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Using the definition of the derivative and limit, compute the derivative of the determinant function on 2 × 2 matrices at the identity (which we consider as a subset of R 4 under the Euclidean norm).
Use the limit definition of partial derivatives to compute the partial derivative of the function f(x,y) = 6 - 6x + 5y - 3x2y at a point (3,4). a. Find f,(3,4). b. Find f(3,4). 1,(3,4)=0 (Simplify your answer.) 12(3,4)=0 (Simplify your answer.)
Use the limit definition of partial derivatives to compute the partial derivative of the function f(x,y) = 2 - 2x + 5y - 3x?y at a point (3.4). a. Find f (3.4). b. Find f (3.4). f|(3.4)=0 (Simplify your answer.) 13(3.4)=0 (Simplify your answer.)
2. Compute the determinant of the following matrices. (a) 2 -1 2 5 -4 A= 3 -11 9 0 (b) 1 2 1 2 1 A= -1 -1 2 1 1 2 (apply row reductions combined with cofactor expansion)
Let f(x) = (4x + 1)2 . Using the limit definition of a derivative f'(a) = limh→0 f(a + h) − f(a) /h find f'(0)
Compute the center of the group GL2(R) of invertible 2 x 2 matrices under multiplication.
*PLEASE DO IN MATHEMATICA* {:1, ifr+ 13. Consider the function f(x)- nction,f(x)-e-r/rifx#0 a. Plot the graph of this function using Mathematica. b. Use the limit definition of the derivative and LHopital's Rule to show that every higher-order derivative of f at r 0 vanishes. c. Find the MacLaurin series for f. Does the series converge to f? {:1, ifr+ 13. Consider the function f(x)- nction,f(x)-e-r/rifx#0 a. Plot the graph of this function using Mathematica. b. Use the limit definition of...
ANSWER c d e ONLY! No need to answer a and b Thanks 2. Determinant function onM 2 (a) Take A E M2. Consider the mapping volA: R2 x R2 - R, which is given by volA(v1, v2) olvA, 2A), for every v1, 02 E R2. Explain why volA is also a volume form (b) Explain why (use section (c) from question 1 above) there is a scalar α(A) E R such that VOLA-α(A) . voi We denote the scalar...
(4 points) Find the derivative of the following function $2 F(2) ds 4 + 754 using the appropriate form of the Fundamental Theorem of Calculus. F'(x) =
4. Consider a particle exposed to the following potential energy (we are using spherical coordinates r, and ф) Use the variation method and the following trial function: to estimate the ground state energy for this system.
Exercice 1 We consider the function f(x) = 2 #0 and for r > 0. let S, = {€ C/2 = r} with positive orientation. For 0 < <R, we denote by r the curve consisting of SRUT-R,-€) US, UL, R), where S = {z E C/121 = } with negative orientation. 1. Prove that o = [513)dz = [5(=)dz + [s()de – [ (dz + 1" $(x)dr.