Calculus Answers

Evaluate the following integrals. (Show your solution)

"1) \\intop 2x\u221ax" "dx"

2) "\\intop" "2\/\u221ax\u00b3" "dx"

. A poster must have 32 sq.in. of printed materials with margins of 4 inches each at the top and 2 inches at each side. Find the dimensions of the whole poster if its area is maximum.

Find the area that us inside r=4 and outside r=4-2sinθ

A particle moves along the parabola y² = 4x with a constant horizontal component velocity of 2 m/s. Find the vertical component of the velocity at the point (1, 2).

Convert (3–√,−1) into polar coordinates (r,θ) so that r≥0 and 0≤θ<2π.

Write z1=3–√+i and z2=1−i in trigonometric form. Then the argument of the quotient z1/z2is given by

Consider the decomposition of

6x3+x−3/x2−2x+1

into partial fractions

Which of the following is part of the set of all possible rational zeros of

f(x)=2x4−x3+6x2−5x−8?

Part I. Differentiation Rules on Algebraic Functions

Apply the appropriate differentiation technique/s to determine the derivative of the following algebraic functions.

1. f(x)=x^(-6)+4x^3-10
2. f(x)=5x^3-4/x^2
3. y=√(5-2x)
4. g(x)=(x^2-5x-7)(x^2-5)
5. g(x)=(3x^2-4x+1)(2x-3)
6. h(x)=(4x^3-2x+1)/(2x-3)
7. h(x)=(4x+3)/(5x^2-3x+6 )
8. y=(3x^2+2x-3)^3
9. y=(5x+3)^2 (2x-1)
10. y=(3x-2)^2/(4x+1)

Part II. Differentiation Rules on Trigonometric Functions and Higher-Order Derivatives

Apply the appropriate differentiation technique/s to determine the derivative of the following trigonometric functions.

1. y=cos〗⁡(x^2+3)
2. y=sin⁡〖(5x^2-7)〗
3. y=cot⁡(6x)
4. y=tan⁡(cos⁡x )
5. y=csc⁡(5x^3 )

Find the first derivative, second derivative, and third derivative for each of the following functions.

1. y=6x^4+3x^3-4x^2-1
2. f(x)=(2x+3)^4

"Solve \t\u222b\u2592\u221bx dx"