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

2024
Views:

487

Pages:

3
Author:

Herbosambx

Question 1 Rewrite as an exponential equation: 𝑒 𝑧 = 2 e z =2 Answer Step 1 The relationship between logarithmic and exponential function is given by: log ⁡ 𝑏 𝑦 = 𝑥 ⟺ 𝑏 𝑥 = 𝑦 log b y=x⟺b x =y Where 𝑏 b is the base. To convert exponential to logarithmic function, find 𝑥 x, 𝑦 y, 𝑏 b, and substitute into the equation. Step 2 Given: 𝑒 𝑧 = 2 e z =2 Here 𝑏 = 𝑒 b=e, 𝑦 = 𝑥 y=x, 𝑥 = 2 x=2. So the logarithmic form is: ln ⁡ 2 = 𝑧 ln2=z Question 2 Solve the given system of equations by matrix equation: 5 𝑥 − 4 𝑦 = 4 5x−4y=4 3 𝑥 − 2 𝑦 = 3 3x−2y=3 Answer Step 1 Given system of equations: 5 𝑥 − 4 𝑦 = 4 5x−4y=4 3 𝑥 − 2 𝑦 = 3 3x−2y=3 The system of equations in matrix form can be written as: 𝐴 𝑋 = 𝐵 ⟹ 𝑋 = 𝐴 − 1 𝐵 AX=B⟹X=A −1 B Step 2 Here: 𝐴 = [ 5 − 4 3 − 2 ] , 𝐵 = [ 4 3 ] , 𝑋 = [ 𝑥 𝑦 ] A=[ 5 3 −4 −2 ],B=[ 4 3 ],X=[ x y ] Formula for 𝐴 − 1 A −1 is: 𝐴 − 1 = 1 det ⁡ ( 𝐴 ) adj 𝐴 A −1 = det(A) 1 adjA Now: det ⁡ ( 𝐴 ) = 5 ( − 2 ) − ( − 4 ) ( 3 ) = − 10 + 12 = 2 det(A)=5(−2)−(−4)(3)=−10+12=2 And: adj 𝐴 = [ − 2 − 4 − 3 5 ] adjA=[ −2 −3 −4 5 ] Thus: 𝐴 − 1 = 1 2 [ − 2 − 4 − 3 5 ] A −1 = 2 1 [ −2 −3 −4 5 ] Step 3 𝑋 = 𝐴 − 1 𝐵 X=A −1 B 𝑋 = 1 2 [ − 2 − 4 − 3 5 ] [ 4 3 ] X= 2 1 [ −2 −3 −4 5 ][ 4 3 ] 𝑋 = 1 2 [ ( − 2 ) ( 4 ) + ( − 4 ) ( 3 ) ( − 3 ) ( 4 ) + ( 5 ) ( 3 ) ] X= 2 1 [ (−2)(4)+(−4)(3) (−3)(4)+(5)(3) ] 𝑋 = 1 2 [ − 8 − 12 − 12 + 15 ] = 1 2 [ − 20 3 ] X= 2 1 [ −8−12 −12+15 ]= 2 1 [ −20 3 ] 𝑋 = [ − 10 1.5 ] X=[ −10 1.5 ] Step 4 The solution of the given system of equations is: 𝑥 = 2 , 𝑦 = 1.5 x=2,y=1.5 Question 3 Find a polar equation for the curve represented by the given Cartesian equation: 𝑦 = 2 y=2 Answer Step 1 Solution: The given Cartesian equation is: 𝑦 = 2 y=2 Finding the polar equation is necessary. Step 2 The given Cartesian equation is: 𝑦 = 2 y=2 From the relationship of Cartesian and polar coordinates, we have: 𝑦 = 𝑟 sin ⁡ 𝜃 y=rsinθ Plug the above values into the equation of the curve: 𝑟 sin ⁡ 𝜃 = 2 rsinθ=2 Apply the trigonometric property to simplify the above equation: 𝑟 = 2 sin ⁡ 𝜃 r= sinθ 2 Thus, the polar equation is: 𝑟 = 2 sin ⁡ 𝜃 r= sinθ 2 Question 4 Write an equation in standard form 𝐴 𝑥 − 𝐵 𝑦 = 𝐶 Ax−By=C of the line that satisfies the given conditions. Use integer values for 𝐴 , 𝐵 , 𝐶 A,B,C. Passes through ( 2 , 8 ) (2,8) and ( − 4 , 16 ) (−4,16). Answer Locate the slope: 𝑚 = 16 − 8 − 4 − 2 = 8 − 6 = − 4 3 m= −4−2 16−8 = −6 8 =− 3 4 Point-slope form of the equation, then simplify it before rewriting it in standard form: 𝑦 − 8 = − 4 3 ( 𝑥 − 2 ) y−8=− 3 4 (x−2) Simplify: 𝑦 − 8 = − 4 3 𝑥 + 8 3 y−8=− 3 4 x+ 3 8 3 𝑦 − 24 = − 4 𝑥 + 8 3y−24=−4x+8 Put both sides in standard form by multiplying by 3: 3 𝑦 = − 4 𝑥 + 32 3y=−4x+32 Put in 𝐴 𝑥 + 𝐵 𝑦 = 𝐶 Ax+By=C form: 4 𝑥 + 3 𝑦 = 32 4x+3y=32 Question 5 0.6 × 0.4 × 0.2 0.6×0.4×0.2 Answer 0.6 × 0.4 = 0.24 0.6×0.4=0.24 0.24 × 0.2 = 0.048 0.24×0.2=0.048 Since both decimals have 1 decimal place, the product has 2 decimal places: Result: 0.048 Result: 0.048