Calculus Answers

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"

Tell whether if the following piecewise function is a continuous at a given point or not. (SHOW THE SOLUTION).

3. at x = 5

x² - 3 if x ≥ 5

4x + 2 if x < 5

Tell whether if the following piecewise function is a continuous at a given point or not. (SHOW THE SOLUTION).

1. at x = 4

x + 1 if x < 4

(x - 4)² + 3 if x ≥ 4

Tell whether if the following piecewise function is a continuous at a given point or not. (SHOW THE SOLUTION).

1. at x = 4

x + 1 if x < 4

(x - 4)² + 3 if x ≥ 4

2. at x = 0

1/x if x ≤ -1

x² - 2 if x > -1

3. at x = 5

x² - 3 if x ≥ 5

4x + 2 if x < 5

CONTINUITY AT A POINT

A. Tell weather if the following is a continuous or discontinuous at a given point.

1. f(x) = 3x² + 2x + 1 at x = -2

2. f(x) = 1/x-2 at x = 2

3. f(x) = x-1/x² -1 at x = 1

4. f(x) = log (x² - 3) at x = 2

5. f(x) = 9x² - 1 at x = 1

Check whether the sequence (an), where

an = 1/ (n+1) + 1/(n+2) +....+1/(2n) is convergent or not

State if the following statement is true or false. Give reasons for your answers in the form of a short proof or a counterexample.

(√2, 1, ½) E Q X Z X R.