University:
Shippensburg University of Pennsylvania
Course:
MAT 211 | Calculus I
Academic year:

2024
Views:

196

Pages:

2
Author:

Herbosambx

Question 1 Find two different sets of parametric equations for the rectangular equation 𝑦 = 6 𝑥 − 5 y=6x−5. Answer Step 1: Given: The rectangular equation is: 𝑦 = 6 𝑥 − 5 y=6x−5 We have to find the two different sets of parametric equations for a given rectangular equation. Step 2: Now consider, 𝑡 t is a parameter for the given rectangular equation. 𝑥 = 𝑡 ⟹ 𝑦 = 6 𝑡 − 5 x=t⟹y=6t−5 Then the set is: ( 𝑥 , 𝑦 ) ∣ 𝑥 = 𝑡 and 𝑦 = 6 𝑡 − 5 (x,y)∣x=t and y=6t−5 Consider: 𝑡 t is a parameter for the given rectangular equation. 𝑥 = 𝑡 + 2 x=t+2 𝑦 = 6 ( 𝑡 + 2 ) − 5 y=6(t+2)−5 𝑦 = 6 𝑡 + 12 − 5 y=6t+12−5 𝑦 = 6 𝑡 + 7 y=6t+7 Then the set is: ( 𝑥 , 𝑦 ) ∣ 𝑥 = 𝑡 + 2 and 𝑦 = 6 𝑡 + 7 (x,y)∣x=t+2 and y=6t+7 Hence, the two different sets of parametric equations for a given rectangular equation are: ( 𝑡 , 6 𝑡 − 5 ) ∣ 𝑥 = 𝑡 and 𝑦 = 6 𝑡 − 5 (t,6t−5)∣x=t and y=6t−5 and ( ( 𝑡 + 2 ) , ( 6 𝑡 + 7 ) ) ∣ 𝑥 = 𝑡 + 2 and 𝑦 = 6 𝑡 + 7 ((t+2),(6t+7))∣x=t+2 and y=6t+7 Question 2 Suppose that a nonhomogeneous system with 10 linear equations in 8 unknowns has a solution with 2 free variables. Is it possible to change some constants on the equations’ right-hand side to make the new system inconsistent? Answer Step 1: The system is equivalent to 𝐴 𝑥 = 𝑏 Ax=b. We have 10 linear equations in 8 unknowns. So, the dimension of 𝐴 = 10 × 8 A=10×8. Step 2: 𝐴 𝑥 = 𝑏 Ax=b has a solution with 2 free variables. So, dim ⁡ Nul 𝐴 = 2 dimNul A=2. So, dim ⁡ Column 𝐴 = 8 − 2 = 6 dimColumn A=8−2=6. Step 3: So, 𝐴 𝑧 → 𝐴 𝑧 is onto 𝑅 6 . So, Az→Az is onto R 6 . Hence, every right-hand side is obtainable. Result: No, it is not possible to change some constants on the equations’ right-hand side to make the new system inconsistent. Question 3 Graph each polynomial function. Estimate the 𝑥 x-coordinate at which the relative maxima and relative minima occur. State the domain and range for each function. 𝑓 ( 𝑥 ) = − 𝑥 3 + 4 𝑥 2 − 2 𝑥 − 1 f(x)=−x 3 +4x 2 −2x−1 Answer The relative maxima: between 2 and 3 The relative minima: between 0 and 1 The domain: 𝑅 R The range: 𝑅 R