_xy = -exp(y-3) F_yy = -exp(y-3)*(x-1) Next, find the discriminant, D = F_xx*F_yy - (F_xy)^2 = exp(x-1)*-exp(y-3)*(x-1)-(-exp(y-3))^2 Calculate it at (1,3) = -1 Since the discriminant D < 0, (1,3) is a saddle point Ans.: (1,3) --------------------------------------------- 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 + 2*y + 9 Find the ABSOLUTE MINIMUM VALUE of f(x,y) INSIDE the TRIANGLE whose VERTICES ARE A = [1, 9], B = [3, 1], C = [1, 3] Here is how I do it (Explain everything) Since it is a linear function, the finalists are automatically the vertices so f(1,9) = 1+2(9) + 9 = 28 f(3,1) = 3 + 2*1 + 9 = 14 f(1,3) = 1 + 2(3) + 9 = 16 The absolute minimum is 14. Ans.: 14 --------------------------------------------- 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*1^2-y^2*1^2)/(x*1-y*1) 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) limit of (x^2 - y^2)/(x-y) as (x,y) goes to (1,1) we can factor the numerator. limit of ((x-y)(x+y))/(x-y) as (x,y) goes to (1,1) we can cancel the (x-y) from the numerator and denominator limit of x + y as (x,y) goes to (1,1) Now we can just plug in the point to get 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, 9*t, 3*t^2] At the point (1,0,0) Here is how I do it (Explain everything) first find r'(t) and r''(t) r'(t) = <0,9,6t> r''(t) = <0,0,6> do cross product of r'(t) x r''(t) After doing it you get r'(t) x r''(t) = 54 i. Find magnitude of r'(t) = 9 at (1,0,0) Find mangitude of r'(t) x r''(t) = 54 Plug into curvature formula K(t) = ||r'(t) x r''(t)||/(||r'(t)||^3) = 54/729 Ans: 54/729 --------------------------------------------- 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 [-1*sin(t), -9*cos(t)] At time t=0 its position is , [0, 9] and its velocity is , [1, 0] Where is it located at time , t = Pi Here is how I do it (Explain everything) first find velocity r'(t) by integrating acceleration function integrating r''(t) = r'(t) = Next, integrate r'(t) to find r(t), the position vector integrating r'(t) = r(t) = plug in time = Pi r(Pi) = =<0,-9> = (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 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, 2 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, 8 How fast is the function changing right now? Here is how I do it (Explain everything) df/dx = 1, df/dy = 2, dx/dt = 1, dy/dt = 8 Formula to find df/dt = (df/dx)*(dx/dt) + (df/dy)*(dy/dt) df/dt = 1*1 + 2*8 = 1 + 16 df/dt = 17 Ans: 17 --------------------------------------------- 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 = 2, z = 3 Here is how I do it (Explain everything) The traditional way of solving for the point of intersection between three planes is by setting up a system of equations x - 1 = 0 y - 2 = 0 z - 3 = 0 Usually, this is followed up with solving for the three variables x,y, and z using the three equations, We can solve these three equations which gives us, x = 1, y = 2, and z = 3. In this case, the values of x,y, and z are already given to us This means that the point of intersection is simply (1,2,3) because the values are already solved and there is no need for a system of equations to set up. Ans: (1,2,3)