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

2024
Views:

263

Pages:

4
Author:

Herbosambx

Question 1 Solve the equation. 20 𝑥 3 + 8 𝑥 2 − 35 𝑥 − 14 = 0 20x 3 +8x 2 −35x−14=0 Answer 1 Step 1: Solve the equation. 20 𝑥 3 + 8 𝑥 2 − 35 𝑥 − 14 = 0 20x 3 +8x 2 −35x−14=0 Factoring step by step: ⇒ ( 5 𝑥 + 2 ) ( 4 𝑥 2 − 7 ) = 0 ⇒(5x+2)(4x 2 −7)=0 Solutions: 𝑥 = − 2 5 , 𝑥 = ± 7 2 x=− 5 2 ,x=± 2 7 Answer 2 Using the Rational Root Theorem and quadratic formula: Roots: 𝑥 = − 2 5 , 𝑥 = 7 2 , 𝑥 = − 7 2 x=− 5 2 ,x= 2 7 ,x=− 2 7 Question 2 Solve the equation. ∣ 4 𝑥 − 1 ∣ = 7 ∣4x−1∣=7 Answer 1 Split into two cases: 4 𝑥 − 1 = 7 ⇒ 𝑥 = 2 4x−1=7⇒x=2 4 𝑥 − 1 = − 7 ⇒ 𝑥 = − 3 2 4x−1=−7⇒x=− 2 3 Solutions: 𝑥 = 2 , 𝑥 = − 3 2 x=2,x=− 2 3 Question 3 With the given general solution, locate a linear homogeneous constant-coefficient equation. 𝑦 ( 𝑥 ) = ( 𝐴 + 𝐵 𝑥 + 𝐶 𝑥 2 ) 𝑒 2 𝑥 y(x)=(A+Bx+Cx 2 )e 2x Answer Given solution corresponds to characteristic equation: ( 𝑟 − 2 ) 3 = 0 (r−2) 3 =0 Differential equation: 𝑦 ( 3 ) − 6 𝑦 ′ ′ + 12 𝑦 ′ − 8 𝑦 = 0 y (3) −6y ′′ +12y ′ −8y=0 Question 4 Write an equation for each transformation of 𝑦 = 𝑥 y=x: vertical compression by a factor of 1 / 4 1/4. Answer Vertical compression means: 𝑦 = 1 4 𝑥 y= 4 1 x Question 5 How do you solve the equation sin ⁡ 2 𝑥 − sin ⁡ 𝑥 = 0 sin2x−sinx=0 on the interval [ 0 , 2 𝜋 ] [0,2π]? Answer Given: sin ⁡ 2 𝑥 − sin ⁡ 𝑥 = 0 sin2x−sinx=0 Using identities and solving step-by-step, solutions are: 𝑥 = 0 , 𝜋 , 𝜋 3 , 5 𝜋 3 x=0,π, 3 π , 3 5π