University:
Florida Gateway College
Course:
MAC 2312 | Calculus II
Academic year:

2024
Views:

112

Pages:

3
Author:

Ruby Bates

Question 1 Find the linearization 𝐿 ( 𝑥 ) L(x) of the function at 𝑎 a. 𝑓 ( 𝑥 ) = 𝑥 4 + 3 𝑥 2 , 𝑎 = − 1 f(x)=x 4 +3x 2 ,a=−1. Answer The equation achieves linearization by tangent line approximation: 𝐿 ( 𝑥 ) = 𝑓 ( 𝑎 ) + 𝑓 ′ ( 𝑎 ) ( 𝑥 − 𝑎 ) L(x)=f(a)+f ′ (a)(x−a) The derivative of 𝑓 ( 𝑥 ) f(x) is: 𝑓 ′ ( 𝑥 ) = 4 𝑥 3 + 6 𝑥 f ′ (x)=4x 3 +6x At 𝑎 = − 1 a=−1: 𝑓 ( − 1 ) = ( − 1 ) 4 + 3 ( − 1 ) 2 = 4 f(−1)=(−1) 4 +3(−1) 2 =4 𝑓 ′ ( − 1 ) = 4 ( − 1 ) 3 + 6 ( − 1 ) = − 4 − 6 = − 10 f ′ (−1)=4(−1) 3 +6(−1)=−4−6=−10 Substitute into the linearization formula: 𝐿 ( 𝑥 ) = 4 − 10 ( 𝑥 + 1 ) L(x)=4−10(x+1) Simplify: 𝐿 ( 𝑥 ) = − 6 − 10 𝑥 L(x)=−6−10x Question 2 Determine whether the given (2×3)(2×3) system of linear equations represents coincident planes (same plane), two parallel planes, or two planes whose intersection is a line. If it is a line, give parametric equations for the line: Equations: 2 𝑥 1 + 𝑥 2 + 𝑥 3 = 3 2x 1 +x 2 +x 3 =3 − 2 𝑥 1 + 𝑥 2 − 𝑥 3 = 1 −2x 1 +x 2 −x 3 =1 Answer The system of equations corresponds to two planes. Let 𝑛 1 = ( 2 , 1 , 1 ) n 1 =(2,1,1) and 𝑛 2 = ( − 2 , 1 , − 1 ) n 2 =(−2,1,−1), which are normal vectors. The planes are not parallel because 𝑛 1 n 1 and 𝑛 2 n 2 are not multiples of each other. Solving for their intersection: Replace 𝑥 3 = 𝑡 x 3 =t in the system: 2 𝑥 1 + 𝑥 2 + 𝑡 = 3 2x 1 +x 2 +t=3 − 2 𝑥 1 + 𝑥 2 − 𝑡 = 1 −2x 1 +x 2 −t=1 Add the equations: 2 𝑥 2 = 4 ⟹ 𝑥 2 = 2 2x 2 =4⟹x 2 =2 Substitute 𝑥 2 = 2 x 2 =2: 2 𝑥 1 + 2 + 𝑡 = 3 ⟹ 2 𝑥 1 = 1 − 𝑡 ⟹ 𝑥 1 = 1 2 − 𝑡 2 2x 1 +2+t=3⟹2x 1 =1−t⟹x 1 = 2 1 − 2 t Parametric equations of the line: 𝑥 1 = 1 2 − 𝑡 2 , 𝑥 2 = 2 , 𝑥 3 = 𝑡 x 1 = 2 1 − 2 t ,x 2 =2,x 3 =t Question 3 If a system of linear equations has infinitely many solutions, it is called a dependent system. If a system has no solution, it is called an inconsistent system. Question 4 What is a system of linear equations in three variables? Answer A system of linear equations in three variables is a set of three linear equations with three unknowns, representing planes in a 3D space. For example: 2 𝑥 + 4 𝑦 + 7 𝑧 = 1 , 3 𝑦 + 5 𝑧 = − 12 , − 𝑥 + 3 𝑦 − 3 𝑧 = 16 2x+4y+7z=1,3y+5z=−12,−x+3y−3z=16 Question 5 Physical meaning of the null space of a matrix. Answer If 𝐴 A is your matrix, the null space is the set of all vectors 𝑣 v such that 𝐴 ⋅ 𝑣 = 0 A⋅v=0. It represents the solutions to the homogeneous equation 𝐴 ⋅ 𝑣 = 0 A⋅v=0. It’s the set of vectors that lose their identity when transformed by 𝐴 A. Example 1 (Intuition): Imagine a map at the entrance of a forest. You apply directions to follow trails, but some trails lead you back to the entrance. These trails represent the null space of the directions. Example 2: The null space complements the rank of the matrix, as Rank + Nullity = Total Columns Rank+Nullity=Total Columns.