Calculus Answers

Find the volume of the region bounded above by the plane z = y/2 and below by the rectangle.

R ∶ 0 ≤ x ≤ 4,0 ≤ y ≤ 2

Evaluate using Green’s Theorem ∮ 3𝑥𝑦𝑑𝑥 + 2𝑥𝑦𝑑𝑦, where 𝐶 is the rectangle bounded by

𝑥 = −2, 𝑥 = 4, 𝑦 = 1 and 𝑦 = 2.

Determine whether the functions ƒ(𝑥) = √4−𝑥2 is continuous on the interval

 [−4,4]. Show your complete solution.

Determine whether the following functions are continuous at a given point. Show your complete solution.

1. ƒ(𝑥) = 𝑥2−4 at 𝑥 = 2 𝑥−2

2. ƒ(𝑥) = 𝑥2−25 at 𝑥 = 2 𝑥−5

Determine whether the following functions are continuous on the given interval. Show your complete solution.

1. ƒ(𝑥)=√4−𝑥2 ;[−2,2]

2. ƒ(𝑥)=3𝑥2 −𝑥+5 ; (−∞,+∞)

Determine whether the following functions are continuous at a given point. Show your complete solution.

1. ƒ(𝑥)=3𝑥2−4𝑥+2at𝑥=2

2. ƒ(𝑥)=𝑥2−6𝑥−3at𝑥=4

Determine whether or not the following are continuous functions.

1. ƒ(𝑥)=5𝑥+3 2. ƒ(𝑥)=−4𝑥+2

3. ƒ(𝑥)=2𝑥2 +𝑥−3

4. ƒ(𝑥)={2𝑥−3 iƒ 𝑥≥2 −2𝑥 + 2 iƒ 𝑥 < 2

5. ƒ(𝑥)={|𝑥+2|iƒ𝑥G−2 4 iƒ 𝑥 = −2

Evaluate the following limits.

1. limX→+∞ 5X+6

2. limX→−∞ e4X+7

3.limX→−∞(0.2)X