Calculus Answers

Consider the function 𝑓(𝑥, 𝑦) = 𝑥𝑦 ( 𝑥 2−𝑦 2 𝑥 2+𝑦2 ) , (𝑥, 𝑦) ≠ (0, 0). Show that lim (𝑥,𝑦)→(0,0) 𝑓(𝑥, 𝑦) = 0

Determine the following integrals:

1. "\\smallint" [x-2/x^2][X+2/x^2]DX

2. "\\smallint" e^5x[e^2x/7+3/e^3x]DX

3."\\smallint" 1/(4-√3x)^3 DX

4. "\\smallint" ^π/4₀ (tan X)^3 (sec X )^3 dx

Find the derivative of the following functions by using the appropriate rultof differentiation:

1. y=1/√x[x^2-2/X]

2. h(X)=sin x/1+cos x

3. G(X)=(cos5x)^sin(x^2)

4. F(X) = "\\int"^x_√x t√t^2+1dt

Write a program to solve the Cauchy-Euler differential equation

Consider the surface S = n (x, y, z) | z = p x 2 + y 2 and 1 ≤ z ≤ 3 o .(a) Sketch the surface S in R 3 . Also show its XY-projection on your sketch. (2) (b) Evaluate the area of S, using a surface integral

A). Determine the following limits (if they exist)

1. Lim X approaches -5 x^2+x-20/3(X+5)

2. Lim t approaches 0 sin5t/t^2+4t

3. Lim approaches -infinity 3-|X|/2|X|+1

4. Lim X approaches 0 2x/3-√x+9

5. Lim X approaches +infinity 2x+x^2+1/1-x+2x^2